Moment Map Research Articles

Symmetry Preservation in Hamiltonian Systems: Simulation and Learning

This work presents a general geometric framework for simulating and learning the dynamics of Hamiltonian systems that are invariant under a Lie group of transformations. This means that a group of symmetries is known to act on the system respecting its dynamics and, as a consequence, Noether’s theorem, conserved quantities are observed. We propose to simulate and learn the mappings of interest through the construction of G-invariant Lagrangian submanifolds, which are pivotal objects in symplectic geometry. A notable property of our constructions is that the simulated/learned dynamics also preserves the same conserved quantities as the original system, resulting in a more faithful surrogate of the original dynamics than non-symmetry aware methods, and in a more accurate predictor of non-observed trajectories. Furthermore, our setting is able to simulate/learn not only Hamiltonian flows, but any Lie group-equivariant symplectic transformation. Our designs leverage pivotal techniques and concepts in symplectic geometry and geometric mechanics: reduction theory, Noether’s theorem, Lagrangian submanifolds, momentum mappings, and coisotropic reduction among others. We also present methods to learn Poisson transformations while preserving the underlying geometry and how to endow non-geometric integrators with geometric properties. Thus, this work presents a novel attempt to harness the power of symplectic and Poisson geometry toward simulating and learning problems.

Separability probability of two-qubit states

Abstract Many works have been devoted to determining the volume of the set of separable states and the separability probability both analytically and numerically. However, computing the exact separability probability remains a highly nontrivial open problem. In these notes, we examine this problem in the simplest context of two-qubit states. A novel aspect of our method is the use of the Duistermaat-Heckman measure, which is the push-forward of Liouville measure on the coadjoint orbit along the moment map. On each coadjoint orbit, the Hilbert-Schmidt measure differs from the Duistermaat-Heckman measure by a constant. The latter measure can be explicitly computed by using tools from symplectic geometry. We confirm the conjecture that the separability probability of two-qubit states with Hilbert-Schmidt measure is 8 33 .

Algebraic Complete Integrability of the $a_4^{(2)}$ Toda Lattice

The aim of this work is focused on the investigation of the algebraic complete integrability of the Toda lattice associated with the twisted affine Lie algebra $a_4^{(2)}$. First, we prove that the generic fiber of the momentum map for this system is an affine part of an abelian surface. Second, we show that the flows of integrable vector fields on this surface are linear. Finally, using the formal Laurent solutions of the system, we provide a detailed geometric description of these abelian surfaces and the divisor at infinity.

Constraints on the Gas-phase C/O Ratio of DR Tau's Outer Disk from CS, SO, and C2H Observations

Millimeter wavelength observations of Class II protoplanetary disks often display strong emission from hydrocarbons and high CS/SO values, providing evidence that the gas-phase C/O ratio commonly exceeds 1 in their outer regions. We present new NOEMA observations of CS 5–4, SO 76–65 and 56–45, C2H N = 3–2, HCN 3–2, HCO+ 3–2, and H13CO+ 3–2 in the DR Tau protoplanetary disk at a resolution of ∼0.″4 (80 au). Estimates for the disk-averaged CS/SO ratio range from ∼0.4 to 0.5, the lowest value reported thus far for a T Tauri disk. At a projected separation of ∼180 au northeast of the star, the SO moment maps exhibit a clump that has no counterpart in the other lines, and the CS/SO value decreases to <0.2 at its location. Thermochemical models calculated with DALI indicate that DR Tau’s low CS/SO ratio and faint C2H emission can be explained by a gas-phase C/O ratio that is <1 at the disk radii traced by NOEMA. Comparisons of DR Tau’s SO emission to maps of extended structures traced by 13CO suggest that late infall may contribute to driving down the gas-phase C/O ratio of its disk.

The HHMP Decomposition of the Permutohedron and Degenerations of Torus Orbits in Flag Varieties

Abstract Let $Z\subset \operatorname{Fl}(n)$ be the closure of a generic torus orbit in the full flag variety. Anderson–Tymoczko express the cohomology class of $Z$ as a sum of classes of Richardson varieties. Harada–Horiguchi–Masuda–Park give a decomposition of the permutohedron, the moment map image of $Z$, into subpolytopes corresponding to the summands of the Anderson–Tymoczko formula. We construct an explicit toric degeneration inside $\operatorname{Fl}(n)$ of $Z$ into Richardson varieties, whose moment map images coincide with the HHMP decomposition, thereby obtaining a new proof of the Anderson–Tymoczko formula.

Cosymplectic Geometry, Reductions, and Energy-Momentum Methods with Applications

Classical energy-momentum methods study the existence and stability properties of solutions of t-dependent Hamilton equations on symplectic manifolds whose evolution is given by their Hamiltonian Lie symmetries. The points of such solutions are called relative equilibrium points. This work devises a new cosymplectic energy-momentum method providing a new and more general framework to study t-dependent Hamilton equations. In fact, cosymplectic geometry allows for using more types of distinguished Lie symmetries (given by Hamiltonian, gradient, or evolution vector fields), relative equilibrium points, and reduction methods, than symplectic techniques. To make our work more self-contained and to fill some gaps in the literature, a review of the cosymplectic formalism and the cosymplectic Marsden–Weinstein reduction is included. Known and new types of relative equilibrium points are characterised and studied. Our methods remove technical conditions used in previous energy-momentum methods, like the Ad∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Ad}^*$$\\end{document}-equivariance of momentum maps. Eigenfunctions of t-dependent Schrödinger equations are interpreted in terms of relative equilibrium points in cosymplectic manifolds. A new cosymplectic-to-symplectic reduction is developed and a new associated type of relative equilibrium points, the so-called gradient relative equilibrium points, are introduced and applied to study the Lagrange points and Hill spheres of a restricted circular three-body system by means of a not Hamiltonian Lie symmetry of the system.

Symplectic Geometry of Teichmüller Spaces for Surfaces with Ideal Boundary

A hyperbolic 0-metric on a surface with boundary is a hyperbolic metric on its interior, exhibiting the boundary behavior of the standard metric on the Poincaré disk. Consider the infinite-dimensional Teichmüller spaces of hyperbolic 0-metrics on oriented surfaces with boundary, up to diffeomorphisms fixing the boundary and homotopic to the identity. We show that these spaces have natural symplectic structures, depending only on the choice of an invariant metric on sl(2,R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {sl}(2,\\mathbb {R})$$\\end{document}. We prove that these Teichmüller spaces are Hamiltonian Virasoro spaces for the action of the universal cover of the group of diffeomorphisms of the boundary. We give an explicit formula for the Hill potential on the boundary defining the moment map. Furthermore, using Fenchel–Nielsen parameters we prove a Wolpert formula for the symplectic form, leading to global Darboux coordinates on the Teichmüller space.

Moment neural network and an efficient numerical method for modeling irregular spiking activity.

Continuous rate-based neural networks have been widely applied to modeling the dynamics of cortical circuits. However, cortical neurons in the brain exhibit irregular spiking activity with complex correlation structures that cannot be captured by mean firing rate alone. To close this gap, we consider a framework for modeling irregular spiking activity, called the moment neural network, which naturally generalizes rate models to second-order moments and can accurately capture the firing statistics of spiking neural networks. We propose an efficient numerical method that allows for rapid evaluation of moment mappings for neuronal activations without solving the underlying Fokker-Planck equation. This allows simulation of coupled interactions of mean firing rate and firing variability of large-scale neural circuits while retaining the advantage of analytical tractability of continuous rate models. We demonstrate how the moment neural network can explain a range of phenomena including diverse Fano factor in networks with quenched disorder and the emergence of irregular oscillatory dynamics in excitation-inhibition networks with delay.

The moment map for the variety of associative algebras

The moment map for the variety of associative algebras

MHONGOOSE: A MeerKAT nearby galaxy H I survey

The MHONGOOSE (MeerKAT H I Observations of Nearby Galactic Objects: Observing Southern Emitters) survey maps the distribution and kinematics of the neutral atomic hydrogen (H I) gas in and around 30 nearby star-forming spiral and dwarf galaxies to extremely low H I column densities. The H I column density sensitivity (3σ over 16 km s−1) ranges from ∼5 × 1017 cm−2 at 90″ resolution to ∼4 × 1019 cm−2 at the highest resolution of 7″. The H I mass sensitivity (3σ over 50 km s−1) is ∼5.5 × 105 M⊙ at a distance of 10 Mpc (the median distance of the sample galaxies). The velocity resolution of the data is 1.4 km s−1. One of the main science goals of the survey is the detection of cold accreting gas in the outskirts of the sample galaxies. The sample was selected to cover a range in H I masses from 107 M⊙ to almost 1011 M⊙ in order to optimally sample possible accretion scenarios and environments. The distance to the sample galaxies ranges from 3 to 23 Mpc. In this paper, we present the sample selection, survey design, and observation and reduction procedures. We compared the integrated H I fluxes based on the MeerKAT data with those derived from single-dish measurement and find good agreement, indicating that our MeerKAT observations are recovering all flux. We present H I moment maps of the entire sample based on the first ten percent of the survey data, and find that a comparison of the zeroth- and second-moment values shows a clear separation in the physical properties of the H I between areas with star formation and areas without related to the formation of a cold neutral medium. Finally, we give an overview of the H I-detected companion and satellite galaxies in the 30 fields, five of which have not previously been cataloged. We find a clear relation between the number of companion galaxies and the mass of the main target galaxy.

Light Field Transformation of Metasurface Based on Arbitrary Jones Matrix

AbstractMetasurfaces are ultra‐thin micro‐nano elements, have abundant freedoms, and are widely used in light field transformation. The Jones matrix of single‐layer birefringent metasurfaces have at most 6 degrees of freedom, which limits the richness and diversity of light field transformation. Bilayer metasurfaces have more design channels and functionality than single‐layer metasurfaces. By mathematical derivation, the decomposition method of the bilayer metasurface unitary Jones matrix is provided. Further, inspired by double‐phase holography, arbitrary complex Jones matrix is decomposed and realized by bilayer metasurface and achieves analytical decoupling of all 4 complex components of the Jones matrix. As the theoretical verification, spin‐orbit angular momentum mapping and four‐channel complex holography are realized by the bilayer metasurface. The decomposition method is completely analytical, universal, and easy to implement. It is of great significance for the transformation of light field, information transmission, and optical encryption based on metasurfaces.

The Weil correspondence and universal special geometry

The Weil correspondence states that the datum of a Seiberg-Witten differential is equivalent to an algebraic group extension of the integrable system associated to the Seiberg-Witten geometry. Remarkably this group extension represents quantum consistent couplings for the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 QFT if and only if the extension is anti-affine in the algebro-geometric sense. The universal special geometry is the algebraic integrable system whose Lagrangian fibers are the anti-affine extension groups; it is defined over a base B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{B} $$\\end{document} parametrized by the Coulomb coordinates and the couplings. On the total space of the universal geometry there is a canonical (holomorphic) Euler differential. The ordinary Seiberg-Witten geometries at fixed couplings are symplectic quotients of the universal one, and the Seiberg-Witten differential arises as the reduction of the Euler one in accordance with the Weil correspondence. This universal viewpoint allows to study geometrically the flavor symmetry of the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SCFT in terms of the Mordell-Weil lattice (with Néron-Tate height) of the Albanese variety AL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {A}_{\\mathbbm{L}} $$\\end{document} of the universal geometry seen as a quasi-Abelian variety YL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {Y}_{\\mathbbm{L}} $$\\end{document} defined over the function field L≡ℂB\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathbbm{L}\\equiv \\mathbb{C}\\left(\\mathcal{B}\\right) $$\\end{document}.

Special representatives of complexified Kähler classes

Motivated by constructions appearing in mirror symmetry, we study special representatives of complexified Kähler classes, which extend the notions of constant scalar curvature and extremal representatives for usual Kähler classes. In particular, we provide a moment map interpretation, discuss a possible correspondence with compactified Landau–Ginzburg models, and prove existence results for such special complexified Kähler forms and their large volume limits in certain toric cases.

Influence of catastrophes and hidden dynamical symmetries on ultrafast backscattered photoelectrons

We discuss the effect of using potentials with a Coulomb tail and different degrees of softening in photoelectron momentum distributions (PMDs) using the recently implemented hybrid forward-boundary CQSFA (H-CQSFA). We show that introducing a softening in the Coulomb interaction influences the ridges observed in the PMDs associated with backscattered electron trajectories. In the limit of a hard-core Coulomb interaction, the rescattering ridges close along the polarization axis, while for a soft-core potential, they are interrupted at ridge-specific angles. We analyze the momentum mapping of the different orbits leading to the ridges. For the hard-core potential, there exist two types of saddle-point solutions that coalesce at the ridge. By increasing the softening, we show that two additional solutions emerge as the result of breaking a hidden dynamical symmetry associated exclusively with the Coulomb potential. Further signatures of this symmetry breaking are encountered in subsets of momentum-space trajectories. Finally, we use scattering theory to show how the softening affects the maximal scattering angle and provide estimates that agree with our observations from the CQSFA. This implies that, in the presence of residual binding potentials in the electron's continuum propagation, the distinction between purely kinematic and dynamic caustics becomes blurred. Published by the American Physical Society 2024

Integrated correlators at strong coupling in an orbifold of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM

We consider the 4dN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 superconformal quiver gauge theory obtained by a ℤ2 orbifold of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super Yang-Mills (SYM). By exploiting supersymmetric localization, we study the integrated correlator of two Coulomb branch and two moment map operators and the integrated correlator of four moment map operators, determining exact expressions valid for any value of the ’t Hooft coupling in the planar limit. Additionally, for the second correlator, we obtain an exact expression also for the next-to-planar contribution. Then, we derive the leading terms of their strong-coupling expansions and outline the differences with respect to the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM theory.

Electronic properties and quantitative mapping of magnetic moments of rare-earth hard magnet SmCo5: First principles computations and inelastic scattering

Looking at the limitations of neutron spectroscopy in exploring properties of rare earth hard magnet SmCo5, inelastic scattering for charge and magnetic Compton profiles are attempted. The experimental momentum densities using 661.65 keV γ-ray energy are compared with the theoretical Compton profiles (CPs) deduced using linear combination of atomic orbitals (LCAO) with different exchange and correlation potentials and also spin polarized relativistic Korringa-Kohn-Rostoker (SPR-KKR) calculations. The anisotropies in the computed CPs, along the principal directions, show high momentum densities along the c-axis [00.1] of SmCo5 than [10.0] and [11.0] directions. An overall better agreement of CP computed using LCAO-GGA and that from experiment is observed which shows more appropriateness of GGA-PBE scheme for the exchange and correlation energies for SmCo5. The small value of charge transfer from Sm → Co and positive value of overlap Mulliken population dictate dominance of covalent bonding in SmCo5. Moreover, available experimental data on c-axis oriented magnetic CP (MCP) of SmCo5 have been compared with the present SPR-KKR calculations. It is seen that present SPR-KKR based MCP is in resemblance with the available experimental MCP. A decomposition of experimental MCP in to its constituent profiles depict that the spin momentum of Co and Sm are aligned anti-parallel while the diffuse contribution aligns opposite to the absolute spin moment. Comparison of amplitude and direction of site specific spin moments in SmCo5 is made with the present SPR-KKR and available experimental and theoretical data.

Geometry of bundle-valued multisymplectic structures with Lie algebroids

We study multisymplectic structures taking values in vector bundles with connections from the viewpoint of the Hamiltonian symmetry. We introduce the notion of bundle-valued n-plectic structures and exhibit some properties of them. In addition, we define bundle-valued homotopy momentum sections for bundle-valued n-plectic manifolds with Lie algebroids to discuss momentum map theories in both cases of quaternionic Kähler manifolds and hyper-Kähler manifolds. Furthermore, we generalize the Marsden-Weinstein-Meyer reduction theorem for symplectic manifolds and construct two kinds of reductions of vector-valued 1-plectic manifolds.

Harmonic metrics for the Hull–Strominger system and stability

We investigate stability conditions related to the existence of solutions of the Hull–Strominger system with prescribed balanced class. We build on recent work by the authors, where the Hull–Strominger system is recasted using non-Hermitian Yang–Mills connections and holomorphic Courant algebroids. Our main development is a notion of harmonic metric for the Hull–Strominger system, motivated by an infinite-dimensional hyperKähler moment map and related to a numerical stability condition, which we expect to exist for families of solutions. We illustrate our theory with an infinite number of continuous families of examples on the Iwasawa manifold.

A moment map for the variety of Jordan algebras

We study the variety of complex [Formula: see text]-dimensional Jordan algebras using techniques from Geometric Invariant Theory. More specifically, we use the Kirwan–Ness theorem to construct a Morse-type stratification of the variety of Jordan algebras into finitely many invariant locally closed subsets, with respect to the energy functional associated to the canonical moment map. In particular we obtain a new, cohomology-free proof of the well-known rigidity of semisimple Jordan algebras in the context of the variety of Jordan algebras.

The symplectic structure of a toric conic transform

Suppose that a compact r-dimensional torus Tr acts in a holomorphic and Hamiltonian manner on polarized complex d-dimensional projective manifold M, with nowhere vanishing moment map Φ. Assuming that Φ is transverse to the ray through a given weight ν, associated to these data there is a complex (d−r+1)-dimensional polarized projective orbifold Mˆν (referred to as the ν-th conic transform of M). Namely, Mˆν is a suitable quotient of the inverse image of the ray in the unit circle bundle of the polarization of M. With the aim to clarify the geometric significance of this construction, we consider the special case where M is toric, and show that Mˆν is itself a Kähler toric orbifold, whose (marked) moment polytope is obtained from the one of M by a certain ‘transform’ operation (depending on Φ and ν).