Moment Map Research Articles

Hole Doping to Perylene on Au(110): Photoelectron Momentum Microscopy

Photoemission spectroscopy of the perylene layers on the Au(110) surface was performed using a momentum microscope. The two-dimensional momentum maps of the perylene monolayer exhibit a “binary star” pattern with two asymmetric bright spots for the electrons from the highest occupied molecular orbital of perylene, reflecting the tilted molecular orientation. This pattern was absent for the perylene multilayer and the Br-doped monolayer because of the random orientation of the molecules in these layers. Br doped to the perylene monolayer facilitated hole injection in perylene, and the electrons attributable to the singly occupied molecular orbital were observed at binding energies of 1.0—1.5 eV.

The Positive Tropical Grassmannian, the Hypersimplex, and them= 2 Amplituhedron

AbstractThe positive Grassmannian $Gr^{\geq 0}_{k,n}$ is a cell complex consisting of all points in the real Grassmannian whose Plücker coordinates are non-negative. In this paper we consider the image of the positive Grassmannian and its positroid cells under two different maps: the moment map$\mu $ onto the hypersimplex [ 31] and the amplituhedron map$\tilde{Z}$ onto the amplituhedron [ 6]. For either map, we define a positroid dissection to be a collection of images of positroid cells that are disjoint and cover a dense subset of the image. Positroid dissections of the hypersimplex are of interest because they include many matroid subdivisions; meanwhile, positroid dissections of the amplituhedron can be used to calculate the amplituhedron’s ‘volume’, which in turn computes scattering amplitudes in $\mathcal{N}=4$ super Yang-Mills. We define a map we call T-duality from cells of $Gr^{\geq 0}_{k+1,n}$ to cells of $Gr^{\geq 0}_{k,n}$ and conjecture that it induces a bijection from positroid dissections of the hypersimplex $\Delta _{k+1,n}$ to positroid dissections of the amplituhedron $\mathcal{A}_{n,k,2}$; we prove this conjecture for the (infinite) class of BCFW dissections. We note that T-duality is particularly striking because the hypersimplex is an $(n-1)$-dimensional polytope while the amplituhedron $\mathcal{A}_{n,k,2}$ is a $2k$-dimensional non-polytopal subset of the Grassmannian $Gr_{k,k+2}$. Moreover, we prove that the positive tropical Grassmannian is the secondary fan for the regular positroid subdivisions of the hypersimplex, and prove that a matroid polytope is a positroid polytope if and only if all 2D faces are positroid polytopes. Finally, toward the goal of generalizing T-duality for higher $m$, we define the momentum amplituhedron for any even $m$.

A robust visible and infrared image matching algorithm for power equipment based on phase congruency and scale-invariant feature

Dense and accurate matching for visible and infrared images of power equipment is crucial to intelligent diagnosis system of power grid, but existing matching methods usually fail in aligning visible and infrared image pairs because of significant intensity, resolution and viewpoint differences. In this paper, we propose a matching algorithm based on phase congruency and scale-invariant feature to address this problem. The proposed method consists of four steps. First, the maximum moment map of phase congruency of input image is computed based on phase congruency theory, which is then used to enhance the raw image. Second, Canny operator and contour tracking method are employed to detect image contours and scale-invariant feature points are extracted by the curvature scale space (CSS) corner detector. Third, the novel histogram of phase congruency orientation (HPCO) descriptors based on phase congruency information are computed for all feature points. Finally, a set of preliminary matches is obtained by the bidirectional matching, and refinement procedures are implemented to achieve dense and accurate matching results. We conduct the experiments on public available dataset. Experimental results show that the proposed method can robustly match feature points in visible and infrared image pairs of power equipment in spite of intensity, resolution and viewpoint differences, and achieve favorable performance compared to state-of-the-art approaches.

Geometric quantization on CR manifolds

Let [Formula: see text] be a compact connected orientable Cauchy–Riemann (CR) manifold with the action of a compact Lie group [Formula: see text]. Under natural pseudoconvexity assumptions we show that the CR Guillemin–Sternberg map is an isomorphism at the level of Sobolev spaces of CR functions, modulo a finite-dimensional subspace. As application we study this map for holomorphic line bundles which are positive near the inverse image of [Formula: see text] by the momentum map. We also show that “quantization commutes with reduction” for Sasakian manifolds.

Momentum maps and the Kähler property for base spaces of reductive principal bundles

AbstractWe investigate the complex geometry of total spaces of reductive principal bundles over compact base spaces and establish a close relation between the Kähler property of the base, momentum maps for the action of a maximal compact subgroup on the total space, and the Kähler property of special equivariant compactifications. We provide many examples illustrating that the main result is optimal.

Attosecond imaging of photoinduced dynamics in molecules using time-resolved photoelectron momentum microscopy

We explore the novel capabilities offered by attosecond extreme ultraviolet and x-ray pulses that can be now generated by free-electron lasers and high-harmonics generation sources for probing photon-induced electron dynamics in molecules. We theoretically analyze how spatial and temporal dependence of charge migration in a pentacene molecule can be followed by means of time-resolved photoelectron microscopy on the attosecond time scale. Performing the analysis, we accurately take into account that an attosecond probe pulse leads to considerable spectral broadening. We demonstrate that the excited-state dynamics of a neutral pentacene molecule in the real space map onto unique features of photoelectron momentum maps.

A kinematically detected planet candidate in a transition disk

Context. Transition disks are protoplanetary disks with inner cavities possibly cleared by massive companions. Observing them at high resolution is ideal for mapping their velocity structure and probing companion–disk interactions. Aims. We present Atacama Large Millimeter/submillimeter Array (ALMA) Band 6 dust and gas observations of the transition disk around RXJ1604.3–2130 A, known to feature nearly symmetric shadows in scattered light, and aim to search for non-Keplerian features. Methods. We studied the 12CO line channel maps and moment maps of the line-of-sight velocity and peak intensity. We fitted a Keplerian model of the channel-by-channel emission to study line profile differences and produced deprojected radial profiles for all velocity components. Results. The 12CO emission is detected out to R ∼ 1.8″ (265 au). It shows a cavity inward of 0.39″ (56 au) and within the dust continuum ring (at ∼0.56″, i.e., 81 au). Azimuthal brightness variations in the 12CO line and dust continuum are broadly aligned with the shadows detected in scattered-light observations. We find a strong localized non-Keplerian feature toward the west within the continuum ring (at R = 41 ± 10 au and PA = 280 ± 2°). It accounts for Δvϕ/vkep ∼ 0.4 or Δvz/vkep ∼ 0.04, depending on if the perturbation is in the rotational or vertical direction. A tightly wound spiral is also detected and extends over 300° in azimuth, possibly connected to the localized non-Keplerian feature. Finally, a bending of the iso-velocity contours within the gas cavity indicates a highly perturbed inner region, possibly related to the presence of a misaligned inner disk. Conclusions. While broadly aligned with the scattered-light shadows, the localized non-Keplerian feature cannot be solely due to changes in temperature. Instead, we interpret the kinematical feature as tracing a massive companion located at the edge of the dust continuum ring. We speculate that the spiral is caused by buoyancy resonances driven by planet–disk interactions. However, this potential planet at ∼41 au cannot explain the gas-depleted cavity, the low accretion rate, and the misaligned inner disk, which suggests the presence of another companion closer in.

Coupled Kähler-Einstein and Hermitian-Yang-Mills equations

We introduce a new system of equations coupling Kähler-Einstein and Hermitian-Yang-Mills equations. We provide a moment map interpretation of these equations. We identify a Futaki type invariant as an obstruction to the existence of solutions to these equations. We also prove a Matsushima-Lichnerowicz type theorem. We prove a deformation result that produces nontrivial solutions of these equations under some conditions. We produce examples on some projective bundles using Calabi ansatz.

Algebraic Symplectic Reduction and Quantization of Singular Spaces

The algebraic method of singular reduction is applied for non regular group action on manifolds which provides singular symplectic spaces. The problem of deformation quantization of the singular surfaces is the focus. For some examples of singular Poisson spaces the deformation quantization is explicitly constructed. In is shown that for the flat phase space with the classical moment map and the orthogonal group action the deformation quantization converges for the entire arguments of exponential type.

Degenerations of bundle moduli

Over a family X of complete curves of genus g, which gives the degeneration of a smooth curve into one with nodal singularities, we build a moduli space which is the moduli space of holomorphic SL(n,C) bundles over the generic smooth curve Xt in the family, and is a moduli space of bundles equipped with extra structure at the nodes for the nodal curves in the family. This moduli space is a quotient by (C⁎)s of a moduli space on the desingularization. Taking a “maximal” degeneration of the curve into a nodal curve built from the glueing of three-pointed spheres, we obtain a degeneration of the moduli space of bundles into a (C⁎)(3g−3)(n−1)-quotient of a (2g−2)-th power of a space associated to the three-pointed sphere. Via the Narasimhan-Seshadri theorem, the moduli space of SL(n,C) bundles on the smooth curve is a space of representations of the fundamental group into SU(n) (the “symplectic picture”). We obtain the degenerations also in this symplectic context, in a way that is compatible with the holomorphic degeneration, so that our limit space is also a (S1)(3g−3)(n−1) symplectic quotient of a (2g−2)-th power of a space associated to the three-pointed sphere.

Bifurcation Diagram of the Model of a Lagrange Top with a Vibrating Suspension Point

The article considers a model system that describes a dynamically symmetric rigid body in the Lagrange case with a suspension point that performs high-frequency oscillations. This system, reduced to axes rigidly connected to the body, after the averaging procedure, has the form of the Hamilton equations with two degrees of freedom and has the Liouville integrability property of a Hamiltonian system with two degrees of freedom, which describes the dynamics of a Lagrange top with an oscillating suspension point. The paper presents a bifurcation diagram of the moment mapping. Using the bifurcation diagram, we presented in geometric form the results of the study of the problem of stability of singular points, in particular, singular points of rank zero and rank one.

Unitary Group Orbits Versus Groupoid Orbits of Normal Operators

We study the unitary orbit of a normal operator $$a\in {{\mathcal {B}}}({{\mathcal {H}}})$$ , regarded as a homogeneous space for the action of unitary groups associated with symmetrically normed ideals of compact operators. We show with an unified treatment that the orbit is a submanifold of the various ambient spaces if and only if the spectrum of a is finite, and in that case, it is a closed submanifold. For arithmetically mean closed ideals, we show that nevertheless the orbit always has a natural manifold structure, modeled by the kernel of a suitable conditional expectation. When the spectrum of a is not finite, we describe the closure of the orbits of a for the different norm topologies involved. We relate these results to the action of the groupoid of partial isometries via the moment map given by the range projection of normal operators. We show that all these groupoid orbits also have differentiable structures for which the target map is a smooth submersion. For any normal operator a, we also describe the norm closure of its groupoid orbit $${{\mathcal {O}}}_a$$ , which leads to necessary and sufficient spectral conditions on a ensuring that $${{\mathcal {O}}}_a$$ is norm closed and that $${{\mathcal {O}}}_a$$ is a closed embedded submanifold of $${{\mathcal {B}}}({{\mathcal {H}}})$$ .

On the Hamilton–Poisson structure and solitons for the Maxwell–Lorentz equations with spinning particle

We construct the Hamilton–Poisson structure for the Maxwell–Lorentz equations with spinning extended charged particle. We calculate the integral kernel of the corresponding Poisson bracket. The calculation relies on the Hamilton least action principle which is proved using the Lie–Poincaré calculus. The Hamilton–Poisson structure turns out to be degenerate and admits a functional family of Casimir invariants. The structure is applied to the calculation of the solitons corresponding to the uniform motion and rotation of the particle. We calculate effective masses and momenta of inertia of the soltions.As an example, the Poisson structure is applied to the construction of the momentum map corresponding to the symmetry group of translations. All calculations are rigorously justified by the proof of the well-posedness and smooth approximations of solutions. The investigation is inspired by the problems of stability of solitons.

Horn conditions for quiver subrepresentations and the moment map

Horn conditions for quiver subrepresentations and the moment map

Topological analysis of pseudo-Euclidean Euler top for special values of the parameters

An analogue of the Euler top is considered for a pseudo-Euclidean space is under consideration. In the cases when the geometric integral or area integral vanishes the bifurcation diagrams of the moment map are constructed and the homeomorphism class of each leaf of the Liouville foliation is determined. For each arc of the bifurcation diagram, for one of the two possible cases of the mutual arrangement of the moments of inertia, the types of singularities in the preimage of a small neighbourhood of this arc (analogues of Fomenko 3-atoms) are determined, and for nonsingular isoenergy and isointegral surfaces an invariant of rough Liouville equivalence (an analogue of a rough molecule) is constructed. The pseudo-Euclidean Euler system turns out to have noncompact noncritical bifurcations. Bibliography: 23 titles.

Moment map for coupled equations of Kähler forms and curvature

Moment map for coupled equations of Kähler forms and curvature

A local normal form for Hamiltonian actions of compact semisimple Poisson–Lie groups

The main contribution of this manuscript is a local normal form for Hamiltonian actions of Poisson-Lie groups $K$ on a symplectic manifold equipped with an $AN$-valued moment map, where $AN$ is the dual Poisson-Lie group of $K$. Our proof uses the delinearization theorem of Alekseev which relates a classical Hamiltonian action of $K$ with $\mathfrak{k}^*$-valued moment map to a Hamiltonian action with an $AN$-valued moment map, via a deformation of symplectic structures. We obtain our main result by proving a ``delinearization commutes with symplectic quotients'' theorem which is also of independent interest, and then putting this together with the local normal form theorem for classical Hamiltonian actions wtih $\mathfrak{k}^*$-valued moment maps. A key ingredient for our main result is the delinearization $\mathcal{D}(\omega_{can})$ of the canonical symplectic structure on $T^*K$, so we additionally take some steps toward explicit computations of $\mathcal{D}(\omega_{can})$. In particular, in the case $K=SU(2)$, we obtain explicit formulas for the matrix coefficients of $\mathcal{D}(\omega_{can})$ with respect to a natural choice of coordinates on $T^*SU(2)$.

Shape analysis via gradient flows on diffeomorphism groups

We study a Riemannian gradient flow on Sobolev diffeomorphisms for the problem of image registration. The energy functional quantifies the effect of transforming a template to a target, while also penalizing non-isometric deformations. The main result is well-posedness of the flow. We also give a geometric description of the gradient in terms of the momentum map.

Dynamical characterization of galaxies up toz∼ 7

Context.The characterization of the dynamical state of galaxies up toz ∼ 7 is crucial for constraining the mechanisms that drive the mass assembly in the early Universe. However, it is unclear whether the data quality of typical observations obtained with current and future facilities is sufficient to perform a solid dynamical analysis at these redshifts.Aims.This paper defines the angular resolution and signal-to-noise ratio (S/N) required for a robust characterization of the dynamical state of galaxies up to the Epoch of Reionization. The final aim is to help design future spatially resolved surveys targeting emission lines of primeval galaxies.Methods.We investigate the [C II]-158 μm emission from sixz ∼ 6 − 7 Lyman break galaxies at three different inclinations from the SERRAzoom-in cosmological simulation suite. The SERRAgalaxies cover a range of dynamical states: from isolated disks to major mergers. We create 102 mock observations with various data quality and apply the kinematic classification methods commonly used in the literature. These tests allow us to quantify the performances of the classification methods as a function of angular resolution and S/N.Results.We find that barely resolved observations, typical of line detection surveys, do not allow the correct characterization of the dynamical stage of a galaxy, resulting in the misclassification of disks and mergers in our sample by 100 and 50%, respectively. However, even when using spatially resolved observations with data quality typical of high-zgalaxies (S/N ∼ 10, and ∼3 independent resolution elements along the major axis), the success rates in the merger identification of the standard kinematic classification methods, based on the analysis of the moment maps, range between 50 and 70%. The high angular resolution and S/N needed to correctly classify disks with these standard methods can only be achieved with current instrumentation for a select number of bright galaxies. We propose a new classification method, called PVsplit, that quantifies the asymmetries and morphological features in position-velocity diagrams using three empirical parameters. We test PVsplit on mock data created from SERRAgalaxies, and show that PVsplit can predict whether a galaxy is a disk or a merger provided thatS/N ≳ 10, and the major axis is covered by ≳3 independent resolution elements.

Optical IFU observations of GOALS sample with KOOLS-IFU on Seimei Telescope: Initial results of nine U/LIRGs at z < 0.04

Abstract We present ionized gas properties of nine local ultra/luminous infrared galaxies (U/LIRGs) at z < 0.04 through Integral Field Unit (IFU) observations with KOOLS-IFU on the Seimei Telescope. The observed targets are drawn from the Great Observatories All-sky LIRG Survey (GOALS), covering a wide range of merger stages. We successfully detect emission lines such as Hβ, [O iii]λ5007, Hα, [N ii]λλ6549, 6583, and [S ii]λλ6717, 6731 with a spectral resolution of R = 1500–2000, which provides (i) a spatially resolved (∼200–700 pc) moment map of ionized gas and (ii) diagnostics for an active galactic nucleus (AGN) within the central ∼3–11 kpc in diameter for our sample. We find that the [O iii] outflow that is expected to be driven by an AGN tends to be stronger (i) towards the galactic center and (ii) as a sequence of the merger stage. In particular, the outflow strength in the late-stage (stage D) mergers is about 1.5 times stronger than that in the early-state (stage B) mergers, which indicates that galaxy mergers could induce AGN-driven outflow and play an important role in the co-evolution of galaxies and supermassive black holes.