Moment Map Research Articles

Generalising G2 geometry: involutivity, moment maps and moduli

We analyse the geometry of generic Minkowski mathcal{N} = 1, D = 4 flux compactifications in string theory, the default backgrounds for string model building. In M-theory they are the natural string theoretic extensions of G2 holonomy manifolds. In type II theories, they extend the notion of Calabi-Yau geometry and include the class of flux backgrounds based on generalised complex structures first considered by Graña et al. (GMPT). Using E7(7) × ℝ+ generalised geometry we show that these compactifications are characterised by an SU(7) ⊂ E7(7) structure defining an involutive subbundle of the generalised tangent space, and with a vanishing moment map, corresponding to the action of the diffeomorphism and gauge symmetries of the theory. The Kähler potential on the space of structures defines a natural extension of Hitchin’s G2 functional. Using this framework we are able to count, for the first time, the massless scalar moduli of GMPT solutions in terms of generalised geometry cohomology groups. It also provides an intriguing new perspective on the existence of G2 manifolds, suggesting possible connections to Geometrical Invariant Theory and stability.

Bootstrapping Coulomb and Higgs branch operators

We apply the numerical conformal bootstrap to correlators of Coulomb and Higgs branch operators in 4d mathcal{N} = 2 superconformal theories. We start by revisiting previous results on single correlators of Coulomb branch operators. In particular, we present improved bounds on OPE coefficients for some selected Argyres-Douglas models, and compare them to recent work where the same cofficients were obtained in the limit of large r charge. There is solid agreement between all the approaches. The improved bounds can be used to extract an approximate spectrum of the Argyres-Douglas models, which can then be used as a guide in order to corner these theories to numerical islands in the space of conformal dimensions. When there is a flavor symmetry present, we complement the analysis by including mixed correlators of Coulomb branch operators and the moment map, a Higgs branch operator which sits in the same multiplet as the flavor current. After calculating the relevant superconformal blocks we apply the numerical machinery to the mixed system. We put general constraints on CFT data appearing in the new channels, with particular emphasis on the simplest Argyres-Douglas model with non-trivial flavor symmetry.

Central limit theorem for toric Kähler manifolds

Associated to the Bergman kernels of a polarized toric \kahler manifold $(M, \omega, L, h)$ are sequences of measures $\{\mu_k^z\}_{k=1}^{\infty}$ parametrized by the points $z \in M$. For each $z$ in the open orbit, we prove a central limit theorem for $\mu_k^z$. The center of mass of $\mu_k^z$ is the image of $z$ under the moment map; after re-centering at $0$ and dilating by $\sqrt{k}$, the re-normalized measure tends to a centered Gaussian whose variance is the Hessian of the \kahler potential at $z$. We further give a remainder estimate of Berry-Esseen type. The sequence $\{\mu_k^z\}$ is generally not a sequence of convolution powers and the proofs only involve \kahler analysis.

Conservation of Generalized Momentum Maps in the Optimal Control of Constrained Mechanical Systems

We show that the optimal control of constrained mechanical systems satisfies an optimal control version of Noether’s theorem. In particular, the symmetry of the uncontrolled mechanical system subject to algebraic constraints is handed down to the optimal control problem. We also show that the corresponding generalized momentum map on the level of the optimal control problem can be preserved under discretization.

Large deviation principle for moment map estimation

Given a representation of a compact Lie group and a state we define a probability measure on the coadjoint orbits of the dominant weights by considering the decomposition into irreducible components. For large tensor powers and independent copies of the state we show that the induced probability distributions converge to the value of the moment map. For faithful states we prove that the measures satisfy the large deviation principle with an explicitly given rate function.

SAR Image Registration Based on ROEWA-Blocks and Multiscale Circle Descriptor

Given the imaging characteristics of synthetic aperture radar (SAR) images and the inherent speckle noise in them, scale-invariant feature transform based algorithms are unable to perform satisfactorily. To improve registration efficiency between SAR images, we propose a robust and efficient registration method with three main contributions. First, considering sudden dark patches appearing in SAR images, we propose the ratio of exponentially weighted average blocks to suppress the sudden dark patches and better adapt to different test images. This new operator called blocks of the ratio of exponentially weighted averages (ROEWA-B) divides the processing windows of ROEWA into blocks, which can not only reduce speckle noise but also retain more edge details compared to ROEWA when sudden dark patches appear. Second, for outlier removal, we present an approach using the minimum moment map to remove erroneous keypoints. Finally, based on the gradient location orientation histogram descriptor, we propose a novel multiscale circle descriptor, which combines scale change information to give weights to feature points at different scales. Experimental results for various thresholds and evaluations demonstrate the advantage and robustness of our method in registration.

Convergence of polarizations, toric degenerations, and Newton–Okounkov bodies

Let $X$ be a smooth irreducible complex algebraic variety of dimension $n$ and $L$ a very ample line bundle on $X$. Given a toric degeneration of $(X,L)$ satisfying some natural technical hypotheses, we construct a deformation $\{J_s\}$ of the complex structure on $X$ and bases $\mathcal{B}_s$ of $H^0(X,L, J_s)$ so that $J_0$ is the standard complex structure and, in the limit as $s \to \infty$, the basis elements approach dirac-delta distributions centered at Bohr-Sommerfeld fibers of a moment map associated to $X$ and its toric degeneration. The theory of Newton-Okounkov bodies and its associated toric degenerations shows that the technical hypotheses mentioned above hold in some generality. Our results significantly generalize previous results in geometric quantization which prove independence of polarization between Kahler quantizations and real polarizations. As an example, in the case of general flag varieties $X=G/B$ and for certain choices of $\lambda$, our result geometrically constructs a continuous degeneration of the (dual) canonical basis of $V_{\lambda}^*$ to a collection of dirac delta functions supported at the Bohr-Sommerfeld fibres corresponding exactly to the lattice points of a Littelmann-Berenstein-Zelevinsky string polytope $\Delta_{\underline{w}_0}(\lambda) \cap \mathbb{Z}^{\dim(G/B)}$.

Supersymmetric black holes and the SJT/nSCFT1 correspondence

We consider 1/4 BPS black hole solutions of mathcal{N} = 2 gauged supergravity in AdS4. The near horizon geometry is AdS2 × S2 and supersymmetry is enhanced. In the first part of the paper we choose a moment map, which allows the embedding of this supergravity solution into a sugra theory with a hypermultiplet. We then perform the s-wave reduction of this theory at the horizon and determine the dilaton multiplet, which couples to both metric and gravitino fluctuations. In the second part we work with Euclidean axial mathcal{N} = (2, 2) JT supergravity and show how to add gauged matter in form of covariantly twisted chiral and anti-chiral multiplets. We demonstrate how to reduce the on-shell action to boundary superspace. We compare both theories and calculate the fourpoint function by integrating out gravitons, gravitini and photons for the s-wave setting and by use of the Super-Schwarzian modes in the JT theory.

Stochastic differential equations for Lie group valued moment maps

The celebrated result by Biane-Bougerol-O'Connell relates Duistermaat-Heckman (DH) measures for coadjoint orbits of a compact Lie group $G$ with the multi-dimensional Pitman transform of the Wiener process on its Cartan subalgebra. The DH theory admits several non-trivial generalizations. In this paper, we consider the case of $G=SU(2)$, and we give an interpretation of DH measures for $SU(2) \cong S^3$ valued moment maps in terms of an interesting stochastic process on the unit disc, and an interpretation of the DH measures for Poisson $\mathbb{H}^3$ valued moment maps (in the sense of Lu) in terms of a stochastic process on the interior of a hyperbola.

Uncertainty Aware Temporal-Ensembling Model for Semi-Supervised ABUS Mass Segmentation.

Accurate breast mass segmentation of automated breast ultrasound (ABUS) images plays a crucial role in 3D breast reconstruction which can assist radiologists in surgery planning. Although the convolutional neural network has great potential for breast mass segmentation due to the remarkable progress of deep learning, the lack of annotated data limits the performance of deep CNNs. In this article, we present an uncertainty aware temporal ensembling (UATE) model for semi-supervised ABUS mass segmentation. Specifically, a temporal ensembling segmentation (TEs) model is designed to segment breast mass using a few labeled images and a large number of unlabeled images. Considering the network output contains correct predictions and unreliable predictions, equally treating each prediction in pseudo label update and loss calculation may degrade the network performance. To alleviate this problem, the uncertainty map is estimated for each image. Then an adaptive ensembling momentum map and an uncertainty aware unsupervised loss are designed and integrated with TEs model. The effectiveness of the proposed UATE model is mainly verified on an ABUS dataset of 107 patients with 170 volumes, including 13382 2D labeled slices. The Jaccard index (JI), Dice similarity coefficient (DSC), pixel-wise accuracy (AC) and Hausdorff distance (HD) of the proposed method on testing set are 63.65%, 74.25%, 99.21% and 3.81mm respectively. Experimental results demonstrate that our semi-supervised method outperforms the fully supervised method, and get a promising result compared with existing semi-supervised methods.

Algebraic and geometric properties of flag Bott–Samelson varieties and applications to representations

We introduce the notion of flag Bott-Samelson variety as a generalization of Bott-Samelson variety and flag variety. Using a birational morphism from an appropriate Bott-Samelson variety to a flag Bott-Samelson variety, we compute Newton-Okounkov bodies of flag Bott-Samelson varieties as generalized string polytopes, which are applied to give polyhedral expressions for irreducible decompositions of tensor products of $G$-modules. Furthermore, we show that flag Bott-Samelson varieties are degenerated into flag Bott manifolds with higher rank torus actions, and find the Duistermaat-Heckman measures of the moment map images of flag Bott-Samelson varieties with the torus action together with invariant closed $2$-forms.

Toric Bruhat interval polytopes

For two elements v and w of the symmetric group Sn with v≤w in Bruhat order, the Bruhat interval polytope Qv,w is the convex hull of the points (z(1),…,z(n))∈Rn with v≤z≤w. It is known that the Bruhat interval polytope Qv,w is the moment map image of the Richardson variety Xw−1v−1. We say that Qv,w is toric if the corresponding Richardson variety Xw−1v−1 is a toric variety. We show that when Qv,w is toric, its combinatorial type is determined by the poset structure of the Bruhat interval [v,w] while this is not true unless Qv,w is toric. We are concerned with the problem of when Qv,w is (combinatorially equivalent to) a cube because Qv,w is a cube if and only if Xw−1v−1 is a smooth toric variety. We show that a Bruhat interval polytope Qv,w is a cube if and only if Qv,w is toric and the Bruhat interval [v,w] is a Boolean algebra. We also give several sufficient conditions on v and w for Qv,w to be a cube.

Using angular momentum maps to detect kinematically distinct galactic components

ABSTRACT In this work, we introduce a physically motivated method of performing disc/spheroid decomposition of simulated galaxies, which we apply to the eagle sample. We make use of the healpix package to create Mollweide projections of the angular momentum map of each galaxy’s stellar particles. A number of features arise on the angular momentum space which allows us to decompose galaxies and classify them into different morphological types. We assign stellar particles with angular separation of less/greater than 30° from the densest grid cell on the angular momentum sphere to the disc/spheroid components, respectively. We analyse the spatial distribution for a subsample of galaxies and show that the surface density profiles of the disc and spheroid closely follow an exponential and a Sérsic profile, respectively. In addition discs rotate faster, have smaller velocity dispersions, are younger and are more metal rich than spheroids. Thus, our morphological classification reproduces the observed properties of such systems. Finally, we demonstrate that our method is able to identify a significant population of galaxies with counter-rotating discs and provide a more realistic classification of such systems compared to previous methods.

Single-hemisphere photoelectron momentum microscope with time-of-flight recording.

Photoelectron momentum microscopy is an emerging powerful method for angle-resolved photoelectron spectroscopy (ARPES), especially in combination with imaging spin filters. These instruments record kx-ky images, typically exceeding a full Brillouin zone. As energy filters, double-hemispherical or time-of-flight (ToF) devices are in use. Here, we present a new approach for momentum mapping of the full half-space, based on a large single hemispherical analyzer (path radius of 225 mm). Excitation by an unfocused He lamp yielded an energy resolution of 7.7 meV. The performance is demonstrated by k-imaging of quantum-well states in Au and Xe multilayers. The α2-aberration term (α, entrance angle in the dispersive plane) and the transit-time spread of the electrons in the spherical field are studied in a large pass-energy (6 eV-660 eV) and angular range (α up to ±7°). It is discussed how the method circumvents the preconditions of previous theoretical work on the resolution limitation due to the α2-term and the transit-time spread, being detrimental for time-resolved experiments. Thanks to k-resolved detection, both effects can be corrected numerically. We introduce a dispersive-plus-ToF hybrid mode of operation, with an imaging ToF analyzer behind the exit slit of the hemisphere. This instrument captures 3D data arrays I (EB, kx, ky), yielding a gain up to N2 in recording efficiency (N being the number of resolved time slices). A key application will be ARPES at sources with high pulse rates such as synchrotrons with 500 MHz time structure.

Covariant momentum map thermodynamics for parametrized field theories

A general-covariant statistical framework capable of describing classical fluctuations of the gravitational field is a thorny open problem in theoretical physics, yet ultimately necessary to understand the nature of the gravitational interaction, and a key to quantum gravity. Inspired by Souriau’s symplectic generalization of the Maxwell–Boltzmann–Gibbs equilibrium in Lie group thermodynamics, we investigate a space–time-covariant formulation of statistical mechanics for parametrized first-order field theories, as a simplified model sharing essential general covariant features with canonical general relativity. Starting from a covariant multi-symplectic phase space formulation, we define a general-covariant notion of Gibbs state in terms of the covariant momentum map associated with the lifted action of the diffeomorphisms group on the extended phase space. We show how such a covariant notion of equilibrium encodes the whole information about symmetry, gauge and dynamics carried by the theory, associated with a canonical spacetime foliation, where the covariant choice of a reference frame reflects in a Lie algebra-valued notion of local temperature. We investigate how physical equilibrium, hence time evolution, emerges from such a state and the role of the gauge symmetry in the thermodynamic description.

Trigonometric Real Form of the Spin RS Model of Krichever and Zabrodin

We investigate the trigonometric real form of the spin Ruijsenaars–Schneider system introduced, at the level of equations of motion, by Krichever and Zabrodin in 1995. This pioneering work and all earlier studies of the Hamiltonian interpretation of the system were performed in complex holomorphic settings; understanding the real forms is a non-trivial problem. We explain that the trigonometric real form emerges from Hamiltonian reduction of an obviously integrable ‘free’ system carried by a spin extension of the Heisenberg double of the {mathrm{U}}(n) Poisson–Lie group. The Poisson structure on the unreduced real phase space {mathrm{GL}}(n,{mathbb {C}})times {mathbb {C}}^{nd} is the direct product of that of the Heisenberg double and dge 2 copies of a {mathrm{U}}(n) covariant Poisson structure on {mathbb {C}}^n simeq {mathbb {R}}^{2n} found by Zakrzewski, also in 1995. We reduce by fixing a group valued moment map to a multiple of the identity and analyze the resulting reduced system in detail. In particular, we derive on the reduced phase space the Hamiltonian structure of the trigonometric spin Ruijsenaars–Schneider system and we prove its degenerate integrability.

Bulk and surface topological indices for a skyrmion string: current-driven dynamics of skyrmion string in stepped samples

The magnetic skyrmion is a topological magnetic vortex, and its topological nature is characterized by an index called skyrmion number which is a mapping of the magnetic moments defined on a two-dimensional space to a unit sphere. In three-dimensions, a skyrmion, i.e., a vortex penetrating though the magnet naturally forms a string, which terminates at the surfaces of the magnet or in the bulk. For such a string, the topological indices, which control its topological stability are less trivial. Here, we study theoretically, in terms of numerical simulation, the dynamics of current-driven motion of a skyrmion string in a film sample with the step edges on the surface. In particular, skyrmion–antiskyrmion pair is generated by driving a skyrmion string through the side step with an enough height. We find that the topological indices relevant to the stability are the followings; (1) skyrmion number along the developed surface, and (2) the monopole charge in the bulk defined as the integral over the surface enclosing a singular magnetic configuration. As long as the magnetic configuration is slowly varying, the former is conserved while its changes is associated with nonzero monopole charge. The skyrmion number and the monoplole charge offer a coherent understanding of the stability of the topological magnetic texture and the nontrivial dynamics of skyrmion strings.

Heterotic backgrounds via generalised geometry: moment maps and moduli

We describe the geometry of generic heterotic backgrounds preserving minimal supersymmetry in four dimensions using the language of generalised geometry. They are characterised by an SU(3) × Spin(6 + n) structure within O(6, 6 + n) × ℝ+ generalised geometry. Supersymmetry of the background is encoded in the existence of an involutive subbundle of the generalised tangent bundle and the vanishing of a moment map for the action of diffeomorphisms and gauge symmetries. We give both the superpotential and the Kähler potential for a generic background, showing that the latter defines a natural Hitchin functional for heterotic geometries. Intriguingly, this formulation suggests new connections to geometric invariant theory and an extended notion of stability. Finally we show that the analysis of infinitesimal deformations of these geometric structures naturally reproduces the known cohomologies that count the massless moduli of supersymmetric heterotic backgrounds.

Revisiting the gas kinematics in SSA22 Lyman-α Blob 1 with radiative transfer modelling in a multiphase, clumpy medium

ABSTRACT We present new observations of Lyman-α (Lyα) Blob 1 (LAB1) in the SSA22 protocluster region (z = 3.09) using the Keck Cosmic Web Imager and Keck Multi-object Spectrometer for Infrared Exploration. We have created a narrow-band Lyα image and identified several prominent features. By comparing the spatial distributions and intensities of Lyα and Hβ, we find that recombination of photo-ionized H i gas followed by resonant scattering is sufficient to explain all the observed Lyα/Hβ ratios. We further decode the spatially resolved Lyα profiles using both moment maps and radiative transfer modelling. By fitting a set of multiphase, ‘clumpy’ models to the observed Lyα profiles, we manage to reasonably constrain many parameters, namely the H i number density in the interclump medium (ICM), the cloud volume filling factor, the random velocity and outflow velocity of the clumps, the H i outflow velocity of the ICM, and the local systemic redshift. Our model has successfully reproduced the diverse Lyα morphologies, and the main results are: (1) the observed Lyα spectra require relatively few clumps per line of sight as they have significant fluxes at the line centre; (2) the velocity dispersion of the clumps yields a significant broadening of the spectra as observed; (3) the clump bulk outflow can also cause additional broadening if the H i in the ICM is optically thick; (4) and the H i in the ICM is responsible for the absorption feature close to the Lyα line centre.

Dipolar magnetic interactions in 3×3 arrays of rectangular Ni nanopillars

A study about magnetic behavior of 3×3 arrays, constituted by Ni squared pillars (SPs), is presented. Micromagnetic simulations using the OOMMF package reveal hysteresis loops that depend on the inter-pillar distance, d . Solid SPs with high L = 120 nm, lateral dimensions D (30 nm and 60 nm) and face to face distance d (20 nm and 30 nm) were used. Hollow pillars with rectangular cavity a (12 nm and 24 nm) were also placed at d = 20 and 30 nm. For hollow pillars, the angular dependence of coercivity was slightly different than that for the solid pillars. The lateral dimension ( D ) was the most relevant geometric parameter acting over magnetic behaviors. The reversal of magnetic moments occurs through magnetization jumps in the hysteresis curves. These jumps are very sensitive to the inter-pillar distance in the array. The maps of moments, at the remanence state, reveal that there are vortexes at the pillars extremes. These maps also reveal that the effect of the internal and external surfaces, in hollow pillars, interact enough to modify the vector configuration. The effects of lateral dimension, D are stronger manifested in hollow pillars, because the distance between the internal and external surfaces changes with a and D . The study presents itself with relevant results, significant to understand the behavior of magnetic objects with a square cross section.