Moment Map Research Articles

High-resolution APEX/LAsMA 12CO and 13CO (3–2) observation of the G333 giant molecular cloud complex

Context. Hub-filament systems are suggested to be the birth cradles of high-mass stars and clusters. Aims. We investigate the gas kinematics of hub-filament structures in the G333 giant molecular cloud complex using 13CO (3–2) observed with the APEX/LAsMA heterodyne camera. Methods. We applied the FILFINDER algorithm to the integrated intensity maps of the 13CO J = 3–2 line to identify filaments in the G333 complex, and we extracted the velocity and intensity along the filament skeleton from moment maps. Clear velocity and density fluctuations are seen along the filaments, allowing us to fit velocity gradients around the intensity peaks. Results. The velocity gradients we fit to the LAsMA and ALMA data agree with each other over the scales covered by ALMA observations in the ATOMS survey (<5 pc). Changes in velocity gradient with scale indicate a funnel structure of the velocity field in position-position-velocity (PPV) space. This is indicative of a smooth, continuously increasing velocity gradient from large to small scales, and thus is consistent with gravitational acceleration. The typical velocity gradient corresponding to a 1 pc scale is ~1.6 km s−1 pc−1. Assuming freefall, we estimate a kinematic mass within 1 pc of ~1190 M⊙, which is consistent with typical masses of clumps in the ATLASGAL survey of massive clumps in the inner Galaxy. We find direct evidence for gravitational acceleration from a comparison of the observed accelerations to those predicted by freefall onto dense hubs with masses from millimeter continuum observations. On large scales, we find that the inflow may be driven by the larger-scale structure, consistent with the hierarchical structure in the molecular cloud and gas inflow from large to small scales. The hub-filament structures at different scales may be organized into a hierarchical system extending up to the largest scales probed through the coupling of gravitational centers at different scales. Conclusions. We argue that the funnel structure in PPV space can be an effective probe for the gravitational collapse motions in molecular clouds. The large-scale gas inflow is driven by gravity, implying that the molecular clouds in the G333 complex may be in a state of global gravitational collapse.

Point vortex dynamics on Kähler twistor spaces

In this paper, we provide tools to study the dynamics of point vortex dynamics on CPn and the flag manifold F1,2(C3). These are the only Kähler twistor spaces arising from four-manifolds. We give an explicit expression for Green’s function on CPn, which enables us to determine the Hamiltonian H and the equations of motions for the point vortex problem on CPn. Moreover, we determine the momentum map μ:F1,2(C3)→su*(3) on the flag manifold.

T-equivariant disc potential and SYZ mirror construction

We develop a G-equivariant Lagrangian Floer theory and obtain a curved A∞ algebra, and in particular a G-equivariant disc potential. We construct a Morse model, which counts pearly trees in the Borel construction LG. When applied to a smooth moment map fiber of a semi-Fano toric manifold, our construction recovers the T-equivariant toric Landau-Ginzburg mirror of Givental. We also study the S1-equivariant Floer theory of a typical singular fiber of a Lagrangian torus fibration (i.e. a pinched torus) and compute its S1-equivariant disc potential via the gluing technique developed in [14,37].

Conditions for symmetry reduction of polysymplectic and polycosymplectic structures

For Hamiltonian field theories on polysymplectic manifolds with a symmetry group action and a momentum map, we explore the redundancy in a set of necessary conditions that has appeared in the literature, for a generalized version of the Marsden–Weinstein symmetry reduction theorem. Next, we prove a necessary and sufficient condition for polycosymplectic reduction. We relate polycosymplectic reduction in a one-to-one way to the reduction of an associated larger polysymplectic manifold. Throughout the paper, we provide examples and discuss special cases.

The planar limit of integrated 4-point functions

We study the planar limit of integrated 4-point functions of moment map operators of \U0001d4a9 = 2 SU(N) SQCD. We do so by considering the planar free energy on S4 of the massive deformation of this theory, and taking advantage of the exact relation between this free energy and the integrated 4-point function. For this planar free energy we derive all the terms with maximal and next-to-maximal transcendentality, and present a procedure to compute terms of lower transcendentality. We also derive the first non-planar corrections, as all order series in the ’t Hooft coupling, and to all orders in transcendentality. Finally, we also apply our approach to the better studied example of \U0001d4a9 = 4 SU(N) SYM integrated 4-point functions, and reproduce their known planar limit.

The 𝑚=2 amplituhedron and the hypersimplex: Signs, clusters, tilings, Eulerian numbers

The hypersimplex Δ k + 1 , n \Delta _{k+1,n} is the image of the positive Grassmannian G r k + 1 , n ≥ 0 Gr^{\geq 0}_{k+1,n} under the moment map. It is a polytope of dimension n − 1 n-1 in R n \mathbb {R}^n . Meanwhile, the amplituhedron A n , k , 2 ( Z ) \mathcal {A}_{n,k,2}(Z) is the projection of the positive Grassmannian G r k , n ≥ 0 Gr^{\geq 0}_{k,n} into the Grassmannian G r k , k + 2 Gr_{k,k+2} under a map Z ~ \tilde {Z} induced by a positive matrix Z ∈ M a t n , k + 2 > 0 Z\in Mat_{n,k+2}^{>0} . Introduced in the context of scattering amplitudes, it is not a polytope, and has full dimension 2 k 2k inside G r k , k + 2 Gr_{k,k+2} . Nevertheless, there seem to be remarkable connections between these two objects via T-duality, as conjectured by Łukowski, Parisi, and Williams [Int. Math. Res. Not. (2023)]. In this paper we use ideas from oriented matroid theory, total positivity, and the geometry of the hypersimplex and positroid polytopes to obtain a deeper understanding of the amplituhedron. We show that the inequalities cutting out positroid polytopes—images of positroid cells of G r k + 1 , n ≥ 0 Gr^{\geq 0}_{k+1,n} under the moment map—translate into sign conditions characterizing the T-dual Grasstopes—images of positroid cells of G r k , n ≥ 0 Gr^{\geq 0}_{k,n} under Z ~ \tilde {Z} . Moreover, we subdivide the amplituhedron into chambers, just as the hypersimplex can be subdivided into simplices, with both chambers and simplices enumerated by the Eulerian numbers. We use these properties to prove the main conjecture of Łukowski, Parisi, and Williams [Int. Math. Res. Not. (2023)]: a collection of positroid polytopes is a tiling of the hypersimplex if and only if the collection of T-dual Grasstopes is a tiling of the amplituhedron A n , k , 2 ( Z ) \mathcal {A}_{n,k,2}(Z) for all Z Z . Moreover, we prove Arkani-Hamed–Thomas–Trnka’s conjectural sign-flip characterization of A n , k , 2 \mathcal {A}_{n,k,2} , and Łukowski–Parisi–Spradlin–Volovich’s conjectures on m = 2 m=2 cluster adjacency and on positroid tiles for A n , k , 2 \mathcal {A}_{n,k,2} (images of 2 k 2k -dimensional positroid cells which map injectively into A n , k , 2 \mathcal {A}_{n,k,2} ). Finally, we introduce new cluster structures in the amplituhedron.

Constructions of b-semitoric systems

In this article, we introduce b-semitoric systems as a generalization of semitoric systems, specifically tailored for b-symplectic manifolds. The objective of this article is to furnish a collection of examples and investigate the distinctive characteristics of these systems. A b-semitoric system is a four-dimensional b-integrable system that satisfies certain conditions: one of its momentum map components is proper and generates an effective global S1-action and all singular points are non-degenerate and devoid of hyperbolic components. To illustrate this concept, we provide five examples of b-semitoric systems by modifying the coupled spin oscillator and the coupled angular momenta, and we also classify their singular points. Additionally, we describe the dynamics of these systems through the image of their respective momentum maps.

Predictions of optimal heating by magnetic reversal behavior of magnetic nanowires (MNWs) with different materials

Objective Magnetic nanowires (MNWs) are potential candidates for heating in biomedical applications that require rapid and uniform heating rates, such as warming cryopreserved organs and hyperthermia treatment of cancer cells. Therefore, it is essential to determine which materials and geometries will provide the optimal heating using available alternating magnetic fields (AMF). Method Micromagnetic simulations are used to investigate the heating ability of MNWs by predicting their hysteretic behavior. MNWs composed of iron (Fe), nickel (Ni), cobalt (Co) or permalloy (FeNi alloy, Py) with different diameters (10-200 nm) are simulated using object oriented micromagnetic framework (OOMMF). Results Hysteresis loops are obtained for each simulated MNW, and the 2D/3D magnetic moment map is simulated to show the reversal mechanism. The heating ability, in terms of specific loss power (SLP), is calculated from the area of the hysteresis loop times frequency for each MNW for comparison with others. Conclusion It is estimated that a theoretical optimal heating ability of 2730 W/g can be provided by isolated Co MNWs with 50 nm diameters using a typical AMF system that can supply 72 kA/m field amplitude and 50 kHz in frequency. Generalized correlation between coercivity and size/material of MNWs is provided as a guidance for researchers to choose the most appropriate MNW as a heater for their AMF system and vice versa.

Reduction theory for singular symplectic manifolds and singular forms on moduli spaces

The investigation of symmetries of b-symplectic manifolds and folded-symplectic manifolds is well-understood when the group under consideration is a torus (see, for instance, [25,21,28] for b-symplectic manifolds and [12,14] for folded symplectic manifolds). However, reduction theory has not been set in this realm in full generality. This is fundamental, among other reasons, to advance in the ‘‘quantization commutes with reduction” programme for these manifolds initiated in [29,30]. In this article, we fill in this gap and investigate the Marsden-Weinstein reduction theory under general symmetries for general bm-symplectic manifolds and other singular symplectic manifolds, including certain folded symplectic manifolds. In this new framework, the set of admissible Hamiltonian functions is larger than the category of smooth functions as it takes the singularities of the differential forms into account. The quasi-Hamiltonian set-up is also considered and brand-new constructions of (singular) quasi-Hamiltonian spaces are obtained via a reduction procedure and the fusion product.

On k-polycosymplectic Marsden–Weinstein reductions

We review and slightly improve the known k-polysymplectic Marsden–Weinstein reduction theory by removing some technical conditions on k-polysymplectic momentum maps by developing a theory of affine Lie group actions for k-polysymplectic momentum maps, removing the necessity of their co-adjoint equivariance. Then, we focus on the analysis of a particular case of k-polysymplectic manifolds, the so-called fibred ones, and we study their k-polysymplectic Marsden–Weinstein reductions. Previous results allow us to devise a k-polycosymplectic Marsden–Weinstein reduction theory, which represents one of our main results. Our findings are applied to study coupled vibrating strings and, more generally, k-polycosymplectic Hamiltonian systems with field symmetries. We show that k-polycosymplectic geometry can be understood as a particular type of k-polysymplectic geometry. Finally, a k-cosymplectic to ℓ-cosymplectic geometric reduction theory is presented, which reduces, geometrically, the space-time variables in a k-cosymplectic framework. An application of this latter result to a vibrating membrane with symmetries is given.

Hofer's metric in compact Lie groups

In this article, we study the Hofer geometry of a compact Lie group K which acts by Hamiltonian diffeomorphisms on a symplectic manifold M . Generalized Hofer norms on the Lie algebra of K are introduced and analyzed with tools from group invariant convex geometry, functional and matrix analysis. Several global results on the existence of geodesics and their characterization in finite-dimensional Lie groups K endowed with bi-invariant Finsler metrics are proved. We relate the conditions for being a geodesic in the group K and in the group of Hamiltonian diffeomorphisms. These results are applied to obtain necessary and sufficient conditions on the moment polytope of the momentum map, for the commutativity of the Hamiltonians of geodesics. Particular cases are studied, where a generalized non-crossing of eigenvalues property of the Hamiltonians hold.

Hypercritical deformed Hermitian-Yang–Mills equation revisited

Abstract In this paper, we study the hypercritical deformed Hermitian-Yang–Mills equation on compact Kähler manifolds and resolve two conjectures of Collins–Yau [Moment maps, nonlinear PDE, and stability in mirror symmetry, preprint (2018), https://arxiv.org/abs/1811.04824].

Quasi Poisson structures, weakly quasi Hamiltonian structures, and Poisson geometry of various moduli spaces

Let G be a Lie group and g its Lie algebra. The aim here is to develop a theory of quasi Poisson structures relative to a not necessarily non-degenerate Ad-invariant symmetric 2-tensor in g⊗g and one of general not necessarily non-degenerate quasi Hamiltonian structures relative to a not necessarily non-degenerate Ad-invariant symmetric bilinear form on g, a quasi Poisson structure being given by a skew bracket of two variables such that suitable data defined in terms of G as symmetry group involving the 2-tensor in the tensor square of the Lie algebra g measure how that bracket fails to satisfy the Jacobi identity. The present approach involves a novel concept of momentum mapping and yields, in the non-degenerate case, a bijective correspondence, in terms of explicit algebraic expressions, between non-degenerate quasi Poisson structures and non-degenerate quasi Hamiltonian structures. The new theory applies to various not necessarily non-singular moduli spaces and yields thereupon, via reduction with respect to an appropriately defined momentum mapping, not necessarily non-degenerate ordinary Poisson structures arising from the data including the symmetric 2-tensor in g⊗g. Among these moduli spaces are representation spaces, possibly twisted, of the fundamental group of a Riemann surface, possibly punctured, and moduli spaces of semistable holomorphic vector bundles as well as Higgs bundle moduli spaces. In the non-degenerate case, such a Poisson structure comes down to a stratified symplectic one of the kind explored in the literature and recovers, e.g., the symplectic part of a Kähler structure introduced by Narasimhan and Seshadri for moduli spaces of stable holomorphic vector bundles on a curve. In the algebraic setting, these moduli spaces arise as not necessarily non-singular affine not necessarily non-degenerate Poisson varieties. Along the way, a side result is an explicit equivalence between extended moduli spaces and quasi Hamiltonian spaces independently of gauge theory.

Runge–Lenz vector as a 3d projection of SO(4) moment map in phase space

We show, using the methods of geometric algebra, that Runge–Lenz vector in the Kepler problem is a 3-dimensional projection of SO(4) moment map that acts on the phase space of 4-dimensional particle motion. Thus, Runge–Lenz vector is a consequence of geometric symmetry of phase space.

Symmetric Toda, gradient flows, and tridiagonalization

The Toda lattice (1967) is a Hamiltonian system given by n points on a line governed by an exponential potential. Flaschka (1974) showed that the Toda lattice is integrable by interpreting it as a flow on the space of symmetric tridiagonal n×n matrices, while Moser (1975) showed that it is a gradient flow on a projective space. The symmetric Toda flow of Deift, Li, Nanda, and Tomei (1986) generalizes the Toda lattice flow from tridiagonal to all symmetric matrices. They showed the flow is integrable, in the classical sense of having d integrals in involution on its 2d-dimensional phase space. The system may be viewed as integrable in other ways as well. Firstly, Symes (1980, 1982) solved it explicitly via QR-factorization and conjugation. Secondly, Deift, Li, Nanda, and Tomei (1986) ‘tridiagonalized’ the system into a family of tridiagonal Toda lattices which are solvable and integrable. In this paper we derive their tridiagonalization procedure in a natural way using the fact that the symmetric Toda flow is diffeomorphic to a twisted gradient flow on a flag variety, which may then be decomposed into flows on a product of Grassmannians. These flows may in turn be embedded into projective spaces via Plücker embeddings, and mapped back to tridiagonal Toda lattice flows using Moser’s construction. In addition, we study the tridiagonalized flows projected onto a product of permutohedra, using the twisted moment map of Bloch, Flaschka, and Ratiu (1990). These ideas are facilitated in a natural way by the theory of total positivity, building on our previous work (2023).

Study of ideal gases in curved spacetimes

The influence of a curved spacetime [Formula: see text] on the physical behavior of an ideal gas of [Formula: see text] particles is analyzed by considering the phase space of the system as a region of the cotangent bundle [Formula: see text] and using Souriau’s Lie group thermodynamics to define the corresponding probability distribution function. While the construction of the phase space respects the separability of the system, by forcing each particle to satisfy the so-called mass-shell constraint, the probability distribution is constructed by mixing the natural symplectic structure of the cotangent bundle with a Hamiltonian description of the system. In this way, the spacetime is introduced into the statistics and its isometries turn out to be of special interest because the distributions are parametrized by the elements of the Lie algebra of the isometry group, through the momentum map of the action of the isometries in [Formula: see text]. We find the Gibbs distribution that, in the simplest case of a flat spacetime, reduces to the so-called modified Jüttner distribution, used to describe ideal gases in the regime of special relativity. We also define a temperature-like function using the norm of a Killing vector, which allows us to recover the so-called Tolman–Ehrenfest effect. As a particular example, we study the outer region of a Schwarzschild black hole, for which a power series expansion of the Schwarzschild radius allows us to represent the partition function and the Gibbs distribution in terms of the corresponding quantities of the Minkowski spacetime.

Cold molecular gas outflow encasing the ionized one in the Seyfert galaxy NGC 3281

ABSTRACT We present ALMA CO (2-1) observations of the Seyfert 2 galaxy NGC 3281 at ∼100 pc spatial resolution. This galaxy was previously known to present a bi-conical ionized gas outflow extending to 2 kpc from the nucleus. The analysis of the CO moment and channel maps, as well as kinematic modelling, reveals two main components in the molecular gas: one rotating in the galaxy plane and another outflowing and extending up to ∼1.8–2.6 kpc from the nucleus, partially encasing the ionized component. The mass of the outflowing molecular gas component is Mmol, out = (2.5 ± 1.6) × 106 $\rm {{\rm M}_{\odot }}$ , representing ∼1.7–2 per cent of the total molecular gas seen in emission within the inner 2.3 kpc. The corresponding mass outflow rate and power are $\dot{M}_{\mathrm{out},\mathrm{mol}}$ = 0.12–0.72 $\rm {{\rm M}_{\odot }\, yr^{-1}}$ and $\dot{E}_{\mathrm{out},\mathrm{mol}}$ = (0.045–1.6) ${\times }\, 10^{40}$ $\rm {erg\, s^{-1}}$ , which translates to a kinetic coupling efficiency with the AGN power of only 10−4–0.02 per cent. This value reaches up to 0.1 per cent when including both the feedback in the ionized and molecular gas, as well as considering that only part of the energy couples kinetically with the gas. Some of the non-rotating CO emission can also be attributed to inflow in the galaxy plane towards the nucleus. The similarity of the CO outflow – encasing the ionized gas one and the X-ray emission – to those seen in other sources, suggests that this may be a common property of galactic outflows.

Orbital-resolved observation of singlet fission

Singlet fission1–13 may boost photovoltaic efficiency14–16 by transforming a singlet exciton into two triplet excitons and thereby doubling the number of excited charge carriers. The primary step of singlet fission is the ultrafast creation of the correlated triplet pair17. Whereas several mechanisms have been proposed to explain this step, none has emerged as a consensus. The challenge lies in tracking the transient excitonic states. Here we use time- and angle-resolved photoemission spectroscopy to observe the primary step of singlet fission in crystalline pentacene. Our results indicate a charge-transfer mediated mechanism with a hybridization of Frenkel and charge-transfer states in the lowest bright singlet exciton. We gained intimate knowledge about the localization and the orbital character of the exciton wave functions recorded in momentum maps. This allowed us to directly compare the localization of singlet and bitriplet excitons and decompose energetically overlapping states on the basis of their orbital character. Orbital- and localization-resolved many-body dynamics promise deep insights into the mechanics governing molecular systems18–20 and topological materials21–23.

Optical and SAR Image Registration Based on Multi-Scale Orientated Map of Phase Congruency

Optical and Synthetic Aperture Radar (SAR) images are highly complementary, and their registrations are a fundamental task for other remote sensing applications. Traditional feature-matching algorithms fail to solve the significant nonlinear radiation difference (NRD) caused by different sensors. To address this problem, a robust registration algorithm with the multi-scale orientated map of phase congruency (MSPCO) is proposed. First, a nonlinear diffusion scale space is established to obtain the scale invariance of feature points. Compared with the linear Gaussian scale space, the nonlinear diffusion scale space can better preserve the edge and texture information. Second, to ensure the quantity and repeatability of features, corner points and edge points are detected on the moment map of phase congruency, respectively, which is the foundation to the next feature matching. Third, the MSPCO descriptor is constructed via the orientation of phase congruency (PCO). PCO is highly robust to NRD, and the different scales of PCOs enhance the robustness of the descriptor. Finally, a feature-matching strategy based on an effective scale ratio is proposed, which reduces the number of comparisons among features and improves computational efficiency. The experimental results show that the proposed method is better than the existing feature-based methods in terms of the number of correct matches and registration accuracy. The registration accuracy is only inferior to that of the most advanced template matching method, and the accuracy difference is within 0.3 pixels, which fully demonstrates the robustness and accuracy of the proposed method in optical and SAR image registration.

Scalar curvature and deformations of complex structures

Abstract We study a system of equations on a compact complex manifold, that couples the scalar curvature of a Kähler metric with a spectral function of a first-order deformation of the complex structure. The system comes from an infinite-dimensional Kähler reduction, which is a hyperkähler reduction for a particular choice of the spectral function. The main tool for studying the system is a flat connection on the space of first-order deformations of the complex structure, that allows to obtain a formal complexification of the moment map equations. Using this connection, we describe a variational characterization of the equations, a Futaki invariant for the system, and a generalization of K-stability that is conjectured to characterize the existence of solutions.