Moment Map Research Articles

The moduli space of Fano manifolds with Kähler–Ricci solitons

We construct a canonical Hausdorff complex analytic moduli space of Fano manifolds with Kähler–Ricci solitons. This enlarges the moduli space of Fano manifolds with Kähler–Einstein metrics. We discover a moment map picture for Kähler–Ricci solitons, and give complex analytic charts on the topological space consisting of Kähler–Ricci solitons, by studying differential geometric aspects of this moment map. Some stacky words and arguments on Gromov–Hausdorff convergence help to glue them together in the holomorphic manner.

Rational Singularities, Quiver Moment Maps, and Representations of Surface Groups

Abstract We prove using jet schemes that the zero loci of the moment maps for the quivers with one vertex and at least two loops have rational singularities. This implies that the spaces of representations of the fundamental group of a compact Riemann surface of genus at least two have rational singularities. This has consequences for the numbers of irreducible representations of the special linear groups over the integers and over the $p$-adic integers.

On Parametrization of the Symplectic Quotient of the Cartesian Product of Coadjoint Orbits of the Complex General Linear Group with Respect to its Diagonal Action

We consider the problem of coordinatizing the manifold constructed via the Marsden–Weinstein quotient. Rational canonical coordinates on the symplectic reduction with respect to the diagonal action of the general linear group on the Cartesian product of coadjoint orbits in the case of the complex general linear group are constructed on an algebraically open subset of the quotient space. The method is based on an iterative process of constructing projection-flag coordinates, and works if the matrices constituting the orbits have a sufficiently rich set of invariant subspaces.

A Construction of Special Lagrangian Submanifolds by Generalized Perpendicular Symmetries

We show a method to construct a special Lagrangian submanifold L' from a given special Lagrangian submanifold L in a Calabi-Yau manifold with the use of generalized perpendicular symmetries. We use moment maps of the actions of Lie groups, which are not necessarily abelian. By our method, we construct some non-trivial examples in the cotangent bundles of the spheres which are non-flat Calabi-Yau manifolds equipped with the Stenzel metrics.

Position and momentum mapping of vibrations in graphene nanostructures.

Propagating atomic vibrational waves-phonons-determine important thermal, mechanical, optoelectronic and transport characteristics of materials. Thus a knowledge of phonon dispersion (that is, the dependence of vibrational energy on momentum) is a key part of our understanding and optimization of a material's behaviour. However, the phonon dispersion of a free-standing monolayer of a two-dimensional material such as graphene, and its local variations, have remained elusive for the past decade because of the experimental limitations of vibrational spectroscopy. Even though electron energyloss spectroscopy (EELS) in transmission has recently been shown to probe local vibrational charge responses1-4, such studies are still limited by momentum space integration due to the focused beam geometry; they are also restricted to polar materials such as boron nitride or oxides1-4, in which huge signals induced by strong dipole moments are present. On the other hand, measurements on graphene performed by inelastic X-ray (neutron) scattering spectroscopy5-7 or EELS in reflection8,9 do not have any spatial resolution and require large microcrystals. Here we provide a new pathway to determine phonon dispersions down to the scale of an individual free-standing graphene monolayer by mapping the distinct vibrational modes for a large momentum transfer. The measured scattering intensities are accurately reproduced and interpreted with density functional perturbation theory10. Additionally, a nanometre-scale mapping of selected momentum-resolved vibrational modes using graphene nanoribbon structures has enabled us to spatially disentangle bulk, edge and surface vibrations. Our results are a proof-of-principle demonstration of the feasibility of studying local vibrational modes in two-dimensional monolayer materials at the nanometre scale.

A New Approach to Simple Modules for Preprojective Algebras

The work of the first author on the moment map for representations of quivers included a classification of the possible dimension vectors of simple modules for deformed preprojective algebras. That classification was later used to solve an additive analogue of the Deligne-Simpson problem. The last step in the proof of the classification involved some general position arguments; here we give a new approach which avoids such arguments.

Quantitative Susceptibility Mapping of Magnetic Quadrupole Moments

We modeled the magnetic field up to the quadrupole term to investigate not only the average susceptibility (dipole), but also the susceptibility distribution (quadrupole) contribution. Expanding the magnetic field up to the 2nd order provides the quadrupole (0th: monopole, 1st: dipole). Numerical simulations were performed to investigate the quadrupole contribution with subvoxel nonuniformity. Conventional dipole and our dipole + quadrupole models were compared in the simulation, the phantom and human brain. Furthermore, the quadrupole field was compared with the anisotropic susceptibility field in the dipole tensor model. In a nonuniformity case, numerical simulations showed a nonnegligible quadrupole field contribution. Our study showed a difference between the two methods in the susceptibility map at the edges; both the phantom and human studies showed sharper structural edges with the dipole + quadrupole model. Quadrupole moments showed contrast mainly at the structural boundaries. The quadrupole moment field contribution was smaller but nonnegligible compared to the anisotropic susceptibility contribution. Nonuniform and uniform source distributions can be separately considered by quadrupole expansion, which were mixed together in the dipole model. In the presence of nonuniformity, the susceptibility maps may be different between the two models. For a comprehensive field model, the quadrupole might need to be considered along with susceptibility anisotropy and microstructure effects.

Clebsch-Lagrange variational principle and geometric constraint analysis of relativistic field theories

Inspired by the Clebsch optimal control problem, we introduce a new variational principle that is suitable for capturing the geometry of relativistic field theories with constraints related to a gauge symmetry. Its special feature is that the Lagrange multipliers couple to the configuration variables via the symmetry group action. The resulting constraints are formulated as a condition on the momentum map of the gauge group action on the phase space of the system. We discuss the Hamiltonian picture and the reduction in the gauge symmetry by stages in this geometric setting. We show that the Yang–Mills–Higgs action and the Einstein–Hilbert action fit into this new framework after a (1 + 3)-splitting. Moreover, we recover the Gauß constraint of Yang–Mills–Higgs theory and the diffeomorphism constraint of general relativity as momentum map constraints.

The phase topology and bifurcation tori of the Hydrogen atom subjected to external fields

In this paper, we performed an adapted canonical transformation, and we analysed the phase space topology and the bifurcation of Liouville tori of the Hydrogen atom subjected to three static external fields: Van der Waals potential, electric and magnetic fields. In particular, for all values of the parameters of the system under consideration, the bifurcation diagrams of the momentum mapping are constructed, bifurcations of the common level sets of the first integrals are described and the all-generic bifurcations are computed for all singular points of the bifurcation diagrams. However no author has combined these three fields and studied their behavior. Numerical investigations are performed for the integrable case by means of Poincaré surfaces of section and the phase space trajectories method, and we observed the chaos-order-chaos transition

Polysymplectic reduction and the moduli space of flat connections

A polysymplectic structure is a vector-valued symplectic form, that is, a closed nondegenerate 2-form with values in a vector space. We first outline the polysymplectic Hamiltonian formalism with coefficients in a vector space , we then apply this framework to show that the moduli space of flat connections on a principal bundle over a compact manifold M is a polysymplectic reduction of the space of all connections by the action of the gauge group with respect to a natural polysymplectic structure with values in an infinite dimensional Banach space. As a consequence, the moduli space inherits a canonical H2(M)-valued presymplectic structure.Along the way, we establish various properties of polysymplectic manifolds. For example, a Darboux-type theorem asserts that every -symplectic manifold locally symplectically embeds in a standard polysymplectic manifold . We also show that both the Arnold conjecture and the well-known convexity properties of the classical moment map fail to hold in the polysymplectic setting.

On the Uniqueness Result of Theorem 6 in “Relative Entropy and the Multivariable Multidimensional Moment Problem”

Matrix-valued covariance extension and multivariate spectral estimation are formulated as generalized moment problems in the "THREE" approach and its extensions. Under this context, we discuss Theorem 6 in \cite{Georgiou-06} concerning the bijectivity of a moment map defined over a parametric family of spectral densities. In particular, we provide a counterexample in which the moment map under consideration is shown to have a critical point, namely a point at which the Jacobian loses rank. Then with standard techniques in bifurcation theory, we conclude further that the computed critical point is a bifurcation point, which means that the moment map is not injective.

Stein kernels and moment maps

We describe a construction of Stein kernels using moment maps, which are solutions to a variant of the Monge–Ampère equation. As a consequence, we show how regularity bounds in certain weighted Sobolev spaces on these maps control the rate of convergence in the classical central limit theorem, and derive new rates in Kantorovitch–Wasserstein distance in the log-concave situation, with explicit polynomial dependence on the dimension.

Symplectic reduction of Sasakian manifolds

When a complex semisimple group G acts holomorphically on a Kahler manifold $$(X,\, \omega )$$ such that a maximal compact subgroup $$K\, \subset \, G$$ preserves the symplectic form $$\omega $$ , a basic result of symplectic geometry says that the corresponding categorical quotient X / G can be identified with the quotient of the zero-set of the moment map by the action of K. We extend this to the context of a semisimple group acting on a Sasakian manifold.

Quantization commutes with singular reduction: Cotangent bundles of compact Lie groups

We analyze the ‘quantization commutes with reduction’ problem (first studied in physics by Dirac, and known in the mathematical literature also as the Guillemin–Sternberg Conjecture) for the conjugate action of a compact connected Lie group [Formula: see text] on its own cotangent bundle [Formula: see text]. This example is interesting because the momentum map is not proper and the ensuing symplectic (or Marsden–Weinstein quotient) [Formula: see text] is typically singular.In the spirit of (modern) geometric quantization, our quantization of [Formula: see text] (with its standard Kähler structure) is defined as the kernel of the Dolbeault–Dirac operator (or, equivalently, the spin[Formula: see text]–Dirac operator) twisted by the pre-quantum line bundle. We show that this quantization of [Formula: see text] reproduces the Hilbert space found earlier by Hall (2002) using geometric quantization based on a holomorphic polarization. We then define the quantization of the singular quotient [Formula: see text] as the kernel of the twisted Dolbeault–Dirac operator on the principal stratum, and show that quantization commutes with reduction in the sense that either way one obtains the same Hilbert space [Formula: see text].

Flavor Symmetries and Unitarity Bounds in N=2 Superconformal Field Theories.

In this Letter I analyze the constraints imposed by unitarity on the flavor central charges of four-dimensional N=2 superconformal field theories with general reductive global symmetry groups. I derive several general and far-reaching consequences of unitarity by computing the norms of flavor singlet Higgs branch operators appearing in the squares of "moment map" operators via the associated vertex operator algebra, and imposing the requirement that they be non-negative.

Injectivity and surjectivity of the Stieltjes moment mapping in Gelfand–Shilov spaces

The Stieltjes moment problem is studied in the framework of general Gelfand-Shilov spaces defined via weight sequences. We characterize the injectivity and surjectivity of the Stieltjes moment mapping, sending a function to its sequence of moments, in terms of growth conditions for the defining weight sequence. Finally, a related moment problem at the origin is studied.

Variational Discretizations of Gauge Field Theories Using Group-Equivariant Interpolation

We describe a systematic mathematical approach to the geometric discretization of gauge field theories that is based on Dirac and multi-Dirac mechanics and geometry, which provide a unified mathematical framework for describing Lagrangian and Hamiltonian mechanics and field theories, as well as degenerate, interconnected, and nonholonomic systems. Variational integrators yield geometric structure-preserving numerical methods that automatically preserve the symplectic form and momentum maps, and exhibit excellent long-time energy stability. The construction of momentum-preserving variational integrators relies on the use of group-equivariant function spaces, and we describe a general construction for functions taking values in symmetric spaces. This is motivated by the geometric discretization of general relativity, which is a second-order covariant gauge field theory on the symmetric space of Lorentzian metrics.

Locally conformally symplectic reduction

We present a reduction procedure for locally conformally symplectic manifolds with an action of a Lie group preserving the conformal structure, with respect to any regular value of the momentum mapping. Under certain conditions, this reduction is compatible with the existence of a locally conformally Kahler structure. As a special consequence, we obtain a compatible contact reduction with respect to any regular value of the contact momentum mapping.

Analysis and test of the central-blue-spot infall hallmark

Aims. The infall of material onto a protostar, in the case of optically thick line emission, produces an asymmetry in the blue- and red-wing line emissions. For an angularly resolved emission, this translates in a blue central spot in the first-order moment (intensity weighted velocity) map. Methods. An analytical expression for the first-order moment intensity as a function of the projected distance was derived, for the cases of infinite and finite infall radius. The effect of a finite angular resolution, which requires the numerical convolution with the beam, was also studied. Results. This method was applied to existing data of several star-forming regions, namely G31.41+0.31 HMC, B335, and LDN 1287, obtaining good fits to the first-order moment intensity maps, and deriving values of the central masses onto which the infall is taking place (G31.41+0.31 HMC: 70–120 M⊙; B335: 0.1 M⊙; Guitar Core of LDN 1287: 4.8 M⊙). The central-blue-spot infall hallmark appears to be a robust and reliable indicator of infall.

Special Lagrangian Submanifolds and Cohomogeneity One Actions on the Complex Projective Space

We construct examples of cohomogeneity one special Lagrangian submanifolds in the cotangent bundle over the complex projective space, whose Calabi-Yau structure was given by Stenzel. For each example, we describe the condition of special Lagrangian as an ordinary differential equation. Our method is based on a moment map technique and the classification of cohomogeneity one actions on the complex projective space classified by Takagi.