In this paper we prequantize the moduli space of non-abelian vortices. We explicitly calculate the symplectic form arising from the metric and we construct a prequantum line bundle whose curvature is proportional to this symplectic form. The prequantum line bundle turns out to be Quillen's determinant line bundle…
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48 results for “symplectic vortices”
New approach proves existence of gravitating vortices on Riemann surfaces.
problem Existence of gravitating vortices on Riemann surfaces.
method Symplectic reduction by stages and reduced -K-energy.
result Existence of solutions implies polystability of effective divisors.
This paper studies symplectic vortices on compact manifolds, proving their asymptotic behavior.
problem Analyzing the asymptotic behavior of finite energy symplectic vortices on compact manifolds.
method Proves the convergence of symplectic vortices to sectors of symplectic reduction at cylinder ends.
result Finite energy symplectic vortices exponentially converge to un/untwisted sectors of the symplectic reduction.
We derive a wall crossing formula for the symplectic vortex invariants of toric manifolds. As an application, we give a proof of Batyrev's formula for the quantum cohomology of a monotone toric manifold with minimal Chern number at least two.
Vortices and coupled vortices arise from Yang-Mills-Higgs theories and can be viewed as generalizations or analogues to Yang-Mills connections and, in particular, Hermitian-Yang-Mills connections. We proved an analytic compactification of the moduli spaces of vortices and coupled vortices on hermitian vector bundles ov…
The paper studies gravitating vortices and Einstein--Bogomol'nyi equations on Riemann surfaces.
problem The existence and properties of gravitating vortices and their relation to the Einstein--Bogomol'nyi equations.
method Analyzes gravitating vortex equations on compact Riemann surfaces, focusing on case and their relation to cosmic strings.
result Establishes a correspondence between gravitating vortex equations and Geometric Invariant Theory, proving non-existence of cosmic strings on .
We present a Hamiltonian framework for higher-dimensional vortex filaments (or membranes) and vortex sheets as singular 2-forms with support of codimensions 2 and 1, respectively, i.e. singular elements of the dual to the Lie algebra of divergence-free vector fields. It turns out that the localized induction approximat…
We completely determine the moduli space M_{N,k} of k-vortices in U(N) gauge theory with N Higgs fields in the fundamental representation. Its open subset for separated vortices is found as the symmetric product (C x CP^{N-1})^k / S_k. Orbifold singularities of this space correspond to coincident vortices and are resol…
Study vortices in Kähler-Yang-Mills equations on complex manifolds.
problem Solving coupled equations for Kähler metrics and connections.
method Dimensional reductions of Kähler-Yang-Mills equations to study vortices.
result Found solutions to Yang-Mills-Higgs equations related to vortices.
Study fluid spacetimes, proving shear-free implies vanishing expansion or vorticity.
problem Understanding shear and vorticity in perfect-fluid spacetimes.
method Analyzing perfect-fluid spacetimes using Weyl tensor and divergence.
result Proves shear-free implies vanishing expansion or vorticity for perfect fluids.
Study connects vortices in abelian Higgs models to discrete conformal maps.
problem Understanding vortices in abelian Higgs models.
method Relating vortices to discrete conformal maps via curvature and volume form.
result Constructing discrete vortex solutions using discrete conformal theory.
We note that the Bogomolny equation for abelian vortices is precisely the condition for invariance of the Hermitian-Einstein equation under a degenerate conformal transformation. This leads to a natural interpretation of vortices as degenerate hermitian metrics that satisfy a certain curvature equation. Using this view…
At critical coupling, the interactions of Ginzburg-Landau vortices are determined by the metric on the moduli space of static solutions. The asymptotic form of the metric for two well separated vortices is shown here to be expressible in terms of a Bessel function. A straightforward extension gives the metric for N vor…
Proposes NSSNNs to predict nonseparable Hamiltonian systems.
problem Predicting nonseparable Hamiltonian systems with coupled dynamics.
method Augmented symplectic time integrator to decouple position and momentum.
result Long-term, accurate, and robust predictions for large-scale Hamiltonian systems.
Elizabethan vorticeshep-th
Vortices on conical surfaces embedded in hyperbolic space.
problem Embedding Abelian vortices on conical surfaces in hyperbolic space.
method Analyzing elliptic sinh-Gordon and Tzitzeica equations, asymptotic analysis of Painleve III ODE.
result Radial solutions can be globally embedded in hyperbolic space.
The paper introduces a new concept of frame vorticity and uses it to find optimal sections in specific geometric settings.
problem Finding optimal sections in geometric settings using frame vorticity.
method Defining frame vorticity, relating it to split pseudo-Riemannian metrics, and using split special Lagrangian calibrations.
result Explicit homologically volume maximizing sections and optimal sections for specific manifolds.
Theory of point vortices extended to closed surfaces.
problem Extending point vortex dynamics to closed surfaces.
method Unified theory of point vortex dynamics on the plane, sphere, and closed surfaces.
result Comprehensive guide to point vortex dynamics on closed surfaces with genus zero and vanishing total vorticity.
We make a detailed study of the moduli space of winding number two (k=2) axially symmetric vortices (or equivalently, of co-axial composite of two fundamental vortices), occurring in U(2) gauge theory with two flavors in the Higgs phase, recently discussed by Hashimoto-Tong (hep-th/0506022) and Auzzi-Shifman-Yung (hep-…
Study vortex moduli space and compute Berry curvature for Ginzburg-Landau vortices.
problem Analyzing vortex moduli space and Berry curvature in Ginzburg-Landau theory.
method Using theorems of Taubes and Bradlow, compute tangent vectors and Berry curvature in the large volume limit.
result Computed Berry curvature and holonomy for Ginzburg-Landau vortices.
Existence and uniqueness of gravitating vortices on Riemann surfaces with specific properties.
problem Existence and uniqueness of gravitating vortices on compact Riemann surfaces.
method Existence via solving a continuity path, proving existence of singular gravitating vortices, and establishing existence of singular Einstein-Bogomol'nyi equations.
result Existence and uniqueness of gravitating vortices on Riemann surfaces with suitable properties.
It is shown that both the sinh--Gordon equation and the elliptic Tzitzeica equation can be interpreted as the Taubes equation for Abelian vortices on a CMC surface embedded in , or on a surface conformally related to a hyperbolic affine sphere in . In both cases the Higgs field and the U(1) vortex conn…
The moduli space metric and its Kahler potential for well-separated non-Abelian vortices are obtained in U(N) gauge theories with N Higgs fields in the fundamental representation.
Uniform convergence of metrics on vortex moduli space in Bradlow limit.
problem Understanding the geometry of vortex moduli spaces.
method Proof of uniform convergence of metrics using normalized metric and Fubini-Study metric.
result Establishes the Fubini-Study metric as the limit of the normalized metric in the Bradlow limit.
We derive general expressions for the Kaehler form of the L^2-metric in terms of standard 2-forms on vortex moduli spaces. In the case of abelian vortices in gauged linear sigma-models, this allows us to compute explicitly the Kaehler class of the L^2-metric. As an application we compute the total volume of the moduli …
We study the gradient flow lines of a Yang-Mills-type functional on the space of gauged holomorphic maps , where is a principal bundle on a Riemann surface and is a Kähler Hamiltonian -manifold. For compact , possibly with boundary, we prove long time existence of the gradient flow. …
Study of decorated surfaces with vortices and their group structures.
problem Understanding group structures of decorated surfaces with vortices.
method Proved isomorphism between cluster braid group, braid twist group, and fundamental group of moduli space.
result Finite presentations of isomorphic groups were given.
Solves existence of gravitating vortices with positive curvature.
problem Existence of gravitating vortices with non-negative topological constant.
method Continuity method, GIT stability condition, Cheeger-Gromov theory.
result Complete solution to existence problem for gravitating vortices with positive curvature.
Complex optical vortex knotsmath.GT
New method generates optical vortices in any knot shape.
problem Generating optical vortices in complex shapes beyond simple knots.
method Mathematical construction and experimental verification.
result Complex optical fields can form any knot shape.
Study of vortex interactions in Ginzburg-Landau models on 2D Riemannian manifolds.
problem Characterize and quantify interactions between vortices in Ginzburg-Landau models.
method Variational Ginzburg-Landau model, Γ-limit analysis, flux quantization constraints.
result Renormalized energy between vortices determined as a Γ-limit.
We review our recent work on solitons in the Higgs phase. We use U(N_C) gauge theory with N_F Higgs scalar fields in the fundamental representation, which can be extended to possess eight supercharges. We propose the moduli matrix as a fundamental tool to exhaust all BPS solutions, and to characterize all possible modu…
Research explores vortices, Painlevé integrability, and projective geometry in Yang-Mills theory.
problem Exploring vortices and their properties in Yang-Mills theory.
method Analyzes Abelian vortices, proposes modified Abelian-Higgs model, applies Painlevé test, and investigates projective differential geometry.
result Identifies integrable cases of soliton equations and derives solutions in terms of third Painlevé transcendents.
A simple property of the Weyl tensor for a shear, vorticity and acceleration-free velocity fieldmath-ph
A simple property of Weyl tensor in shear-free, vorticity-free, acceleration-free velocity fields.
problem Proving a property of the Weyl tensor in specific velocity fields.
method Analyzing the Weyl tensor's divergence and contraction properties in shear-free, vorticity-free, acceleration-free velocity fields.
result The covariant divergence of the Weyl tensor is zero if the contraction of the Weyl tensor with the velocity is zero, and vice versa.
Shape modes of vorticeshep-th
The paper finds shape modes for vortices in a specific sigma model.
problem Existence of internal modes in vortices.
method Developed a geometric formalism based on the Bogomol'nyi decomposition of the energy functional.
result Proved the existence of at least one shape mode for a general vortex solution.
Local well-posedness proved for 3D compressible Euler equations with rough vorticity.
problem Proving local well-posedness for compressible Euler equations with rough vorticity.
method Decomposing velocity into irrotational and wave components, using cancellations, trilinear estimates, and Strichartz estimates.
result Local well-posedness achieved for compressible Euler equations in with for rough vorticity.
We solve surface Navier-Stokes using DEC and compare with vorticity-stream function.
problem Solving the surface Navier-Stokes equation numerically.
method Discrete exterior calculus (DEC) in space and semi-implicit time discretization.
result Second order convergence demonstrated in flat space discretization.
On the interplay between vortices and harmonic flows: Hodge decomposition of Euler's equations in 2dmath-ph
This paper explores vortices and harmonic flows on compact surfaces, using Hodge decomposition.
problem Understanding the interplay between vortices and harmonic flows on compact surfaces.
method Hodge decomposition of Euler's equations, focusing on point vortices on compact Riemann surfaces.
result The harmonic part of the flow is constant on flat tori but not on non-flat tori.
Calculates the Hilbert space dimension for vortex moduli spaces.
problem Determining the dimension of the Hilbert space for vortex moduli spaces.
method Kahler quantization of the moduli space of vortices on a Riemann surface.
result The dimension is given by the holomorphic Euler characteristic of the quantum line bundle.
We study the problem of coupling Einstein's equations to a relativistic and physically well-motivated version of the Navier-Stokes equations. Under a natural evolution condition for the vorticity, we prove existence and uniqueness in a suitable Gevrey class if the fluid is incompressible, where this condition is given …
Researchers find non-abelian symmetric gravitating vortices on a sphere.
problem Finding solutions to non-abelian gravitating vortex equations on a sphere.
method Dimensional reduction of Kähler-Yang-Mills-Higgs equations, reduction to ODEs, method of continuity.
result Existence and admissible volumes of solutions proved.
Topologically protected vortex knots and linkscond-mat.quant-gas
Topologically protected vortex knots and links are proposed and proven.
problem Decaying of tangled vortex structures through local reconnections and strand crossings.
method Proposed and proven topologically protected vortex structures using non-Abelian topological vortices.
result Existence of topologically protected -colored links and classification using the -invariant.
In this note we show that for the group G = U(N) the space of Hecke modifications of a rank N vector bundle over a Riemann surface C coincides with the moduli space of solutions of certain non-abelian vortex equations over C . Through the recent work of Kapustin and Witten this then leads to an isomorphism between the …
The paper constructs new solutions to the Kahler-Yang-Mills equations and finds cosmic string solutions.
problem Solving the Kahler-Yang-Mills equations and understanding cosmic strings.
method Dimensional reduction and applying the Kahler-Yang-Mills equations to the product of the complex projective line with a compact Riemann surface.
result Found solutions to the Einstein-Bogomol'nyi equations corresponding to cosmic strings.
For Ginzburg-Landau vortices, energy quantization holds only when density is less than 2.
problem Energy quantization in Ginzburg-Landau vortices for higher dimensions.
method Analyzing normalized energy measures and vorticity sets.
result Energy quantization only holds when density is less than 2.
This article provides an explicit construction for a family of singular instantons on S^4 S^2 with arbitrary real holonomy parameter α. This family includes the original α= 1/4, c_2 = 3/2 solution discovered by P. Forgacs, Z. Horvath, and L. Palla, and our approach is modeled on that of their 1981 paper. Our primary to…
Study SO(3) vortices over orbifold Riemann surfaces and monopoles.
problem Characterize solutions of SO(3) monopole equations.
method Use moduli spaces of SO(3) vortices over orbifold Riemann surfaces.
result Compute Morse-Bott indices of a function on moduli space.
Helicoids and vorticesmath.DG
New minimal surfaces found from vortex crystals.
problem Minimal surfaces and vortex crystals.
method Gluing helicoids into minimal surfaces.
result New minimal surfaces and vortex crystals discovered.
Study vortex flows on Riemann surfaces, proving dominated splitting and Anosov properties.
problem Investigate flow properties on Riemann surfaces.
method Associate flow to vortex equations, investigate properties of flow.
result Show that flow always admits a dominated splitting and identify special cases of Anosov flow.
New instantons found on Euclidean Schwarzschild manifold, challenging existing theories.
problem Challenging the understanding of instantons on the Euclidean Schwarzschild manifold.
method Exploring a correspondence between planar Abelian vortices and spherically symmetric instantons.
result Complete description of a connected component of the moduli space of unit energy SU(2) instantons.