Paper proves solvability condition for complex equation on special submanifolds.
problem Solvability condition for supercritical deformed Hermitian-Yang-Mills equation.
method Used integrals on subvarieties to provide necessary and sufficient condition.
result Confirms mirror version of Thomas-Yau conjecture about special Lagrangian submanifolds.
Study of deformed Hermitian Yang-Mills equations with variable Kähler metrics.
problem Solving special Lagrangian type equations with variable metrics.
method Introducing extended gauge group to couple moment maps and scalar curvature.
result Solutions satisfy a mixture of K-stability and Bridgeland-type stability.
Stability of Morse index for Yang-Mills connections in 4D.
problem Stability of critical points in Yang-Mills energy relaxation.
method Establishing lower semi-continuity of Morse index and upper continuity of Morse index plus nullity.
result Yang-Mills fields are more stable than harmonic maps in 4D.
The study extends weak connection space to high dimensions, overcoming a key obstacle.
problem Obstacle in studying Yang-Mills Lagrangian in high dimensions.
method Generalized space of weak connections and proved weak sequential closure.
result Established strong approximation property of weak connections by smooth connections.
Yang-Mills theory is growing at the interface between high energy physics and mathematics. It is well known that Yang-Mills theory and Gauge theory in general had a profound impact on the development of modern differential and algebraic geometry. One could quote Donaldson invariants in four dimensional differential top…
From string theory, the notion of deformed Hermitian Yang-Mills connections has been introduced by Mariño, Minasian, Moore and Strominger. After that, Leung, Yau and Zaslow proved that it naturally appears as mirror objects of special Lagrangian submanifolds via Fourier-Mukai transform between dual torus fibrations. In…
Study deformed Hermitian-Yang-Mills equation via GIT and prove existence of geodesics.
problem Existence and regularity of solutions to deformed Hermitian-Yang-Mills equation.
method Variational approach via infinite dimensional GIT problem, scale estimates, Fourier-Mukai transform.
result Existence of smooth and weak geodesics with C1,α regularity. Study special Lagrangian sections in Calabi-Yau threefolds, showing stability conditions imply isomorphism to special Lagrangians.
problem Understanding stability conditions on Fukaya-Seidel categories of Calabi-Yau threefolds.
method Analyzing sections of special Lagrangian fibrations, constructing Bridgeland stability conditions, and relating to deformed Hermitian Yang-Mills connections.
result Semistability of L[2] implies isomorphism to special Lagrangian sections. Study weak geodesics in deformed Hermitian-Yang-Mills equation space.
problem Geodesics in the space of potentials for deformed Hermitian-Yang-Mills equation.
method Formulated as degenerate elliptic equation, used nonlinear Dirichlet duality theory, constructed continuous solutions.
result Continuous solutions constructed for Dirichlet problem.
The Lagrangian and Hamiltonian structures for an ideal gauge-charged fluid are determined. Using a Kaluza-Klein point of view, the equations of motion are obtained by Lagrangian and Poisson reductions associated to the automorphism group of a principal bundle. As a consequence of the Lagrangian approach, a Kelvin-Noeth…
This paper is motivated by a relatively recent work by Joyce in special Lagrangian geometry, but the basic idea of the present paper goes back to an earlier pioneering work of Donaldson in Yang--Mills gauge theory; Donaldson discovered a global structure of a (compactified) moduli space of Yang--Mills instantons, and a…
We exhibit a transformation taking special Lagrangian submanifolds of a Calabi-Yau together with local systems to vector bundles over the mirror manifold with connections obeying deformed Hermitian-Yang-Mills equations. That is, the transformation relates supersymmetric A- and B-cycles. In this paper, we assume that th…
Langmuir-Blodgett films (LB-films) consist from few LB-monolayers which are high structured nanomaterials that are very promising materials for applications. We use a geometrical approach to describe structurization into LB-monolayers. Consequently, we develop on the 1-jet space J^1([0,\infty),R^2) the single-time Lagr…
The paper finds explicit instantons on a specific 6-manifold.
problem Finding explicit solutions to the Hermitian Yang-Mills equations.
method Dimensional reduction and reformulation of the equations on quotient spaces.
result Explicit abelian instantons and deformed Hermitian Yang-Mills connections are found.
Develops a unified theory of Yang-Mills and GR using generalized principal bundles.
problem Combining Yang-Mills theories and General Relativity into a single framework.
method Using generalized principal bundle theory, the authors develop a new approach to field theories.
result Recover General Relativity within the framework of generalized principal connections.
Unified solution to M-theory problems using super-exceptional geometry.
problem M-theory's Lagrangian construction, gauge field origin, and supersymmetric enhancement.
method Combining homotopy theory, supergeometry, and super-homotopy theory.
result Unified description of M-theory Lagrangians and emergence of gauge fields.
Paper confirms conjecture for projective manifolds in supercritical phase.
problem Stability condition for deformed Hermitian-Yang-Mills equation.
method Establishes stability result not involving uniform constants.
result Confirms conjecture for projective manifolds in supercritical phase.
New tan-concavity property for Lagrangian phase operators helps in studying dHYM metrics.
problem Lack of concavity in Lagrangian phase operator for dHYM metrics.
method Introduce tangent Lagrangian phase flow (TLPF) on almost calibrated (1,1)-forms.
result TLPF exists for all positive time and converges to dHYM metrics under certain conditions.
Defines a new conformally invariant Yang-Mills type energy for 6-manifolds.
problem No specific problem stated; focuses on defining a new energy.
method Defines a conformally invariant action S on gauge connections on a 6-manifold M, leading to higher-order conformally invariant Yang-Mills equations.
result The Euler-Lagrange equations of S provide a conformally invariant analogue of Yang-Mills equations, with special cases recovering known invariants.
Let L be a holomorphic line bundle over a compact Kähler manifold X. Motivated by mirror symmetry, we study the deformed Hermitian-Yang-Mills equation on L, which is the line bundle analogue of the special Lagrangian equation in the case that X is Calabi-Yau. We show that this equation is the Euler-Lagrange equ…
Using the notion of vacuum pairs we show how the (square of the) mass matrix of the fermions can be considered geometrically as curvature. This curvature together with the curvature of space-time, defines the total curvature of the Clifford module bundle representing a ``free'' fermion within the geometrical setup of s…
We develop a unifed theory to study geometry of manifolds with different holonomy groups. They are classified by (1) real, complex, quaternion or octonion number they are defined over and (2) being special or not. Specialty is an orientation with respect to the corresponding normed algebra A. For example, special Riema…
The paper studies how adding a 'Gauge Mass' term breaks gauge symmetry in Yang-Mills-Higgs systems and analyzes the resulting behavior.
problem Breaking gauge symmetry in Yang-Mills-Higgs systems.
method Analyzing the asymptotic behavior of the system with a 'Gauge Mass' term added.
result The system's behavior is characterized by concentration phenomena and convergence to harmonic maps and minimal energies.
Moduli spaces of real bundles over a real curve arise naturally as Lagrangian submanifolds of the moduli space of semi-stable bundles over a complex curve. In this paper, we adapt the methods of Atiyah-Bott's "Yang-Mills over a Riemann Surface" to compute Z/2-Betti numbers of these spaces, proving formulas recently obt…
Expanding on supersymmetric Yang-Mills theory, this work focuses on cohomological field theory aspects.
problem Exploring the mathematical aspects of supersymmetric Yang-Mills theory on 4D manifolds.
method Constructing a cohomological field theory using the Atiyah-Jeffrey construction and demonstrating the transversally elliptic complex.
result The deformation theory of flipping instantons is controlled by a transversally elliptic complex, crucial for non-degeneracy and calculability.
The paper studies singularities in a complex flow related to mean curvature.
problem Investigating singularities in a complex flow related to mean curvature.
method Constructing two distinct examples of singularities using the line bundle mean curvature flow.
result Found a finite time singularity, ruling out long time existence of the flow.
Develops Lagrange-Hamilton geometry for COVID-19 disease dynamics.
problem Modeling the spread of COVID-19 disease.
method Least squares variational method, nonlinear connections, d-torsions, Lagrangian Yang-Mills.
result Jacobi stability of the dynamical system.
`Gluing' is a technique of constructing solutions to non-linear (elliptic) partial differential equations such as Yang--Mills equations, minimal surface equations and Einstein equations. Calibrated submanifolds are a certain class of minimal surfaces, and there are various examples of them constructed by the gluing tec…
Develops methods to solve complex and real Hessian equations.
problem Solving complex and real Hessian equations on various domains.
method Introduces an ansatz to reduce PDEs to systems of ODEs, integrating via abelian integrals.
result Constructs entire solutions of arbitrary subcritical phase for dHYM/LYZ and special Lagrangian equations.
Sp(n)-instantons linked to complex Lagrangian graphs via Fourier-Mukai transform.
problem Understanding Sp(n)-instantons on hyperkahler manifolds with conical singularities.
method Relating Sp(n)-instantons to deformed instantons and studying their properties on hyperkahler manifolds.
result Sp(n)-instantons on hyperkahler manifolds correspond to tri-contact instantons on the 3-Sasakian link.
Proves stability condition for Lagrangian sections in toric weak Fano manifolds.
problem Stability of Lagrangian sections in Calabi-Yau fibrations.
method SYZ transform, toric gamma theorem, Nakai-Moishezon criterion.
result Hamiltonian isotopy of Lagrangian sections under stability condition.
By formulating N = 1, 2, 4, 8, D = 3, Yang-Mills with a single Lagrangian and single set of transformation rules, but with fields valued respectively in R,C,H,O, it was recently shown that tensoring left and right multiplets yields a Freudenthal-Rosenfeld-Tits magic square of D = 3 supergravities. This was subsequently…
Develops geometry for Lotka-Volterra model of species competition.
problem Population dynamics of competing species.
method Least squares variational method, Lagrange-Hamilton geometry.
result Jacobi stability discussed for the Lotka-Volterra system.
Let (X,α) be a Kähler manifold of dimension n, and let [ω]∈H1,1(X,R). We study the problem of specifying the Lagrangian phase of ω with respect to α, which is described by the nonlinear elliptic equation \[ \sum_{i=1}^{n} \arctan(λ_i)= h(x) \] where λi are the eigenvalues of ω with respect …
Study shows convergence of Yang-Mills connections on K3 surfaces under fiber collapse.
problem Analyzing Yang-Mills connections on K3 surfaces as fibers collapse.
method Proves convergence of a sequence of anti-self-dual connections under specific conditions.
result Connections converge to a limiting flat connection with holomorphic structure.
The paper solves the dHYM equation on rational homogeneous varieties using Lie theory.
problem Solving the deformed Hermitian Yang-Mills equation on rational homogeneous varieties.
method Using Lie theory to describe the Lagrangian phase and characterize solutions.
result Characterization of all supercritical and hypercritical homogeneous solutions of the dHYM equation.
In this article we discuss the geometry of moduli spaces of (1) flat bundles over special Lagrangian submanifolds and (2) deformed Hermitian-Yang-Mills bundles over complex submanifolds in Calabi-Yau manifolds. These moduli spaces reflect the geometry of the Calabi-Yau itself like a mirror. Strominger, Yau and Zaslow c…
The thesis explores dualities and gaugings in supergravity theories.
problem Exploring dualities and gaugings in supergravity theories.
method Analyzing the space of local deformations and the BV-BRST deformation of scalar-vector coupled Lagrangians.
result Only Yang-Mills type deformations are possible for a large class of theories.
The geometry of submanifolds is intimately related to the theory of functions and vector bundles. It has been of fundamental importance to find out how those two objects interact in many geometric and physical problems. A typical example of this relation is that the Picard group of line bundles on an algebraic manifold…
The paper solves pseudo-convexity for special Lagrangian equations, with applications in mirror symmetry.
problem Existence of solutions to the Special Lagrangian Potential Equation.
method Explicitly computing boundary conditions for pseudo-convexity in potential theory.
result Interesting pseudo-convexity results for various related equations, including deformed Hermitian-Yang-Mills and curvature equations.
Paper extends Simons theorem to F-Yang-Mills connections for instability.
problem Tackles instability of F-Yang-Mills connections. method Extends Simons theorem to F-Yang-Mills connections using Kobayashi-Ohnita-Takeuchi's method. result Derives a sufficient condition for instability of non-flat F-Yang-Mills connections. Moduli spaces of semi-stable real and quaternionic vector bundles of a fixed topological type admit a presentation as Lagrangian quotients, and can be embedded into the symplectic quotient corresponding to the moduli variety of semi-stable holomorphic vector bundles of fixed rank and degree on a smooth complex projecti…
Since its inception, Floer homology has been an important tool in low-dimensional topology. Floer theoretic invariants of 3-manifolds tend to be either gauge theoretic or symplecto-geometric in nature, and there is a general philosophy that each gauge theoretic Floer homology should have a corresponding symplectic Fl…
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.
The paper studies stability of F-Yang-Mills connections on complex projective spaces.
problem Stability of F-Yang-Mills connections on complex projective spaces.
method Inspired by Lawson-Simons, the paper proves stability conditions and structures for F-Yang-Mills connections.
result Conditions for weakly stable F-Yang-Mills connections on complex projective spaces.
The paper examines Yang-Mills-Higgs pairs on vector bundles and proves stability and energy identity.
problem Stability and energy identity of Yang-Mills-Higgs pairs on vector bundles.
method Bubble-neck decomposition and analysis of weakly stable pairs.
result A sequence of Yang-Mills-Higgs pairs converges to a Yang-Mills-Higgs pair with uniformly bounded energy.
Removes singularities for Yang-Mills-Higgs fields in higher dimensions.
problem Yang-Mills-Higgs fields with isolated singularities.
method Establishes decay estimates and conformally invariant energy bounds.
result Removable singularity theorem for Yang-Mills-Higgs fields.
Compactifies moduli spaces of Hermitian-Yang-Mills connections on balanced manifolds.
problem Analyzing Ω-Yang-Mills connections on Riemannian manifolds. method Extending known results on Yang-Mills connections to Ω-Yang-Mills connections, proving weak compactness and removable singularity theorems. result Compactification of moduli spaces of smooth Hermitian-Yang-Mills connections on unitary bundles over balanced manifolds.