The paper examines sequences of solutions to Seiberg-Witten systems in 4D manifolds.
problem Analyzing sequences of solutions to Seiberg-Witten systems in 4D manifolds.
method Investigates the behavior of sequences of solutions to Seiberg-Witten-like equations for Hermitian connections and spinor sections.
result Provides insights into the behavior of sequences of solutions to Seiberg-Witten systems in 4D manifolds.
This work generalizes a formula linking Seiberg-Witten prepotential and topological recursion.
problem Analyzing the relationship between Seiberg-Witten curves and topological recursion.
method Analytical approach using Seiberg-Witten family of curves.
result A generalized formula relating Seiberg-Witten prepotential to the genus zero part of topological recursion on a Seiberg-Witten curve.
The paper solves Seiberg-Witten equations in dimensions 5, 6, and 8.
problem Solving Seiberg-Witten equations in higher dimensions.
method Elliptic system of equations, constructing explicit solutions, relating to vortices.
result Explicit solutions in dimensions 5, 6, and 8, linking to vortices.
The paper describes Seiberg-Witten equations in all dimensions.
problem Formulating Seiberg-Witten equations in all dimensions.
method Develops elliptic systems for Seiberg-Witten equations in various dimensions, using connections, spinors, and odd degree forms.
result Proves a system of equations for Seiberg-Witten in all dimensions, modulo gauge.
Study Seiberg-Witten theory on manifolds with codimension-3 foliations.
problem Analyzing Seiberg-Witten theory on manifolds with codimension-3 Riemannian foliations.
method Construct basic Seiberg-Witten invariant and monopole Floer homologies for each transverse spin structure.
result Monopole Floer homologies are independent of the bundle-like metric and generic perturbation.
In this article, we consider a gauge-theoretic equation on compact symplectic 6-manifolds, which forms an elliptic system after gauge fixing. This can be thought of as a higher-dimensional analogue of the Seiberg-Witten equation. By using the virtual neighbourhood method by Ruan, we define an integer-valued invariant, …
New equations use Pin(2) symmetry to study spinor and connection solutions.
problem Study of spinor and connection solutions on 4-manifolds.
method Define and analyze Seiberg-Witten-like equations with Rarita-Schwinger operators.
result Moduli space of solutions is non-compact and non-empty under certain conditions.
Formula connects surgeries to Seiberg-Witten invariants.
problem Understanding how surgeries affect Seiberg-Witten invariants.
method Proves surgery formulas for Seiberg-Witten invariants and families.
result Expresses new invariants in terms of original ones.
Formula for combining Seiberg-Witten invariants of 4-manifolds.
problem Combining invariants of 4-manifolds obtained by connected sum.
method Proved a gluing formula for families Seiberg-Witten invariants.
result Formula expresses invariants of connected sum in terms of one summand.
In this thesis we study the Seiberg-Witten theory of an oriented homology 3-sphere. The goal is to extract topological invariants - the Seiberg-Witten invariants - by counting the solutions to the Seiberg-Witten equations on the manifold. The first question we consider is whether the Seiberg-Witten invariants depend on…
We give the definition of the Seiberg-Witten-Floer homology group for a homology 3-sphere. Its Euler characteristic number is a Casson-type invariant. For a four-manifold with boundary a homology sphere, a relative Seiberg-Witten invariant is defined taking values in the Seiberg-Witten-Floer homology group, these relat…
Formula calculates Seiberg-Witten invariants after surgery.
problem Calculating Seiberg-Witten invariants for surgeries on 4-manifolds.
method Surgery formula for Seiberg-Witten invariants of 4-manifolds.
result Formula expresses invariants of the manifold after surgery.
We investigate Seiberg-Witten theory in the presence of real structures. Certain conditions are obtained so that integer valued real Seiberg-Witten invariants can be defined. In general we study properties of the real Seiberg-Witten projection map from the point of view of Fredholm map degrees.
Abstract: Proves compactness theorem for generalized Seiberg-Witten equations in 3D.
problem Compactness of solutions to generalized Seiberg-Witten equations in 3D.
method Abstract compactness theorem for a family of generalized Seiberg-Witten equations in dimension three.
result Recovers and extends known compactness theorems for stable flat connections and Seiberg-Witten equations.
Abstract relates G2 instantons to Seiberg-Witten monopoles.
problem Understanding the connection between G2 instantons and Seiberg-Witten monopoles.
method Relates Seiberg-Witten monopoles, Fueter sections, and G2 instantons.
result Describes a conjectural relation between G2 instantons and Seiberg-Witten monopoles.
New formula connects Bauer-Furuta and Seiberg-Witten invariants of families of 4-manifolds.
problem Understanding invariants of families of 4-manifolds.
method Cohomological formula linking Bauer-Furuta and Seiberg-Witten invariants.
result New divisibility properties of families Seiberg-Witten invariants.
Proof of Donaldson's theorem using Seiberg-Witten equations for multiple spinors.
problem Donaldson's Diagonalization Theorem
method Abelian gauge-theoretic variant of Seiberg-Witten equations for multiple spinors
result Proof of Donaldson's Diagonalization Theorem
Seiberg-Witten theory connects 3-manifold connections to spinor solutions.
problem Constructing Seiberg-Witten moduli spaces and understanding blow-up sets.
method Interpreting flat PSL(2;R)-connections as Seiberg-Witten solutions with two spinors.
result Explicit examples of Seiberg-Witten moduli spaces constructed.
Proves simple type conjecture for mod 2 Seiberg-Witten invariants.
problem Proving simple type conjecture for mod 2 Seiberg-Witten invariants.
method Analyzes cohomology ring conditions to prove simple type for all smooth structures.
result Shows existence of topological 4-manifolds with mod 2 simple type.
Researchers compute mod 2 Seiberg-Witten invariants for spin structures and families.
problem Computing mod 2 Seiberg-Witten invariants for spin structures and families.
method Using Pin(2)-symmetry and localisation in equivariant cohomology, the researchers enhance and compute the invariants.
result The mod 2 Seiberg-Witten invariants are computed for spin structures and families, confirming the simple type conjecture mod 2.
Seiberg-Witten invariant vanishes for certain 4-manifolds with specific foliations.
problem Analyzing Seiberg-Witten invariants on manifolds with rank 2 foliations.
method Using foliations with positive scalar curvature and \spinc structures.
result Seiberg-Witten invariant vanishes for specified manifolds.
New hyperbolic 4-manifolds found with zero Seiberg-Witten invariants.
problem Existence of hyperbolic 4-manifolds with vanishing Seiberg-Witten invariants.
method Using geometric and arithmetic group theory, showing certain hyperbolic 4-manifolds contain L-spaces.
result Existence of hyperbolic 4-manifolds with vanishing Seiberg-Witten invariants.
Compact moduli space shown for Seiberg-Witten on flat scalar curvature manifold.
problem Compactification of Seiberg-Witten moduli space on manifolds with flat scalar curvature.
method Analysis of Seiberg-Witten equations on non-compact manifolds with periodic ends, scalar curvature zero, and vanishing cohomologies.
result Moduli space is compact under given conditions.
Invariants from Seiberg-Witten equations link scalar curvature to diffeomorphisms.
problem Linking positive scalar curvature metrics to diffeomorphisms of 4-manifolds.
method Introducing an invariant using families of Seiberg-Witten equations.
result Invariant yields applications to homotopy groups of positive scalar curvature metrics.
We introduce a new class of perturbations of the Seiberg-Witten equations. Our perturbations offer flexibility in the way the Seiberg-Witten invariants are constructed and also shed a new light to LeBrun's curvature inequalities.
Given a three-manifold with b_1=1 and a nontorsion spin^c structure, we use finite dimensional approximation to construct from the Seiberg-Witten equations two invariants in the form of a periodic pro-spectra. Various functors applied to these invariants give different flavors of Seiberg-Witten Floer homology. We also …
Variant of Seiberg-Witten equations for multiple-spinors connects to stability of holomorphic bundles.
problem Detecting stability of holomorphic vector bundles using Seiberg-Witten equations.
method Abelian gauge-theoretic variant of Seiberg-Witten equations for multiple-spinors.
result Constructs a numerical invariant related to φ−stability of SU(n)−holomorphic vector bundles. A formula is given for the Seiberg-Witten invariants of a 4-manifold that is cut along certain kinds of 3-dimensional tori. The formula involves a Seiberg-Witten invariant for each of the resulting pieces.
Proves a general connected sum formula for families Seiberg-Witten invariants.
problem Limited connected sum formulae for families Seiberg-Witten theory.
method Develops a general connected sum formula incorporating previous results.
result Proves a new connected sum formula for Seiberg-Witten families.
New invariant from Seiberg-Witten equations constrains embedded surfaces.
problem Constraints on embedded surfaces in 4-manifolds.
method Constructs an invariant using Seiberg-Witten equations on families of embedded surfaces.
result Non-vanishing invariant provides new constraints on Seiberg-Witten vanishing.
New constraints on embedded spheres and projective planes in 4-manifolds from Seiberg-Witten theory.
problem Constraints on configurations of embedded spheres and real projective planes in 4-manifolds.
method Equivariant Seiberg-Witten invariants and gluing formula for relative Seiberg-Witten invariants.
result Existence of certain configurations of surfaces leads to 4-manifolds of non-simple type.
In the paper we formulate and derive the family blowup formula of family Seiberg-Witten invariants. The formula has been used in the enumerative application of counting singular curves on algebraic surfaces. We first give a topological derivation of the formula by using family index theorem. Then we define the algebrai…
Proves a specific knot is not smoothly slice using real invariants.
problem Determining the smooth sliceness of (2n,1)-cables of the figure-eight knot. method Used real Seiberg-Witten Frøyshov invariant and developed an equivariant lattice homotopy type.
result Proves the (2n,1)-cable of the figure-eight knot is not smoothly slice when n is odd. New estimates are derived concerning the behavior of self-dual hamonic 2-forms on a compact Riemannian 4-manifold with non-trivial Seiberg-Witten invariants. Applications include a vanishing theorem for certain Seiberg-Witten invariants on compact 4-manifolds of constant negative sectional curvature.
The monopole map defines an element in an equivariant stable cohomotopy group refining the Seiberg-Witten invariant. This first of two articles presents the details of the definition of the stable cohomotopy invariant and discusses its relation to the integer valued Seiberg-Witten invariant.
Deforms Seiberg-Witten solutions in 3D Lie group representations.
problem Non-compact moduli spaces of Seiberg-Witten solutions.
method Constructs Kuranishi models and discusses deformations.
result Fueter sections can be deformed to Seiberg-Witten solutions.
Let Y be a closed and oriented 3-manifold. We define different versions of unfolded Seiberg-Witten Floer spectra for Y. These invariants generalize Manolescu's Seiberg-Witten Floer spectrum for rational homology 3-spheres. We also compute some examples when Y is a Seifert space.
This is a survey article on the stable cohomotopy refinement of Seiberg-Witten invariants containing also new results, for example: - Stable cohomotopy groups describe path components of certain mapping spaces. - Relation of stable cohomotopy invariants to Seiberg-Witten invariants without restriction on Betti numbers.…
Proves additivity of Casson-Seiberg-Witten invariant for a specific 4-manifold.
problem Additivity of Casson-Seiberg-Witten invariant for integral homology S1imesS3. method Proves additivity under fiber sum along embedded curves and embedded tori.
result Establishes the 4-dimensional analogue of the Casson invariant additivity. We introduce the gradient flow of the Seiberg-Witten functional on a compact, orientable Riemannian 4-manifold and show the global existence of a unique smooth solution to the flow. The flow converges uniquely in C∞ up to gauge to a critical point of the Seiberg-Witten functional.
Using Furuta's idea of finite dimensional approximation in Seiberg-Witten theory, we refine Seiberg-Witten Floer homology to obtain an invariant of homology 3-spheres which lives in the S^1-equivariant graded suspension category. In particular, this gives a construction of Seiberg-Witten Floer homology that avoids the …
Paper extends Miyazawa's construction to more 4-manifolds, finding exotic involutions and embeddings.
problem Finding exotic involutions and embeddings in 4-manifolds.
method Real Seiberg-Witten theory.
result Many infinite families of exotic involutions and embeddings.
Formulae for Seiberg-Witten invariants derived from graph combinatorics.
problem Calculating Seiberg-Witten invariants for 3-manifolds.
method Combining Poincaré series and counting functions of a graph to derive surgery formulae.
result The periodic constant difference of invariants for surgeries on 3-manifolds.
Paper shows Seiberg-Witten invariants vanish for Davis hyperbolic 4-manifold.
problem Proving vanishing of Seiberg-Witten invariants for a specific 4-manifold.
method Using adjunction inequalities for embedded surfaces in the Davis hyperbolic 4-manifold.
result All Seiberg-Witten invariants vanish for the Davis hyperbolic 4-manifold.
New framework for Seiberg-Witten map on non-compact 4-manifolds.
problem L^2 self-dual forms on non-compact 4-manifolds cannot exhaust d^+ images.
method Functional analytic framework for Seiberg-Witten map.
result Construction of new functional analytic framework.
The Seiberg-Witten equations are defined on certain complex line bundles over smooth oriented four manifolds. When the base manifold is a complex Kahler surface, the Seiberg-Witten equations are essentially the Abelian vortex equations. Using known non-abelian generalizations of the vortex equations as a guide, we expl…
In this note, we give an exposition of the construction of Seiberg-Witten invariants.
Recent developments in Seiberg-Witten theory and relations with Complex Geometry.