A new Witten deformation modifies Dolbeault complex properties.
problem Modifying Dolbeault complex properties.
method Introducing a Witten-Novikov type perturbation of the Dolbeault complex.
result Heat invariants of lower order are zero.
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Holomorphic Witten instanton complexes on stratified pseudomanifolds with Kähler wedge metricsmath.DG
This paper extends Witten's holomorphic Morse inequalities to singular spaces.
problem Applying Witten's holomorphic Morse inequalities to singular spaces.
method Constructing Witten instanton complexes for Kähler Hamiltonian Morse functions on stratified pseudomanifolds.
result Extends Witten's holomorphic Morse inequalities to singular spaces.
Study Witten deformation on noncompact manifolds with bounded geometry.
problem Cohomology of Witten deformation on noncompact manifolds.
method Witten deformation, Agmon estimate, Witten's instanton complex.
result Cohomology of Witten deformation is isomorphic to Thom-Smale and relative cohomology.
Introduces Witten deformation and its applications in topology.
problem Analyzing and applying Witten deformation in topology.
method Deformation of Dirac operators and analytic proofs.
result Analytic proofs of Poincaré-Hopf index theorem, Real Morse inequalities, Thom-Smale complex quasi-isomorphism, and Atiyah vanishing theorem.
Recent developments in Seiberg-Witten theory and relations with Complex Geometry.
This expository article introduces the Kapustin-Witten equations to mathematicians. We discuss the connections between the Complex Yang-Mills equations and the Kapustin-Witten equations. In addition, we show the relation between the Kapustin-Witten equations, the moment map condition and the gradient Chern-Simons flow.…
Complex manifolds and their associative submanifolds are studied via Seiberg-Witten equations.
problem Understanding isotropic associative deformations in complex manifolds.
method Introducing complex manifolds and their complexification, applying Seiberg-Witten equations to associative submanifolds.
result Isotropic associative deformations in complex manifolds are described by Seiberg-Witten equations.
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…
We compute the Dijkgraaf-Witten invariants of surfaces in terms of projective representations of groups. As an application we prove that the complex Dijkgraaf-Witten invariants of surfaces of positive genus are positive integers.
The Homology Groups of math.DG
Computed homology groups of real Grassmann manifold.
problem Calculating homology groups of real Grassmann manifold.
method Used Witten complex approach.
result Homology groups of computed.
We construct a generalized Witten genus for spin manifolds, which takes values in level 1 modular forms with integral Fourier expansion on a class of spin manifolds called string manifolds. We also construct a mod 2 analogue of the Witten genus for dimensional spin manifolds. The Landweber-Stong type…
We provide a differential cocycle model for elliptic cohomology with complex coefficients and use analytic methods to construct a cocycle representative for the Witten class in this language. Our motivation stems from the conjectural connection between 2-dimensional field theories and elliptic cohomology originally due…
New structures derived from equivariant de Rham complex for -action.
problem Deriving new structures from equivariant de Rham complex.
method Applying Witten's deformation and homological perturbation to get new structures; extending and proving Fukaya's conjecture.
result Proved Fukaya's conjecture relating Witten's deformed equivariant de Rham complexes to new Morse theoretical complexes.
As is well-known, the Witten deformation of the De Rham complex computes the De Rham cohomology. In this paper we study the Witten deformation on a noncompact manifold and restrict it to differential forms which behave polynomially near infinity. Such polynomial differential forms naturally appear on manifolds with a c…
For a complex projective manifold Gromov-Witten invariants can be constructed either algebraically or symplectically. Using the versions of Gromov-Witten theory by Behrend and Fantechi on the algebraic side and by the author on the symplectic side, we prove that both points of view give the same results. A similar stat…
Localization reveals geometric and analytic properties of the Witten genus.
problem Understanding the Witten genus and its modularity properties.
method Equivariant localization techniques and geometric interpretations.
result Identifies the elliptic Bismut-Chern character as a candidate target.
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.
Study generalizes Seiberg-Witten equations on hyperKahler manifolds.
problem Generalizing Seiberg-Witten equations on hyperKahler manifolds.
method Construction of a non-linear Dirac operator and derivation of transformation formula for the Dirac operator under conformal changes.
result Generalized Seiberg-Witten equations can be expressed as a second order PDE in terms of almost-complex structure and a conformal factor.
A formula is given which computes the Seiberg-Witten invariant of a 3-orbifold from the invariant of the underlying manifold. As an application, we derive a formula for the Seiberg-Witten invariant of a non-Kähler complex surface, which was originally due to O. Biquard \cite{Biq} and S.R. Williams \cite{W} independentl…
In this note, we prove that the Witten genus of nonsingular string complete intersections in product of complex projective spaces vanishes. Our result generalizes a known result of Landweber and Stong (cf. [HBJ]).
The paper constructs instanton complexes on stratified pseudomanifolds.
problem Analyzing functions with non-isolated critical points on singular spaces.
method Constructing Witten instanton complexes and Hilbert complexes.
result Proves Morse inequalities for stratified pseudomanifolds.
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 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.
On a compact oriented four-manifold with an orientation preserving involution c, we count solutions of Seiberg-Witten equations, which are moreover symmetrical in relation to c, to construct "real" Seiberg-Witten invariants. Using Taubes' results, we prove that on a symplectic almost complex manifold with an antisymple…
We propose an explicit formula connecting Donaldson invariants and Seiberg-Witten invariants of a 4-manifold of simple type via Nekrasov's deformed partition function for the N=2 SUSY gauge theory with a single fundamental matter. This formula is derived from Mochizuki's formula, which makes sense and was proved when t…
Analytic realization of Thom-Smale complex for G-manifolds.
problem Realizing Thom-Smale complex for G-manifolds with Lie group action.
method Using G-invariant Witten instanton complex associated with a Morse-Bott function.
result Generalized Thom-Smale complex for G-manifolds including horizontal direction influence.
Given a smooth compact manifold with boundary, we show that the subcomplex of the deformed de Rham complex consisting of eigenspaces of small eigenvalues of the Witten Laplacian is canonically isomorphic to the Thom-Smale complex constructed by Laudenbach. Our proof is based on Bismut-Lebeau's analytic localization tec…
Introduces a Morse complex on symplectic manifolds using gradient flows and proves its cohomology is independent of metrics and Morse functions.
problem Cohomology of symplectic manifolds under different metrics and Morse functions.
method Symplectic Morse complex with gradient flows and Witten deformation.
result Cohomology of the complex is isomorphic to Tsai, Tseng, and Yau's cohomology and independent of metrics and Morse functions.
Witten deformation connects manifold spectra to Morse functions.
problem Understanding spectral properties of Riemannian manifolds.
method Rellich-Kato theorem applied to Witten deformation.
result Relates spectral package to Morse complex and harmonic oscillators.
Accurate asymptotic expressions are given for the exponentially small eigenvalues of Witten Laplacians acting on p-forms. The key ingredient, which replaces explicit formulas for global quasimodes in the case p = 0, is Barannikov's presentation of Morse theory.
Computes Seiberg-Witten invariants for Kähler families of 4-manifolds.
problem Computing invariants for families of Kähler 4-manifolds.
method Generalized Seiberg-Witten invariants for smooth families of 4-manifolds with Kähler structures.
result Computed invariants for specific Kähler families in terms of characteristic classes.
This paper explores adiabatic solutions of Haydys-Witten equations for knot homology.
problem Investigating instanton Floer homology and its relation to Khovanov homology.
method Analyzes decoupled Haydys-Witten equations and their equivalence to EBE solutions.
result Proposes an equivalence between adiabatic solutions of decoupled Haydys-Witten equations and non-vertical paths in EBE moduli space.
We treat the Witten operator on the de Rham complex with semiclassical heat kernel methods to derive the Poincaré-Hopf theorem and degenerate generalizations of it. Thereby, we see how the semiclassical asymptotics of the Witten heat kernel are related to approaches using the Thom form of Mathai and Quillen.
The paper studies obstructions to solutions of the Wess-Zumino-Witten equation and its generalizations.
problem Existence of solutions to the Wess-Zumino-Witten equation and its generalizations.
method Identification of algebraic obstructions and construction of approximate solutions using Monge-Ampère type equations.
result Approximate solutions to the generalized Wess-Zumino-Witten equation are shown to be the closest to true solutions when the latter do not exist.
Formula derived for Gromov-Witten invariants of smooth curves.
problem Calculating Gromov-Witten invariants for smooth curves in genus zero.
method Closed formula derived from solving the degree zero limit of the loop equation for the complex projective line.
result Closed formula for generating series of Gromov-Witten invariants in genus zero.
Survey on SYZ mirror symmetry using Morse theory.
problem Understanding SYZ mirror symmetry with quantum corrections.
method Using Witten-Morse theory and Fukaya's combinatorial structures.
result Explicit relation between geometric and combinatorial structures.
Machine learning classifies complex geometric patterns with high accuracy.
problem Classifying extension degree of dessins d'enfants over the rationals.
method Deep feed-forward neural network trained on machine learning.
result 0.92 accuracy in classification with 0.03 standard error.
The main result is a version of Morse inequalities for the minimum and maximum ideal boundary conditions of the de Rham complex on strata of compact Thom-Mather stratifications, endowed with adapted metrics. An adaptation of the analytic method of Witten is used in the proof, as well as certain perturbation of the harm…
Wedge product on deRham complex of a Riemannian manifold can be pulled back to via explicit homotopy, constructed using Green's operator, to give higher product structures. We prove Fukaya's conjecture which suggests that Witten deformation of these higher product structures have semiclassical limits as op…
New examples of hyperbolic 3-manifolds where Seiberg-Witten equations fail.
problem Finding hyperbolic 3-manifolds without irreducible Seiberg-Witten solutions.
method Combining hyperbolic geometry, upper bound for eigenvalues, Selberg trace formula, and precise numerical bounds.
result First examples of hyperbolic 3-manifolds where Seiberg-Witten equations do not admit irreducible solutions.
Let a compact connected orientable 4-manifold. We study the space of -structures of fixed fundamental class, as an infinite dimensional principal bundle on the manifold of riemannian metrics on . In order to study perturbations of the metric in Seiberg-Witten equations, we study the transversality of…
The paper presents a chain complex for 3-manifold covers, including surface bundles and surgeries.
problem Calculating linking forms and Dijkgraaf-Witten invariants for specific 3-manifold covers.
method Presentation of cellular chain complexes for universal covers of 3-manifolds in a specified class.
result Established a formula for linking forms and developed procedures for Dijkgraaf-Witten invariants.
We give a direct calculation of the curvature of the Hitchin connection, in geometric quantization on a symplectic manifold, using only differential geometric techniques. In particular, we establish that the curvature acts as a first-order operator on the quantum spaces. Projective flatness follows if the Kähler struct…
The main result of this paper asserts that if a Seifert fibered 4-manifold has nonzero Seiberg-Witten invariant, the homotopy class of regular fibers has infinite order. This is a nontrivial obstruction to smooth circle actions; as applications, we show how to destroy smooth circle actions on a 4-manifold by knot surge…
We prove that every Einstein metric on the unit ball B^4 of C^2, asymptotic to the Bergman metric, is equal to it up to a diffeomorphism. We need a solution of Seiberg--Witten equations in this infinite volume setting. Therefore, and more generally, if M^4 is a manifold with a CR-boundary at infinity, an adapted spinc-…
The abstract conjectures a formula for Higgs sheaves on complex surfaces.
problem Understanding moduli spaces of Higgs sheaves on surfaces with holomorphic 2-forms.
method Interpolates between K-theoretic Donaldson and Vafa-Witten invariants.
result Verlinde formula for moduli space of Higgs sheaves verified in examples.
On finite energy monopoles on math.DG
Finite energy solutions classified for Seiberg-Witten equations on complex plane and Riemann surface.
problem Classifying solutions to Seiberg-Witten equations with finite energy.
method Established a classification theorem for solutions on with finite analytic energy.
result Finite energy solutions correspond to polynomial maps from to .
We consider a claim mentioned in \cite{Witten} pp 187 about the relation between a symplectic chain complex with compatible bases and Reidemeister Torsion of it. This is an explanation of it.