We generalize Witten's conjectured formula relating Donaldson and Seiberg-Witten invariants to manifolds of non-simple type, via equivariant localization techniques. This approach does not use the theory of non-abelian monopoles, but works directly on the Donaldson-Witten and Seiberg-Witten moduli spaces. We give a for…
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Optimal lower bounds for eigenvalues of Dirac-Witten operator on certain submanifolds.
problem Estimating eigenvalues of the Dirac-Witten operator on specific submanifolds.
method Optimal lower bounds derived using intrinsic and extrinsic expressions.
result Limiting-cases of eigenvalues studied and optimal bounds obtained.
The Witten class is derived from equivariant cohomology of a conformal loop space.
problem Deriving the Witten class using equivariant cohomology.
method Applying equivariant localization formula to a conformal loop space.
result The Witten class is obtained from the conformal loop space.
The paper characterizes boundaries in Turaev-Viro TQFTs and Dijkgraaf-Witten theories.
problem Characterizing boundaries in Turaev-Viro TQFTs and Dijkgraaf-Witten theories.
method Identifying explicit boundary locality conditions and proving consistency with state sum models.
result Turaev-Viro and Dijkgraaf-Witten theories with boundary defects admit a state sum description.
The paper constructs local solutions concentrating near singular points of spinors.
problem Constructing solutions near singular points of spinors.
method Constructs local solutions parameterized by ε, concentrating near singular points.
result Local solutions concentrate in tubular neighborhoods of singular points, converging to original spinors after renormalization.
Researchers derived heat kernel expansions for non-compact spaces using Witten deformation.
problem Heat kernel expansions on non-compact spaces, especially for Witten Laplacians.
method Introduced parabolic distance and used it to derive asymptotic expansions.
result Derived an asymptotic expansion of trace of heat kernel for small-time .
By using the equivariant localization formula of toric varieties. We prove the vanishing of the Witten genus of some string complete intersections in smooth toric varieties.
Local tri-Hamiltonian structure for Ablowitz-Ladik hierarchy established.
problem Establishing a tri-Hamiltonian structure for the Ablowitz-Ladik hierarchy.
method Constructing a local tri-Hamiltonian structure and computing central invariants.
result Central invariants of one bi-Hamiltonian structure are 1/24, and dispersionless limit matches Frobenius manifold.
We use localization formulas in the theory of equivariant cohomology to rederive the wall crossing formulas of Li-Liu and Okonek-Teleman for Seiberg-Witten invariants.
Equivariant localization techniques give a rigorous interpretation of the Witten genus as an integral over the double loop space. This provides a geometric explanation for its modularity properties. It also reveals an interplay between the geometry of double loop spaces and complex analytic elliptic cohomology. In part…
Two lectures on local riemannian geometry, $Spin^{\C}$ - structures and Seiberg - Witten equationmath.DG
In the present text we discuss basic aspects of the Seiberg - Witten theory mainly focusing the attantion on some geometrical details which could make the introduction to the subject more illustrative. At the same time we list there natural problems arise in this framework mostly interesting to the author. This text co…
We show that moduli spaces of stable maps admits virtual orbifold structure. The symplectic version of virtual localization formula is obtained.
Let be a compact Riemannian manifold endowed with an isometric action of a compact Lie group. The method of the Witten deformation is used to compute the virtual representation-valued equivariant index of a transversally elliptic, first order differential operator on . The multiplicities of irreducible represent…
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.
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.
Study of 5D SYM theory on toric surfaces yields refined Vafa-Witten invariants.
problem Computing partition functions in 5D SYM theory on toric surfaces.
method Supersymmetric localization, Nekrasov partition functions, moduli space of sheaves.
result Found correspondence between poles and torus-fixed points in moduli space, leading to refined Vafa-Witten invariants.
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…
Alternative proof and description of orientations for instanton moduli spaces.
problem Describing the relation between different orientations of -instanton moduli spaces.
method Using Witten localization of the index of a Dirac type operator.
result Alternative construction and proof of orientability of instanton moduli spaces.
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.
Generalizing local Gromov-Witten theory, in this paper we define a local version of symplectic field theory. When the symplectic manifold with cylindrical ends is four-dimensional and the underlying simple curve is regular by automatic transversality, we establish a transversality result for all its multiple covers and…
Limiting measures and energy growth for sequences of solutions to Taubes's Seiberg-Witten equationsmath.AP
Study sequences of solutions to Taubes's Seiberg-Witten equations with unbounded energy.
problem Behavior of solutions with unbounded energy and their limiting nodal sets.
method Novel maximum principle for unbounded energy solutions, connection to vector field dynamics.
result Limiting nodal set converges to invariant set of vector field for slow energy growth.
We present an approach to Gromov-Witten invariants that works on arbitrary (closed) symplectic manifolds. We avoid genericity arguments and take into account singular curves in the very formulation. The method is by first endowing mapping spaces from (prestable) algebraic curves into the symplectic manifold with the st…
New model for Witten-Reshetikhin-Turaev invariants using Lagrangian intersections.
problem Categorification of Witten-Reshetikhin-Turaev invariants for 3-manifolds.
method Constructing a topological model from quantum group U_q(sl(2)) using Lagrangian intersections in configuration spaces.
result Witten-Reshetikhin-Turaev invariants are encoded by intersections of Lagrangian submanifolds in a fixed configuration space.
The Weinstein conjecture is extended to a new class of manifolds.
problem Existence of Reeb 2-curves in locally conformally symplectic manifolds.
method Extended Gromov-Witten theory and elliptic curve counts.
result Partial verification of the conjecture in higher dimensions.
In this article, we study the twisting procedure of orbifold cohomology. We introduce local system and construct twisted orbifold cohomology. Then, we generalize Vafa-Witten's notion of discrete torsion to general orbifold and examine its relation to local system. Finally, some examples are computed.
We conjecture a formula for the generating function of virtual -genera of moduli spaces of rank 2 sheaves on arbitrary surfaces with holomorphic 2-form. Specializing the conjecture to minimal surfaces of general type and to virtual Euler characteristics, we recover (part of) a formula of C. Vafa and E. Witten. The…
We investigate the perturbative aspects of Rozansky-Witten's 3d -model using Costello's approach to the Batalin-Vilkovisky (BV) formalism. We show that the BV quantization (in Costello's sense) of the model, which produces a perturbative quantum field theory, can be obtained via the configuration space method of reg…
Paper uses algebraic methods to prove an equivariant Poincaré-Hopf theorem.
problem Equivariant index theory on manifolds.
method Localization algebras and Witten deformation techniques in K-homology.
result Established an equivariant version of the Poincaré-Hopf theorem.
In the present paper, we establish a gluing construction for the Nahm pole solutions to the Kapustin-Witten equations over manifolds with boundaries and cylindrical ends. Given two Nahm pole solutions with some convergence assumptions on the cylindrical ends, we prove that there exists an obstruction class for gluing t…
This article studies the nonabelian localization results of Beasley and Witten, and considers the analogue of these results when the gauge group is U(1). It compares these results with results of Manoliu on abelian Chern-Simons theory, showing that the dependence on the coupling constant is the same.
The paper proves properties of a specific Seiberg-Witten equation over 3-manifolds.
problem Analyzing the -Seiberg-Witten equation over 3-manifolds.
method Analyzes the -Seiberg-Witten equation over closed hyperbolic 3-manifolds and .
result The canonical irreducible solution is infinitesimally rigid under certain conditions.
We characteristize those Einstein four manifolds which are locally symmetric spaces of noncompact type. Namely they are four manifolds which admit solutions to the (non-Abelian) Seiberg Witten equations and satisty certain characterisitc number equality.
We propose localization techniques for computing Gromov-Witten invariants of maps from Riemann surfaces with boundaries into a Calabi-Yau, with the boundaries mapped to a Lagrangian submanifold. The computations can be expressed in terms of Gromov-Witten invariants of one-pointed maps. In genus zero, an equivariant ver…
Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov - Witten invariantsmath.DG
We present a project of classification of a certain class of bihamiltonian 1+1 PDEs depending on a small parameter. Our aim is to embed the theory of Gromov - Witten invariants of all genera into the theory of integrable systems. The project is focused at describing normal forms of the PDEs and their local bihamiltonia…
New results on localization of exotic diffeomorphisms in 4-manifolds.
problem Understanding when groups of exotic diffeomorphisms can be localized to smaller embedded submanifolds.
method Analyzing families Seiberg-Witten theory, Dehn twists, and properties of Seifert fibered homology spheres.
result Exotic diffeomorphisms cannot be localized to topologically embedded rational homology balls or homology spheres.
Study invariants of elliptic curves in LCS manifolds, leading to new phenomena in Riemann-Finsler geometry.
problem Defining and studying invariants of elliptic curves in locally conformally symplectic manifolds.
method Using -holomorphic curves and Gromov-Witten theory to define and study invariants.
result Found new phenomena in Riemann-Finsler geometry and an analogue of the Weinstein conjecture.
We initiate here the study of Gromov-Witten theory of locally conformally symplectic manifolds or $\lcs$ manifolds, $\lcsm$'s for short, which are a natural generalization of both contact and symplectic manifolds. We find that the main new phenomenon (relative to the symplectic case) is the potential existence of holom…
The paper applies -localization to symplectic cohomology.
problem Equivariant symplectic cohomology relations.
method Localisation by pseudocycles and moduli space lifting.
result Relations between equivariant symplectic classes and Gromov-Witten invariants.
We establish a product formula for Gromov-Witten invariants for closed, connected, relatively semi-positive Hamiltonian fibrations over any symplectic base. Furthermore, we show that the fibration projection induces a locally trivial (orbi-)fibration map from the moduli space of pseudo-holomorphic maps with marked poin…
This is my Habilitation a Diriger des Recherches (French thing). In this paper I summarize the work I have done to realize the program of Witten called -non abelian localization-. This work deals first with problems of localization in equivariant cohomology. The second part of this paper concerns the Guillemin-Sternber…
The paper studies heat kernel asymptotics and proves Morse inequalities.
problem Analyzing the asymptotic behavior of heat kernels near critical points.
method Localization and scaling techniques in semi-classical analysis.
result The heat kernel near critical points is approximated by harmonic oscillator kernels, leading to Morse inequalities.
Study Lefschetz fibrations with 4D fibers using Seiberg-Witten theory.
problem Constraints on the topology of Lefschetz fibrations with 4D fibers.
method Using the family Bauer-Furuta invariant and framed bordism class of 1D Seiberg-Witten moduli spaces.
result New obstructions to the smooth isotopy of compositions of Dehn twists.
We introduce a variant of the Seiberg-Witten equations, Pin^-(2)-monopole equations, and give its applications to intersection forms with local coefficients of 4-manifolds. The first application is an analogue of Froyshov's results on 4-manifolds which have definite forms with local coefficients. The second is a local …
The paper analyzes heat trace asymptotics for de Rham and Dolbeault complexes in both real and complex settings.
problem Examining heat trace asymptotics for de Rham and Dolbeault complexes in different geometric settings.
method Analyzing the derived heat trace asymptotics for generalized Witten perturbations in both real and complex settings.
result The integral of the local density for the derived heat trace asymptotics is related to the Euler characteristic and characteristic numbers of the tangent and twisting vector bundles.
Formula calculates Riemann-Roch number for singular symplectic quotients.
problem Computing Riemann-Roch number for singular symplectic quotients.
method Complete singular stationary phase expansion of Witten integral.
result New explicit local invariant of singularities in symplectic quotients.
In this paper, we first prove a local family version of the Atiyah-Bott-Segal-Singer Lefschetz fixed point formula, then we extend the famous Witten's rigidity Theorems to the family case. Several family vanishing theorems for elliptic genera are also proved.
Calculates spectral flow bounds for reducible solutions to Vafa-Witten equations.
problem Bounding spectral flow between diverging reducible solutions.
method Localization and excision techniques to calculate spectral flow.
result Bounds on spectral flow are given for reducible solutions.
This paper constructs and studies the Gromov-Witten invariants and their properties for noncompact geometrically bounded symplectic manifolds. Two localization formulas for GW-invariants are also proposed and proved. As applications we get solutions of the generalized string equation and dilation equation and their var…