Lecture notes for talk by the first author at the International Georgia Topology Conference 2001 (Athens, Georgia) and the Mathematische Arbeitstagung 2001 (Bonn, Germany). 11 pages
发表信息：
Topology and Geometry of Manifolds (Athens, GA 2001), Proceedings of Symposia in Pure Mathematics 71, American Mathematical Society, Providence, RI, 2003, pp. 237-248
This article is based on a lecture by the first author at the International Georgia Topology Conference 2001 (Athens, Georgia) and the Mathematische Arbeitstagung 2001 (Bonn, Germany). We sketch a proof of Witten's formula relating the Donaldson and Seiberg-Witten series modulo powers of degree c+2, with c = -{1/4}(7 chi + 11 sigma), for four-manifolds obeying some mild conditions, where chi and sigma are their Euler characteristic and signature. We use the moduli space of SO(3) monopoles as a cobordism between a link of the Donaldson moduli space of anti-self-dual SO(3) connections and links of the moduli spaces of Seiberg-Witten monopoles. Gluing techniques allow us to compute contributions from Seiberg-Witten moduli spaces lying in the first (or `one-bubble') level of the Uhlenbeck compactification of the moduli space of SO(3) monopoles.
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 formal derivation of Witten's conjecture and its generalization, making use of an infinite dimensional version of the abelian localization theorem.
Using Morse-Bott techniques adapted to the gauge-theoretic setting, we show that the limiting boundary values of the space of finite energy monopoles on a connected 3-manifold with at least two cylindrical ends provides an immersed Lagrangian submanifold of the vortex moduli space at infinity. By studying the signed intersections of such Lagrangians, we supply the analytic details of Donaldson's TQFT construction of the Seiberg-Witten invariants of a closed 3-manifold.
We prove Witten's formula relating the Donaldson and Seiberg-Witten series modulo powers of degree $c+2$, with $c=-{1/4}(7χ+11σ)$, for four-manifolds obeying some mild conditions, where $χ$ and $σ$ are their Euler characteristic and signature. We use the moduli space of SO(3) monopoles as a cobordism between a link of the Donaldson moduli space of anti-self-dual SO(3) connections and links of the moduli spaces of Seiberg-Witten monopoles. Gluing techniques allow us to compute contributions from Seiberg-Witten moduli spaces lying in the first (or `one-bubble') level of the Uhlenbeck compactific…
This article provides a summary of arXiv:1701.08899 and arXiv:1701.08902 where the authors studied the enumerative geometry of nested Hilbert schemes of points and curves on algebraic surfaces and their connections to threefold theories, and in particular relevant Donaldson-Thomas, Vafa-Witten and Seiberg-Witten theories.
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 the 4-manifold is complex projective. Assuming our formula is true for a 4-manifold of simple type, we prove Witten's conjecture and sum rules for Seiberg-Witten invariants (superconformal simple type condition), conjectured by Mariño, Moore and Peradze.
This paper gives a detailed construction of Seiberg-Witten-Floer homology for a closed oriented 3-manifold with a non-torsion $\spinc$ structure. Gluing formulae for certain 4-dimensional manifolds splitting along an embedded 3-manifold are obtained.
We use Donaldson invariants of regular surfaces with p_g >0 to make quantitative statements about modulispaces of stable rank 2 sheaves. We give two examples: a quantitative existence theorem for stable bundles, and a computation of the rank of the canonical holomorphic two forms on the moduli space. The results are in some sense dual to the Donaldson and O'Grady non vanishing theorems because they use the Donaldson series of the surface as _input_. Results in purely algebraic geometric terms can be obtained by using the explicit form of the Donaldson series of the surface. The Donaldson serie…
We prove an analogue of the Kotschick-Morgan conjecture in the context of SO(3) monopoles, obtaining a formula relating the Donaldson and Seiberg-Witten invariants of smooth four-manifolds using the SO(3)-monopole cobordism. The main technical difficulty in the SO(3)-monopole program relating the Seiberg-Witten and Donaldson invariants has been to compute intersection pairings on links of strata of reducible SO(3) monopoles, namely the moduli spaces of Seiberg-Witten monopoles lying in lower-level strata of the Uhlenbeck compactification of the moduli space of SO(3) monopoles [arXiv:dg-ga/9710…
We study the path integral of a twisted $N=2$ supersymmetric Yang-Mills theory coupled with hypermultiplet having the bare mass. We explicitly compute the topological correlation functions for the $SU(2)$ theory on a compact oriented simply connected simple type Riemann manifold with $b_2^+ \geq 3$. As the corollaries, we determine the topological correlation functions of the theory without the bare mass and those of the theory without coupling to the hypermultiplet. This includes a concrete field theoretic proof of the relation between the Donaldson and the Seiberg-Witten invariants.
I describe a relation (mostly conjectural) between the Seiberg-Witten monopoles, Fueter sections, and G2 instantons. In the last part of this article I gathered some open questions connected with this relation.
It is shown that the non-homogeneous Dirichlet and Neuman problems for the $2^{nd}$-order Seiberg-Witten equation admit a regular solution once the $\mathcal{H}$-condition (described in the article) is satisfied. The approach consist in applying the elliptic techniques to the variational setting of the Seiberg-Witten equation.
The $2^{nd}$ variation formula of the Seiberg-Witten functional is obtained in order to estimate the Morse index of redutible solutions $(A,0)$. It is shown that their Morse index is given by the dimension of the largest negative eigenspace of the operator $\triangle_{A} +\frac{k_{g}}{4}$, hence it is finite.
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.
We prove a gluing formula for the families Seiberg-Witten invariants of families of $4$-manifolds obtained by fibrewise connected sum. Our formula expresses the families Seiberg-Witten invariants of such a connected sum family in terms of the ordinary Seiberg-Witten invariants of one of the summands, under certain assumptions on the families. We construct some variants of the families Seiberg-Witten invariants and prove the gluing formula also for these variants. One variant incorporates a twist of the families moduli space using the charge conjugation symmetry of the Seiberg-Witten equations.…
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.
We construct Bott-type and equivariant Seiberg-Witten Floer homology and cohomology for 3-manifolds, in particular rational homology spheres, and prove their diffeomorphism invariance. This paper is a revised version of math.DG/9701010. Some typos are removed.
We study the problem of counting instantons with coassociative boundary condition in (almost) G_(2)-manifolds. This is analog to the open Gromov-Witten theory for counting holomorphic curves with Lagrangian boundary condition in Calabi-Yau manifolds. We explain its relationship with the Seiberg-Witten invariants for coassociative submanifolds.
Kontsevich-Soibelman (2017) reformulated Eynard-Orantin topological recursion (2007) in terms of Airy structure which provides some geometrical insights into the relationship between the moduli space of curves and topological recursion. In this work, we investigate the analytical approach to this relationship using the Seiberg-Witten family of curves as the main example. In particular, we are going to show that the formula computing the Hitchin systems' Special Kahler's prepotential from the genus zero part of topological recursion as obtained by Baraglia-Huang (2017) can be generalized for a …
We prove that the Seiberg-Witten invariants of a rational homology sphere are determined in a very explicit fashion by the Casson-Walker invariant and the Reidemeister torsion
We establish a surgery formula for 3-dimensional Seiberg-Witten monopoles under (+1) Dehn surgery on a knot in a homology 3-sphere. (substantial revision)
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 construct stable homotopy versions of the relative Seiberg-Witten invariants for certain four-manifolds with boundary.
We give a surgery formula for the torsions and Seiberg-Witten invariants associated with $Spin^c$-structures on 3-manifolds. We use the technique of Reidemeister-type torsions and their refinements.
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.
For two nearby disjoint coassociative submanifolds C and C' in a G_2-manifold, we construct thin instantons with boundaries lying on C and C' from regular J-holomorphic curves in C. We explain their relationship with the Seiberg-Witten invariants for C.
We show how the families Seiberg-Witten invariants of a family of smooth $4$-manifolds can be recovered from the families Bauer-Furuta invariant via a cohomological formula. We use this formula to deduce several properties of the families Seiberg-Witten invariants. We give a formula for the Steenrod squares of the families Seiberg-Witten invariants leading to a series of mod $2$ relations between these invariants and the Chern classes of the spin$^c$ index bundle of the family. As a result we discover a new aspect of the ordinary Seiberg-Witten invariants of a $4$-manifold $X$: they obstruct t…
We show the existence of a symplectic structure on the moduli space of the Seiberg-Witten equations on $Σ\times Σ$ where $Σ$ is a compact oriented Riemann surface. To prequantize the moduli space, we construct a Quillen-type determinant line bundle on it and show its curvature is proportional to the symplectic form.
For a rational homology 3-sphere $Y$ with a $\spinc$ structure $\s$, we show that simple algebraic manipulations of our construction of equivariant Seiberg-Witten Floer homology lead to a collection of variants which are topological invariants. We establish exact sequences relating them, we show that they satisfy a duality under orientation reversal, and we explain their relation to ou previous construction of equivariant Seiberg-Witten Floer (co)homologies. We conjecture the equivalence of these versions of equivariant Seiberg-Witten Floer homology with the Heegaard Floer invariants introduce…
These notes provide an introductory exposition of the Seiberg-Witten gauge theory. They collect the material presented in a series of seminars given by the author at the University of Milano.
We construct equivariant and Bott-type Seiberg-Witten Floer homology and cohomology for 3-manifolds, in particular rational homology spheres, and prove their diffeomorphism invariance. We present several versions of the equivariant theory: the singular version, the de Rham version and the Cartan version, with the first playing the most important role. These versions are shown to be equivalent to each other. A few typos are removed.
We prove a gluing formula for Seiberg--Witten invariants which describes in particular the behaviour of the invariant under blow-up and rational blow-down.
We prove that, under a simple condition on the cohomology ring, every closed 4-manifold has mod 2 Seiberg-Witten simple type. This result shows that there exists a large class of topological 4-manifolds such that all smooth structures have mod 2 simple type, and yet some have non-vanishing (mod 2) Seiberg-Witten invariants. As corollaries, we obtain adjunction inequalities and show that, under a mild topological condition, every geometrically simply connected closed 4-manifold has the vanishing mod 2 Seiberg-Witten invariant for at least one orientation.
This is the second part of the proof of the exact traiangles in Seiberg-Witten Floer theory. We analyse the splitting and gluing of flow lines of the Chern-Simons-Dirac functional when the underlying three-manifold splits along a torus. (two corrections added)
We introduce an invariant of tuples of commutative diffeomorphisms on a 4-manifold using families of Seiberg-Witten equations. This is a generalization of Ruberman's invariant of diffeomorphisms defined using 1-parameter families of Seiberg-Witten equations. Our invariant yields an application to the homotopy groups of the space of positive scalar curvature metrics on a 4-manifold. We also study the extension problem for families of 4-manifolds using our invariant.
In this paper, we investigate the Seiberg-Witten gauge theory for Seifert fibered spaces. The monopoles over these three-manifolds, for a particular choice of metric and perturbation, are completely described. Gradient flow lines between monopoles are identified with holomorphic data on an associated ruled surface, and a dimension formula for such flows is calculated.
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.
We show that for manifolds of dimension $m\geq5$, the flow of a Seiberg-Witten-type functional admits a global smooth solution on $M \times [0,\infty)$.
In this note, we present a proof of Donaldson's Diagonalization Theorem via an abelian gauge-theoretic variant of the Seiberg-Witten equations for multiple spinors. Like the other proof of Donaldson's theorem using the standard Seiberg-Witten theory, Elkies' theorem also plays a key role in our argument.
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 $C^\infty$ up to gauge to a critical point of the Seiberg-Witten functional.
We extend the vanishing theorem for the Seiberg-Witten invariants of a manifold with positive scalar curvature to the case when the curvature is allowed to be negative on a set of small volume. (The precise curvature bounds are described in the paper.) The idea is to combine the method of `semigroup domination' with the Weitzenbock formula. The same curvature hypothesis implies the vanishing of the Seiberg-Witten invariant of any finite covering space. We also show that, in high dimensions, a spin manifold with mostly positive scalar curvature in fact admits a metric of positive scalar curvatu…
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 the geometric or analytic data involved in their definition. In the first main result of this thesis, we completely determine the dependence of the Seiberg-Witten invariants on the data involved in their definition. In particular, we show that even for the simplest manifold, the…
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.
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.
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 relative Seiberg-Witten invariants are applied to certain homology spheres bounding Stein surfaces.
In this paper, we study the Seiberg-Witten equations on the product R x Y, where Y is a compact 3-manifold with boundary. Following the approach of Salamon and Wehrheim in the instanton case, we impose Lagrangian boundary conditions for the Seiberg- Witten equations. The resulting equations we obtain constitute a nonlinear, nonlocal boundary value problem. We establish regularity, compactness, and Fredholm properties for the Seiberg- Witten equations supplied with Lagrangian boundary conditions arising from the monopole spaces studied in [20]. This work therefore serves as an analytic foundati…
We show that solutions of the Seiberg-Witten equations lead to non-trivial lower bounds for the L2-norm of the Weyl curvature of a compact Riemannian 4-manifold. These estimates are then used to derive new obstructions to the existence of Einstein metrics. These results considerably refine those previously obtained using scalar-curvature estimates alone.
Let $M$ be a closed oriented $4$-manifold admitting a rank-$2$ oriented foliation with a metric of leafwise positive scalar curvature. If $b^+>1$, then we will show that the Seiberg-Witten invariant vanishes for all \spinc structures.
We study the Seiberg-Witten equations on surfaces of logarithmic general type. First, we show how to construct irreducible solutions of the Seiberg-Witten equations for any metric which is "asymptotic" to a Poincaré type metric at infinity. Then we compute a lower bound for the $L^{2}$-norm of scalar curvature on these spaces and give non-existence results for Einstein metrics on blow-ups.
I show that flat PSL(2;R)-connections on three-manifolds satisfying certain 'stability condition' can be interpreted as solutions of the Seiberg-Witten equations with two spinors. This is used to construct explicit examples of the Seiberg-Witten moduli spaces. Also, I show that in this setting blow up sets satisfy certain non-trivial topological restrictions.
We prove an abstract compactness theorem for a family of generalized Seiberg-Witten equations in dimension three. This result recovers Taubes' compactness theorem for stable flat $\mathbf{P}\mathrm{SL}_2(\mathbf{C})$-connections as well as the compactness theorem for Seiberg-Witten equations with multiple spinors. Furthermore, this result implies a compactness theorem for the ADHM$_{1,2}$ Seiberg-Witten equation, which partially verifies a conjecture by Doan and Walpuski.
We construct Bott-type and stable equivariant Seiberg-Witten Floer homology and cohomology for rational homology spheres, and prove their diffeomorphism invariance.
The Euler-Lagrange equations for the variational approach to the Seiberg-Witten equations always admit reducible solutions. In this context, the existence of unstable reducible solutions is achieved by assuming the existence of a parallel spinor or the negativeness of a Perelman-Yamabe type of invariant defined for a $\spinc$-structure.
