Seiberg-Witten Invariants-An Expository Account

Seiberg-Witten-Floer Homology and Gluing Formulae

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.

2002-06-06math-phmath.DGmath.GT

Intersection theory of coassociative submanifolds in G_(2)-manifolds and Seiberg-Witten invariants

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.

2004-01-29math-phmath.DGmath.MP

Seiberg-Witten Theory and Topological Recursion

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 …

2020-12-02math-phmath.AGmath.DG

Thin instantons in G_2-manifolds and Seiberg-Witten invariants

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.

2013-03-27math-phmath.DGmath.MP

Quillen-type bundle and geometric prequantization on moduli space of the Seiberg-Witten equations on product of Riemann surfaces

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.

2022-03-30math-phmath.DGmath.MP

Notes on Seiberg-Witten Gauge Theory

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.

1995-09-20dg-gahep-thmath.DG

A proof via the Seiberg-Witten moduli space of Donaldson's theorem on smooth 4-manifolds with definite intersection forms

A proof via the Seiberg-Witten moduli space of Donaldson's theorem on smooth 4-manifolds with definite intersection forms.

2012-07-26math-phmath.APmath.AT

Recent Developments in Seiberg-Witten Theory and Complex Geometry

Recent developments in Seiberg-Witten theory and relations with Complex Geometry.

1997-03-04alg-geomdg-gahep-th

Orientability and real Seiberg-Witten invariants

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.

2009-05-03math.AGmath.DG

Hilbert Schemes, Donaldson-Thomas Theory, Vafa-Witten and Seiberg Witten theories

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.

2019-11-06hep-thmath.AGmath.DG

Seiberg-Witten Flow in Higher Dimensions

We show that for manifolds of dimension $m\geq5$, the flow of a Seiberg-Witten-type functional admits a global smooth solution on $M \times [0,\infty)$.

2010-07-31math.DG

The perturbation of the Seiberg-Witten equations revisited

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.

2016-06-21math.DGmath.GT

Global Existence for the Seiberg-Witten Flow

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.

2010-08-19math.DG

Equivariant and Bott-type Seiberg-Witten Floer Homology: Part I

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.

1999-01-26hep-thmath.AGmath.DG

Equivariant Cohomology and Wall Crossing Formulas in Seiberg-Witten Theory

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.

1998-04-29math.DG

The Seiberg-Witten theory of homology 3-spheres

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…

2000-02-01dg-gamath.DG

A gluing formula for families Seiberg-Witten invariants

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.…

2018-12-31math.DGmath.GT

Hyperbolic Manifolds, Harmonic Forms, and Seiberg-Witten Invariants

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.

2001-11-26math.DGmath.GT

A stable cohomotopy refinement of Seiberg-Witten invariants: I

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.

2002-04-29math.ATmath.DG

Seiberg-Witten-Floer Theory for Homology 3-Spheres

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.

1996-07-29dg-gamath.DG

The Seiberg-Witten Equations on Manifolds with Boundary II: Lagrangian Boundary Conditions for a Floer Theory

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…

2016-06-01hep-thmath.APmath.DG

Seiberg-Witten equations on tubes

We establish the adiabatic dissapearance of Seiberg-Witten tunnelings on tubes R x N, where N is an S^1 fibration over a Riemann surface.

1997-01-10dg-gamath.DG

Donaldson = Seiberg-Witten from Mochizuki's formula and instanton counting

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.

2010-01-27hep-thmath.AGmath.DG

Seiberg-Witten invariants of rational homology spheres

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

2001-03-04math.DGmath.GT

Seiberg-Witten equation on a manifold with rank $2$-foliation

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.

2020-03-07math.DG

Seiberg-Witten equations on surfaces of logarithmic general type

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.

2011-12-03math.DG

Seiberg-Witten monopoles and flat PSL(2;R)-connections

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.

2022-08-26math.DG

Mod p equality theorem for Seiberg-Witten invariants under Z_p-actions

When a cyclic group G of prime order acts on a 4-manifold X, we prove a formula which relates the Seiberg-Witten invariants of X to those of X/G.

2009-05-19math.DG

The geometric triangle for 3-dimensional Seiberg-Witten monopoles

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)

2002-03-21math.DGmath.GT

Periodic Floer pro-spectra from the Seiberg-Witten equations

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.

2014-02-03math.DGmath.GT

On the compactness problem for a family of generalized Seiberg-Witten equations in dimension three

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.

2021-03-15math.DG

Surgery formula for torsions and Seiberg-Witten invariants of 3-manifolds

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.

2001-01-12math.DGmath.GT

The Seiberg-Witten invariants and 4-manifolds with essential tori

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.

2001-05-25math.DGmath.GT

A non-abelian Seiberg-Witten invariant for integral homology 3-spheres

A new diffeomorphism invariant of integral homology 3-spheres is defined using a non-abelian 'quaternionic' version of the Seiberg-Witten equations.

2004-01-04math.DG

Localisation of the Donaldson's invariants along Seiberg-Witten classes

This article is a first step in establishing a link between the Donaldson polynomials and Seiberg-Witten invariants of a smooth 4-manifold.

1995-07-24alg-geomdg-gamath.AG

On the Bauer-Furuta and Seiberg-Witten invariants of families of $4$-manifolds

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…

2021-12-20math.DGmath.GT

A symplectic proof of Seiberg-Witten blow-up formula

withdrawn by authors

2009-01-07math.AGmath.DGmath.SG

Instability of Reducible Critical Points of the Seiberg-Witten Functional

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.

2015-01-03math.DG

Equivariant and Bott-type Seiberg-Witten Floer Homology: Part II

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.

1999-01-26hep-thmath.AGmath.DG

Variants of equivariant Seiberg-Witten Floer homology

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…

2002-11-15math.DGmath.GT

The Family Blowup Formula of the Family Seiberg-Witten Invariants

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 algebraic (family) Seiberg-Witten invariants for algebraic surfaces and then give an algebraic derivation of the family blowup formula for the algebraic family Seiberg-Witten invariants.

2003-05-20math.AGmath.DG

Constraints for Seiberg-Witten basic classes of glued manifolds

We use rudiments of the Seiberg-Witten gluing theory for trivial circle bundles over a Riemann surface to relate de Seiberg-Witten basic classes of two $4$-manifolds containing Riemann surfaces of the same genus and self-intersection zero with those of the $4$-manifold resulting as a connected sum along the surface. We study examples in which this is enough to describe completely the basic classes.

1995-11-23dg-gamath.DG

Bott-type and equivariant Seiberg-Witten Floer homology I

We construct Bott-type and stable equivariant Seiberg-Witten Floer homology and cohomology for rational homology spheres, and prove their diffeomorphism invariance.

1997-01-29alg-geomdg-gamath.AG

Notes on of Seiberg-Witten map on manifold with flat scalar curvature

In this paper, we focus on the moduli space of Seiberg-Witten equation on non-compact manifold with periodic end. Suppose that the scalar curvature on the periodic end is identically zero and the topological conditions: the first de-Rham cohomology and the self-dual cohomology restricting on the periodic end vanish. Then, we will show that the moduli space of the perturbed Seiberg-Witten equation is compact.

2022-09-19math.DG

Seiberg-Witten invariants and real curves

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 antisymplectic and antiholomorphic involution, this invariants are not all trivial, and that the canonical bundle is represented by a real holomorphic curve.

2004-04-30math.CVmath.DGmath.SG

Exact triangles in Seiberg-Witten Floer theory. Part III: proof of exactness

This is the third part of the work on the exact triangles. We construct chain homomorphisms and show exactness of the resulting sequence.

2000-09-15math.DG

Edges, Orbifolds, and Seiberg-Witten Theory

Seiberg-Witten theory is used to obtain new obstructions to the existence of Einstein metrics on 4-manifolds with conical singularities along an embedded surface. In the present article, the cone angle is required to be of the form 2(pi)/p, where p is a positive integer, but we conjecture that similar results will also hold in greater generality.

2013-05-08math.DG

Seiberg-Witten invariants for manifolds with $b_+=1$, and the universal wall crossing formula

In this paper we describe the Seiberg-Witten invariants, which have been introduced by Witten, for manifolds with $b_+=1$. In this case the invariants depend on a chamber structure, and there exists a universal wall crossing formula. For every Kähler surface with $p_g=0$ and $q$=0, these invariants are non-trivial for all $Spin^c(4)$-structures of non-negative index.

1996-04-19alg-geomdg-gahep-th

A generalized blow-up formula for Seiberg--Witten invariants

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.

2006-04-19math.DGmath.GT

Yamabe Constants and the Perturbed Seiberg-Witten Equations

Among all conformal classes of Riemannian metrics on ${\Bbb CP}_2$, that of the Fubini-Study metric is shown to have the largest Yamabe constant. The proof, which involves perturbations of the Seiberg-Witten equations, also yields new results on the total scalar curvature of almost-Kähler 4-manifolds.

1996-05-29dg-gamath.DG

PU(2) monopoles. II: Top-level Seiberg-Witten moduli spaces and Witten's conjecture in low degrees

In this article we complete the proof---for a broad class of four-manifolds---of Witten's conjecture that the Donaldson and Seiberg-Witten series coincide, at least through terms of degree less than or equal to c-2, where c is a linear combination of the Euler characteristic and signature of the four-manifold. This article is a revision of sections 4--7 of an earlier version, while a revision of sections 1--3 of that earlier version now appear in a separate companion article (math.DG/0007190). Here, we use our computations of Chern classes for the virtual normal bundles for the Seiberg-Witten …

2001-02-09alg-geomdg-gahep-th

The simple type conjecture for mod 2 Seiberg-Witten invariants

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.

2021-03-29math.DGmath.GT

Seiberg-Witten invariants for manifolds with $b_+=1$

In this paper we describe the Seiberg-Witten invariants, which have been introduced by Witten, for manifolds with $b_+=1$. In this case the invariants depend on a chamber structure, and there exists a universal wall crossing formula. We take into account the contribution of the 1-homology of the base-manifold. For every Kähler surface with $p_g=0$ and $q$=0, these invariants are non-trivial for all $Spin^c(4)$-structures of non-negative index.

1996-06-17alg-geomdg-gahep-th

Mod 2 Seiberg-Witten invariants of homology tori

We show that the mod 2 Seiberg-Witten invariant can be determined for a spin manifold X which has the same homology groups as the 4-torus. The value depends on the structure of the cohomology ring of X, and in particular on the 4-fold cup product on H^1(X). We also consider some examples of homology tori.

2000-03-15math.DG