# Top PDF The Seiberg-Witten equations on 3-manifolds with boundary

### The Seiberg-Witten equations on 3-manifolds with boundary

We define a family of admissible perturbations and show, for a generic perturbation, the moduli space is a compact , smooth manifo l d of half the dimension of moduli[r]

### Boundary value problems for thend-order Seiberg-Witten equations

1. Introduction Let X be a compact smooth 4-manifold with nonempty boundary. In our context, the Seiberg-Witten equations are the 2nd-order Euler-Lagrange equation of the functional defined in Definition 2.3. When the boundary is empty, their variational aspects were first studied in [3] and the topological ones in [1]. Thus, the main aim here is to obtain the existence of a solution to the nonhomogeneous equations whenever ∂X = ∅ . The nonemptiness of the boundary inflicts boundary conditions on the problem. Classically, this sort of problem is classified according to its boundary conditions in Dirichlet problem (Ᏸ) or Neumann problem (ᏺ).

### On the treewidth of triangulated 3-manifolds

Graph splittings and fork complexes. The decomposition described above exhibits a linear structure. Here we introduce a more general approach of decomposing a 3-manifold into compression bodies following a graph structure [56]. The main difference is that now we allow the lower boundary components of a com- pression body to be glued to lower boundary components of distinct compression bodies. The top boundary of a compression body, however, is still identified with the top boundary of a single other compression body. This structure can be represented by a so-called fork complex (which is essentially a labeled graph) in which the compression bodies of the de- composition are modeled by forks. More precisely, an n-fork is a tree F with n + 2 nodes V (F ) = {g, p, t 1 , . . . , t n } with p being of degree n + 1 and all other nodes being leaves. The nodes g, p, and the t i are called the grip, the root, and the tines of F , respectively (Figure 1(i) shows a 0- and a 3-fork). We think of a fork F = F N as an abstraction of a compression body N , such that the grip of F corresponds to ∂ + N , whereas the tines correspond to the connected components of ∂−N .

### Geometry of random 3-manifolds

The structure of the collar Ω will be modelled on the geometry of a hy- perbolic mapping torus, or pseudo-Anosov mapping class, with a short pants decomposition. Such object is one that is obtained from the following proce- dure: Let P be a pants decomposition of Σ. Let φ ∈ Mod(Σ) be a mapping class such that no curve in P is isotopic to a curve in φP . For example, a large power of any pseudo-Anosov suffices. Consider the convex core Q of the maximally cusped structure Q(P, φP ). The boundary ∂Q consists of to- tally geodesic hyperbolic three punctured spheres that are paired according to φ. We glue them together isometrically as prescribed by the pairing. The glued manifold is a finite volume hyperbolic 3-manifold diffeomorphic to

### Minimal entropy of 3-manifolds

3.2 Non-positively curved C 2 metrics on Seifert mani- folds We observed in Chapter 2 that Seifert components of irreducible, orientable, closed 3-manifolds having at leat a hyperbolic piece in their JSJ decomposition are either of hyperbolic type or homeomorphic to K ˜×I, as D 2 ×S 1 and T 2 ×I cannot appear as J SJ components of closed, irreducible 3-manifolds. In §2.2.3 we explained how a non- positively curved metric with totally geodesic boundary g 0 on a Seifert fibred manifold S with base orbifold O S provides us with a non-positively curved metric with totally geodesic boundary ¯g 0 on the base orbifold. Moreover, the quotient map collapsing the fibres to points is a Riemannian orbifold submersion p : (S, g 0 ) → (O S , g ¯ 0 ). This section is devoted to the proof of the following proposition:

### CiteSeerX — The Seiberg-Witten equations and 4–manifold topology

THE SEIBERG-WITTEN EQUATIONS AND 4-MANIFOLD TOPOLOGY S. K. DONALDSON Since 1982 the use of gauge theory, in the shape of the Yang-Mills instanton equations, has permeated research in 4-manifold topology. At first this use of dif- ferential geometry and differential equations had an unexpected and unorthodox flavour, but over the years the ideas have become more familiar; a body of tech- niques has built up through the efforts of many mathematicians, producing results which have uncovered some of the mysteries of 4-manifold theory, and leading to substantial internal conundrums within the field itself. In the last three months of 1994 a remarkable thing happened: this research area was turned on its head by the introduction of a new kind of differential-geometric equation by Seiberg and Witten: in the space of a few weeks long-standing problems were solved, new and unexpected results were found, along with simpler new proofs of existing ones, and new vistas for research opened up. This article is a report on some of these devel- opments, which are due to various mathematicians, notably Kronheimer, Mrowka, Morgan , Stern and Taubes, building on the seminal work of Seiberg [S] and Seiberg and Witten [SW]. It is written as an attempt to take stock of the progress stemming from this initial period of intense activity. The time period being comparatively short, it is hard to give complete references for some of the new material, and perhaps also to attribute some of the advances precisely. The author is grateful to a number of mathematicians, but most particularly to Peter Kronheimer, for explaining these new developments as they unfolded.

### CiteSeerX — Generalized Seiberg–Witten equations and hyperKähler geometry

It was shown in [59] that the SeibergWitten moduli space associated to the generalized SeibergWitten equations is compact provided M admits a permuting H ∗ action or, equivalently, a hyperK¨ ahler potential. In this section we study hyperK¨ ahler manifolds M with hyperK¨ ahler po- tential and triholomorphic action of S 1 . The triholomorphic action of S 1 is necessary for the definition of the SeibergWitten equations. First we show how to reconstruct the manifold M by its hyperK¨ ahler reduction ˜ M with re- spect to a nonzero value of momentum map. This is done in Theorem 2.1.3, which simultaneously provides a classification of hyperK¨ ahler manifolds with hyperK¨ ahler potential and triholomorphic action of S 1 . Then we describe cor- respondent quaternionic K¨ ahler manifolds (see Theorem 2.1.8). In Section 2.3 we describe new examples of hyper– and quaternionic–Kahler manifolds mak- ing use of a certain freedom in choice of parameters of the construction. We prove for example that the total spaces of O P 1 (m), m ≥ 1 admit an Ein- stein and self–dual metric with positive scalar curvature. The last section is devoted to K¨ ahler structure on quaternionic K¨ ahler manifolds with positive scalar curvature naturally invoked by the action of S 1 . Theorem 2.4.1 was known earlier from twistor theory, however our proof is new. We not only prove the existence of K¨ ahler structure, but also compute it explicitly.

### The relation between Maxwell, Dirac, and the Seiberg Witten equations

Crucial to that proposed equivalence is the possibility of solving for ψ (a repre- sentative on a given spinorial frame of a Dirac-Hestenes spinor ﬁeld) the equation F = ψγ 21 ψ, where ˜ F is a given electromagnetic ﬁeld. Such task is presented and it permits to clarify some objections to the MDE which claim that no MDE may exist because F has six (real) degrees of freedom and ψ has eight (real) degrees of freedom. Also, we review the generalized Maxwell equation describing charges and monopoles. The enterprise is worth, even if there is no evidence until now for magnetic monopoles, because there are at least two faithful ﬁeld equations that have the form of the generalized Maxwell equations. One is the generalized Hertz potential ﬁeld equation (which we discuss in detail) associated with Maxwell theory and the other is a (nonlinear) equation (of the generalized Maxwell type) satisﬁed by the 2-form ﬁeld part of a Dirac-Hestenes spinor ﬁeld that solves the Dirac-Hestenes equation for a free electron. This is a new result which can also be called MDE of the second kind. Finally, we use the MDE of the ﬁrst kind together with a reasonable hypothesis to give a derivation of the famous Seiberg-Witten equations on Minkowski spacetime. A physical interpretation for those equations is proposed.

### Seiberg-Witten Theory

So the eﬀective Lagrangian, which only describes the remaining massless ﬁelds, is still N=2 massless supersymmetric, and has gauge transformations based on U(1), which is Abelian. Then the only remaining degree of freedom in the eﬀective Lagrangian is a branch cut of a multivalued function 𝑓. Such a function 𝑓 is ﬁxed, up to SL(2, ℤ), using: quantum ﬁeld theoretic perturbation theory, and a family of cubic curves as manifolds together with some complex analysis on holomorphic functions; thereby ﬁxing the eﬀective Lagrangian.

### CiteSeerX — Perturbations of the metric in Seiberg-Witten equations Luca Scala

Proof. By (4.3) the cokernel of the differential D (A,ψ,ξ) (F Met(M) ) 2,r p coincides with the kernel of the formal adjoint of the differential D (A,ψ,ξ) F Met(M) on sections of class C ∞ . The equations (4.7) of the kernel of (D (A,ψ,ξ) F Met(M) ) ∗ are now equivalent, by lemma 4.4.1, to equations (4.11). Remark 4.4.3. We discuss now the gaps in the proof of the transversality with generic metrics by Eichhorn and Friedrich. The two authors (in [5, Proposition 6.4] and Friderich alone in [8, page 141]) try to prove directly that the differential D (A,ψ,g ξ ) F ˜ of the perturbed Seiberg-Witten functional is surjective. A first source of unclearness is that they never give a precise expression of the variation of the Dirac operator, which we have seen as being a fundamental difficulty in the question; in particular no mention is made about the term −ρ ξ ◦ s ∗ ◦ ∇ W A ψ. The authors take into account variations of the metric which are orthogonal to the orbits of the action the diffeomorphism group Diff(M ) on Met(M ): this condition is precisely expressed by div s = 0. They now remark that the variation of the second equation involves just the traceless part of the tensor s 0 : as a consequence, they now claim that they can deal with conformal perturbations separately from volume preserving ones. Thanks to this uncorrect argument, as we will see, they get to the two separate conditions, reading, our notations:

### Defects and quantum Seiberg-Witten geometry.

When the theory is on the manifold with boundary, the S transformation should be carefully taken. If there exists the effective Chern-Simons term of the background gauge field A, the naive gauging procedure fails to work due to the gauge anomaly. The Chern- Simons term makes the gauge invariance anomalous on the boundary. To keep the gauge invariance one need to couple relevant boundary degrees of freedom such that the flavor anomaly of the 2d theory compensates the gauge anomaly on the boundary. In addition, the mixed Chern-Simons term introduced by the S transformation also breaks the gauge invariance. The 2d theory should be chosen to cancel this anomaly as well.

### CiteSeerX — SEIBERG-WITTEN FLOER HOMOLOGY AND

that solves the corresponding version of the equations in (3.1 ) at each t ∈ I ′ and is such that a F θ (t) = a F (c θ,I (t)) at each t ∈ I ′ . Proof. The proof is, but for notational changes and two additional remarks, identical to that of Proposition 2.5 in [T4]. To set the stage for the first remark, fix a base point 0 ∈ S 1 \ S r . The identifications of the Seiberg- Witten Floer homology groups given by adapting what is done in [T4] may result in the following situation: As t increases from 0, these identifications results at t = 2π in an automorphism, U, on the t = 0 version of the Seiberg- Witten Floer homology. This automorphism need not obey a F Uθ = a F θ . If not, then it follows using Proposition 3.3 that the identifications made at t < 2π to define U can be changed if necessary as t crosses points in T r so that the new version of U does obey a F Uθ = a F θ . The second remark concerns the fact that any given c θ,I is unique up to gauge equivalence. This follows from Proposition 3.3’s assertion that the function a F distinguishes the Seiberg-Witten solutions when t ∈ S 1 \ T r .  When E = C, we need to augment what is said in Proposition 3.4 with the following:

### The Kapustin Witten Equations with Singular Boundary Conditions

F A − Φ ∧ Φ + ?d A Φ = h 1 , d ? A Φ = h 2 . (1.2) Here is the outline of the paper. In Section 2, we introduce some preliminaries on the Kapustin-Witten equations, including the Kuranishi complex and some examples of the Nahm pole solutions. In Section 3, we introduce a gauge fixing condition and the elliptic system associated to the equations. In Section 4, we study the gradient flow of the Kapustin-Witten equations, and the structure of the linearization operator over Y × R . In Section 5, we establish the Fredholm theory for the linearization operator over manifolds with boundaries and cylindrical ends. In Section 6, we build up a slicing theorem and Kuranishi model for the Nahm pole solutions. In Section 7, after assuming the solution over cylindrical ends is simple and L 1 p converges to a flat SL ( 2; C ) connection over the cylindrical end for p > 2, we prove that the solution will exponentially decay to the SL (2; C ) flat connection in the cylindrical ends. In Section 8, we describe the obstruction in the second homology group of the Kuranishi complex to the existence of solutions when gluing along the cylindrical ends. In Section 9, we build up a local Kuranishi model for the gluing picture.

### CiteSeerX — Donaldson and Seiberg-Witten invariants of algebraic surfaces

basic classes κ i ∈ H 2 (M ; Z) and rational numbers a i . Here the κ i are characteristic elements of H 2 (M ; Z), i.e. for all α ∈ H 2 (M ; Z), α 2 ≡ α · κ mod 2. 3. Definition of the Seiberg-Witten invariants One basic motivation which led to the definition of the Seiberg-Witten invariants was the attempt to find an a priori description of the Kronheimer-Mrowka basic classes. To define the Seiberg-Witten invariants, we need to recall the definition of a Spin c structure on M . Let M be an oriented Riemannian 4-manifold (actually we can define a Spin c structure in every dimension). Then the tangent bundle of M corresponds to a principal SO(4)-bundle. Now SO(4) has a unique double cover Spin(4), which by one of the coincidences of Lie group theory in low dimensions is isomorphic to SU (2) × SU(2), and one can ask if the tangent bundle of M lifts to a Spin(4) bundle. The answer is that such a lift exists if and only if the second Stiefel-Whitney class w 2 (M ) is zero, and in case M is simply connected (or more generally if H 2 (M ; Z) has no 2-torsion) this condition is equivalent to assuming that H 2 (M ; Z) has an even intersection form. In this case a lift of T M to Spin(4) is called a Spin structure on M . Of course, most 4-manifolds do not have a Spin structure. However, if one does exist, then from the isomorphism Spin(4) ∼ = SU (2) × SU(2), there are two associated rank two complex vector bundles S + , S − with trivial determinant. A fundamental fact is that the Levi-Civita connection on T M induces a differential operator, the Dirac operator / ∂ : S + → S − . Here /∂ is a first order elliptic formally self-adjoint operator.

### Mod 2 Seiberg-Witten invariants of real algebraic surfaces with the opposite orientation

1. Introduction It is a delicate problem to calculate the Seiberg-Witten invariants of closed smooth 4-manifolds. In his paper [8], Furuta introduced a new method of cal- culating the Seiberg-Witten invariants as a certain equivariant degree of a map constructed from the Seiberg-Witten equations. The goal of this paper is to show that under certain topological assumptions the Seiberg-Witten invariants of real algebraic surfaces with the opposite orientation are zero mod 2.

### Slow manifolds and invariant sets of the primitive equations.

For boundary conditions, we assume periodicity in the horizontal directions and free-slip › z v 5 0, w 5 0, and b 5 0, at the bottom and top. Spectrally, this can be implemented by doubling the domain in the z direction and imposing appropriate symmetries (see, e.g., Bartello 1995). These boundary conditions are not needed for the regularity of the solutions [note that Cao and Titi (2007), Kobelkov (2007), and Kukavica and Ziane (2007) used different boundary conditions], but they are essential for the existence of our exponential slow manifold. With no loss of generality, we also assume that y has zero integral over the domain.

### Mathematical Research Letters 2, 9 13 (1995) MORE CONSTRAINTS ON SYMPLECTIC FORMS FROM SEIBERG-WITTEN INVARIANTS. Clifford Henry Taubes

Using the Seiberg-Witten equations, I proved in [T] the following: Theorem 1. Let X be a compact, oriented, 4 dimensional manifold with b 2+ ≥ 2. Let ω be a symplectic form on X with ω∧ω giving the orientation. Then the ﬁrst Chern class of the canonical bundle of a compatible, almost complexstructure on X has Seiberg-Witten invariant equal to ±1.

### 4-Manifolds With Inequivalent Symplectic Forms and 3-Manifolds With Inequivalent Fibrations

The SeibergWitten norm manifests the rigidity of the smooth structure on X, allowing us to check that the Chern classes c 1 (ω 1 ), c 1 (ω 2 ) lie in different orbits of Diff(X). On the other hand, using Freedman’s work one can see that these two Chern classes are related by a homeomorphism of X. In fact, using the 3-torus we can write H 2 (X, Z ) with its intersection form as a direct sum

### Gluing manifolds with boundary and bordisms of positive scalar curvature metrics

APPENDIX GEOMETRIC ANALYSIS BACKGROUND Let us first describe the structure of this Appendix. In Section A.1, we state the basic regularity facts for solutions of elliptic equations both with and without boundary conditions. These facts are used in both Part One and Part Two. In Section A.2, we recall relevant facts on the minimal graph equation and provide the Schauder estimates we use in the proof of Main Lemma from Part Two. Section A.3 is dedicated to Theorem 1.5.4. Here we recall necessary results on currents and state well-known facts on their compactness and regularity, adapted to our setting. Section A.31 describes a simple doubling method which is a convenient technical tool in the remaining sections. In Section A.32, we justify Step 2 from the proof of Claim 3.2.3. In Section A.33, we discuss regularity issues in dimension 8 and prove Theorem 1.5.6 for n = 7.