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]

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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).

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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 .

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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

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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:

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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.

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It was shown in [59] that the **Seiberg**–**Witten** moduli space associated to the generalized **Seiberg**–**Witten** **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 **Seiberg**–**Witten** **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.

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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.

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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.

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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:

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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.

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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:

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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.

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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.

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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.

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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.

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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.

The **Seiberg**–**Witten** 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

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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.

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M 2 = X −1 M 1 X , M **3** = X −2 M 1 X 2 . (41) A solution to this monodramy problem is given by
M 1 = ST S −1 , X = T . (42)
We see that M 1 ≡ M + , thus in the large µ regime the local physics around two of the singularities is exactly the same as in the pure SU (2) case: a monopole and a dyon become light respectively.

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