Concentrating Local Solutions of the Two-Spinor Seiberg-Witten Equations on 3-Manifolds

Given a compact 3-manifold $Y$ and a $\Z_2$-harmonic spinor $(\mathcal Z_0, A_0,\Phi_0)$ with singular set $\mathcal Z_0$, this article constructs a family of local solutions to the two-spinor Seiberg-Witten equations parameterized by $\e \in (0,\e_0)$ on tubular neighborhoods of $\mathcal Z_0$. These solutions concentrate in the sense that the $L^2$-norm of the curvature near $\mathcal Z_0$ diverges as $\e\to 0$, and after renormalization they converge locally to the original $\Z_2$-harmonic spinor. In a sequel to this article, these model solutions are used in a gluing construction showing that any $\Z_2$-harmonic spinor satisfying some mild assumptions arises as the limit of a family of two-spinor Seiberg-Witten solutions on $Y$.

§ INTRODUCTION

Equations of generalized Seiberg-Witten type are conjectured to have deep connections to the geometry and topology of manifolds. Examples include the Vafa-Witten equations [38, 30, 29] and the Kapustin-Witten equations [15, 41, 40], which are predicted to connect 3 and 4-dimensional topology to other areas. Another example is the ADHM Seiberg-Witten equations, which are expected to play a key role in Donaldson-Segal's program to construct invariants of manifolds with special holonomy in dimensions 6,7, and 8  [6, 3, 10]. In contrast to the standard Seiberg-Witten (SW) equations, the moduli spaces of solutions to generalized Seiberg-Witten equations may not be compact, and the lack of well-understood compactifications is one of the main barriers in the study of these equations.

Pioneering work of Taubes [31, 33, 36, 37, 35, 34], Haydys-Walpuski [13], and Walpuski-Zhang [39] has shown that sequences of solutions to generalized SW equations on a manifold $Y$ can diverge, but after renormalization must converge to $\Z_2$-harmonic spinors or more general types of Fueter sections—solutions of a different, in general non-linear PDE on $Y-\mathcal Z$ where $\mathcal Z$ is a codimension 2 singular set. On a compact Riemannian 3-manifold $(Y,g_0)$, the first of these is defined for our purposes as follows. Given an embedded submanifold $\mathcal Z\subset Y$ of dimension 1 and a spinor bundle $S_0\to Y$, fix a real line bundle $\ell \to Y-\mathcal Z$ equipped with its unique flat connection $A_0$ with holonomy in $\Z_2$. A $\Z_2$-harmonic spinor consists of a triple $(\mathcal Z, A_0,\Phi)$ where $\Phi \in \Gamma(S_0\otimes_\R \ell)$ is a spinor satisfying D_A_0Φ=0     on    Y-𝒵_0  and  ∫_Y-𝒵 |∇_A_0Φ|^2 < ∞,where $\slashed D_{A_0}$ denotes the Dirac operator twisted by the connection $A_0$ on $\ell$. Said more simply, a $\Z_2$-harmonic spinor is a harmonic spinor on the open manifold $Y-\mathcal Z$ in a spin structure which does not necessarily extend over $\mathcal Z$ and whose covariant derivative is $L^2$.

The above convergence results suggest that the moduli spaces of solutions to a specific generalized Seiberg-Witten equation should admit natural compactifications obtained by including $\Z_2$-harmonic spinors or, more generally, the type of Fueter section arising for that equation as boundary strata. Constructing these compactifications requires addressing the converse to the convergence question:

which $\Z_2$-harmonic spinors or Fueter sections arise as the limit of a sequence of Seiberg-Witten solutions?

In terms of PDE, this question is a gluing problem, requiring the patching together of solutions of the two different equations. When the singular set $\mathcal Z=\emptyset$ is empty, this gluing problem was solved in a general setting by Doan-Walpuski [4], though in this case the “gluing” is rather trivial because no model solutions are required. Indeed, they instead refer to this case as a “deformation” problem. In the situation that $\mathcal Z\neq \emptyset $, which is a stable condition under perturbations [5, 28, 25], the gluing problem requires model solutions near the singular set.

The present work provides two crucial steps towards solving the gluing problem in the case of the two-spinor Seiberg-Witten equations on a closed 3-manfiold. In this case, the $\Z_2$-harmonic spinors that arise are as defined in (Z2harmonicpreliminary). Given a $\Z_2$-harmonic spinor $(\mathcal Z_0, A_0,\Phi_0)$ satisfying some mild assumptions, the first step accomplished in this article is to construct a 1-parameter family of model solutions to the two-spinor Seiberg-Witten equations in a neighborhood of the singular set $\mathcal Z_0$ which converge locally to $(A_0,\Phi_0)$. The second step is to analyze the linearized equations at this family of model solutions whose limiting linearization at $(A_0,\Phi_0)$ is a degenerate elliptic operator whose symbol vanishes along $\mathcal Z_0$.
In a sequel to this article [26], these model solutions are used to solve the gluing problem and answer Question <ref> in this setting.

§.§ Main Results

To state the main results, let us first describe the set-up briefly. Additional details are given in Section <ref>. Let $(Y,g_0)$ denote a closed, oriented Riemannian 3-manifold, and let $S\to Y$ be the spinor bundle associated to a $\text{Spin}^c$ structure. Furthermore, let $E\to Y$ be a rank 2 complex vector bundle with structure group $SU(2)$ bundle endowed with a fixed connection $B_0$. The two-spinor Seiberg-Witten equations are the following system of equations for a pair $(\Psi,A)\in \Gamma(S\otimes_\C E) \times \mathcal A_{U(1)}$ of an $E$-valued spinor and a $U(1)$-connection lifted from $\text{det}(S)$:

\begin{eqnarray}
\slashed D_{A}\Psi&=&0 \label{SW1intro}\\
\star F_A +\tfrac{1}{2}\mu(\Psi, \Psi)& =& 0
\label{SW2intro}
\end{eqnarray}

where $\slashed D_{A}$ is the Dirac operator on $S\otimes E$ twisted by $A$ and the fixed connection $B_0$ on $E$, $F_A$ is the curvature of $A$, and $\mu$ a point-wise quadratic map. The equations are invariant under $U(1)$-gauge transformations.

As mentioned above, there may be sequences $(\Psi_i, A_i)$ of solutions to (SW1intro)-(SW2intro) that have no convergent sub-sequences modulo gauge. For such sequences, the $L^2$ norm $\|\Psi_i\|_{L^2}\to \infty$ must diverge. To highlight the role of the $L^2$ norm for such sequences, one can renormalize the spinor by setting

$$ \e:= \frac{1}{\|\Psi\|_{L^2}} \hspace{3cm} \Phi:= \e \Psi $$

and instead consider the equations (SW1intro)-(SW2intro) for the pair $(\tfrac{\Phi}{\e}, A)$. The results of Haydys-Walpuski in [13] show that if the sequence $(\Psi_i, A_i)$ has no subsequences for which $\|\Psi_i\|$ remains bounded, then the renormalized sequence $(\Phi_i, A_i)$ converges subsequentially to a $\Z_2$-harmonic spinor as $\e\to 0$ modulo gauge transformations on the complement of the singular set $\mathcal Z$ (see Theorem <ref> in Section <ref> for a precise statement). At present, it is not known that the singular set of a $\Z_2$-harmonic spinor arising in this way  necessarily has more regularity than being a closed, rectifiable subset of Hausdorff codimension 2. We do not attempt to address these regularity issues here.

Reversing the convergence statement to address the gluing question, let $(\mathcal Z_0, A_0, \Phi_0)$ be a $\Z_2$-harmonic spinor on $(Y,g_0)$ with respect to a perturbation induced by $B_0$. We assume that it satisfies the following.

1(Smoothness) the singular set $\mathcal Z_0\subseteq Y$ is a smooth, embedded link, and $A_0$ does not extend smoothly across any component of $\mathcal Z_0$.

2 (Non-degeneracy)
the spinor $\Phi_0$ has non-vanishing leading-order, i.e. there is a constant $c_1$ such that
$$|\Phi_0| \geq c_1\text{dist}(-,\mathcal Z_0)^{1/2}.$$

3 (Isolated)
$\Phi_0$ is the unique $\Z_2$-harmonic spinor with respect to $(\mathcal Z_0, A_0,g_0,B_0)$ up to scaling.

The main result is the following construction of model solutions:

Given a $\Z_2$-harmonic spinor $(\mathcal Z_0, A_0,\Phi_0)$ as in (Z2harmonicpreliminary) satisfying Assumptions <ref>–<ref> and an orientation on $\mathcal Z_0$, there exists a $\text{Spin}^c$ structure $S$ on $Y$ such that the following hold.

(i)  $S$ extends $S_0\otimes_\R \ell$ to $Y$ in the sense that $S |_{Y-\mathcal Z_0}\simeq  S_0\otimes_\R \ell$, and is determined by a complex line bundle $\mathcal L\to Y$ such that
$$ S=S_0\otimes_\C \mathcal L \hspace{1.5cm} \text{ and } \hspace{1.5cm} c_1(\det S)= -\text{PD}[\mathcal Z_0].$$

Additionally, $\Phi_0$ is naturally a section of a rank 4 real subbundle of $(S\otimes_\C E)|_{Y-\mathcal Z_0}$.

(ii) For this $\text{Spin}^c$ structure $S$, there is an $\e_0>0$ such that for $\e\in (0, \e_0)$ there exist model solutions $(\Phi_\e, A_\e)$ on the tubular neighborhood $N_{\lambda}(\mathcal Z_0)$ of radius $\lambda=\tfrac{1}{2}\e^{1/2}$ satisfying the two-spinor Seiberg-Witten equations

\begin{eqnarray}
\slashed D_{A_\e}\Phi_\e &=& 0 \\
\star F_{A_\e} + \tfrac{1}{2}\tfrac{\mu(\Phi_\e,\Phi_\e)}{\e^2} &=&0
\end{eqnarray}

for spinors in $S\otimes_\C E$.

(iii) $\tfrac{\Phi_\e}{\e}$ extends via a cut-off function to a smooth section of $S\otimes_\C E$ on the closed manifold $Y$ that is equal to $\tfrac{\Phi_0}{\e}$ away from $N_{\lambda/2}(\mathcal Z_0)$ and has $L^2$ norm $\tfrac{1}{\e}+ O(\e^{-1/4})$ on $Y$.

The second main result shows that these model solutions approach $(\Phi_0, A_0)$ as $\e\to 0$. This follows from applying the main results of [24] to the model solutions constructed in Theorem <ref>.

The model solutions $(\Phi_\e, A_\e)$ converge to $(\Phi_0, A_0)$ in the following sense. Fix a family of compact subsets  $K_\e \Subset  N_{\lambda}(\mathcal Z_0)-\mathcal Z_0$ such that $\text{dist}(K_\e, \mathcal Z_0) \geq c_1 \e^{2/3}$ for a positive constants $c_1$. Then there are constants $C,c$ independent of $\e$ such that the un-renormalized difference (_, a_)=(Φ_/, A_)-(Φ_0/, A_0)obeys the following properties.

(i) There is a half-dimensional subbundle $S^\text{Im}\subseteq S\otimes_\C E$ such that the components of the spinor in $S^\text{Im}$ and the connection decay to $(\tfrac{\Phi_0}{\e},A_0)$ exponentially on $K_\e$.  That is,

$$\|(\ph^\text{Im}_\e, a_\e)\|_{C^0(K_\e)} \ \leq \  \frac{C}{|\text{dist}(K_\e, \mathcal Z)|^{3/2}\e}\ \text{Exp}\left(-\frac{c}{\e}\text{dist}(K_\e,\mathcal Z)^{3/2}\right).$$

(ii) The remaining spinor components decay to $(\tfrac{\Phi_0}{\e}, A_0)$ like  $\text{dist}(-,\mathcal Z_0)^{-\nu}$ for any $0<\nu<\frac{1}{4}$; i.e. there is an $\e$-independent constant $\gamma'<<1$ such that

$$\|\text{dist}(-,\mathcal Z_0)^\nu\ph_\e\|_{L^{1,2}(K_\e)} \ \leq \ C_\nu\, \e^{1/12-\gamma'}.$$

Note that $\text{dist}(-,\mathcal Z_0)$ is a function on $Y$, whereas $\text{dist}(K_\e,\mathcal Z)= \inf_{y\in K_\e} \text{dist}(y,\mathcal Z_0)$ is a constant depending on $\e$. In particular, if $\text{dist}(K_\e,\mathcal Z_0)\geq \e^{2/3-\gamma_1}$ for some $\gamma_1>0$, then $(\ph^\text{Im}_\e,a_\e)$ decays to 0 faster than any polynomial on $K_\e$ as $\e\to 0$.

The third and final main result is about the linearization of the Seiberg-Witten equations at the model solutions. Since in the eventual gluing construction these model solutions are pasted onto the manifold using a cut-off function to form global approximate solutions on $Y$, the statement of the theorem is given for these. Let $\chi(r)$ denote a cut-off function supported on $N_{\lambda}(\mathcal Z_0)$ equal to 1 for radii $r\leq \lambda/2$. The Approximate solutions are defined as

(Φ^App_/, A^App_):= (Φ_0/, A_0) + χ(_, a_),

where $(\ph_\e, a_\e)$ is the un-renormalized difference defined in  unrenormalizeddiff.

Let $\mathcal L_\e$ denote the “extended gauge-fixed” Seiberg-Witten equations at these approximate solutions, which are defined precisely in Section <ref>. This operator is viewed as a first-order boundary value problem on a tubular neighborhood $N_{2\lambda}(\mathcal Z_0)$ (with twice the radius of the one in Theorem <ref>) by introducing a Hilbert space $H$ and a projection Π^ℒ: L^1,2(N_2λ(𝒵_0))→H so that $\ker(\Pi^\mathcal L)$ is the subspace of sections satisfying certain boundary and orthogonality conditions. The precise definitions of $H$ and  $\Pi^\mathcal L$ are given in Section <ref>. The statement also references certain weighted norms $\| -\|_{H^1_{\e,\nu}}$ and $\|-\|_{L^2_{\e, \nu}}$ defined in Section <ref>; these are equivalent (not uniformly) to the standard norms on $L^{1,2}(N_{2\lambda}(\mathcal Z_0))$ and $L^{2}(N_{2\lambda}(\mathcal Z_0))$ respectively.

Subject to the boundary and orthogonality conditions defined by bdorthogonalityconstraints, the extended gauge-fixed linearization of the two-spinor Seiberg-Witten equations at the approximate solutions approx

ℒ_: (Π^ℒ)⊆L^1,2(N_2λ(𝒵_0)) L^2(N_2λ(𝒵_0))

is Fredholm of Index 0. Additionally, there is an $\e_0>0$ such that for $\e<\e_0$,  $\mathcal L_\e$ is invertible, and there are positive constants $C,\gamma^\text{in}\!<\!<1$ independent of $\e$ such that  the bound

(,a)_H^1_,ν ≤C_ν/^1/12+γ^in   ℒ_(,a)_L^2_,νholds.

The model solutions constructed in part (ii) of Theorem <ref> will sometimes be referred to as “fiducial solutions”. This terminology is taken from [19] where fiducial solutions of a similar nature were found for Hitchin's equations (see Section <ref> and Section <ref> for further discussion). The model solutions of Theorem <ref> actually solve the extended Seiberg-Witten equations as defined in Section 2.4. That is, they include an auxiliary 0-form component $a_0$.

Assumptions <ref>–<ref> can be shown to be open conditions (see [14]), though we do not prove such a statement here. It is conjectured that Assumption <ref> also holds generically within the set of $(g_0, B_0)$ that admit $\Z_2$-harmonic spinors. The genericity of the embedding condition is the subject of ongoing work by other authors [12]. This and other questions on the regularity of the singular set $\mathcal Z_0$ involve significant detours into geometric measure theory (see [34, 43, 11]) and are beyond the scope of the present article, thus we choose to impose Assumption <ref>. This assumption could readily be weakened (e.g. $\mathcal Z_0$ is an $L^{k,2}$-embedding for  $k=3$), but the required analysis would distract the main goals of the present article.

Notice that the size of the tubular neighborhood $N_{\lambda}(\mathcal Z_0)$ on which the model solutions exist shrinks as $\e^{1/2}$. This is perhaps displeasing, but is unavoidable. The choice of a power of $\e$ plays a crucial role in the proofs of Theorems <ref> and <ref>; in particular it has a nuanced role in the definition of $\Pi^\mathcal L$ and the proof of the bound Linvertibleboundmainc. Additionally, the lack of a uniform bound in Theorem <ref> is not merely a shortcoming of the methods: explicit counterexamples to Linvertibleboundmainc without the power of $\e$ are constructed during the proof.

§.§ Motivation for Approach

This section briefly motivates and summarizes the approach taken to the proofs of Theorem <ref> and Theorem <ref>.

1.2.1. Degenerating Linearizations

The gluing problem for $\Z_2$-harmonic spinors does not fit into the standard framework used in many other gluing problems. These differences are due to the existence of the singular set $\mathcal Z_0$, near which the equations degenerate and standard elliptic theory breaks down. Indeed, under the assumption that $\mathcal Z_0=\emptyset$, Doan-Walpuski  [4] solved the gluing problem in great generality using standard elliptic theory. Unfortunately, none of their approach extends to the case that $\mathcal Z\neq \emptyset$.

To be more precise, the standard elliptic theory breaks down in the following way. The linearized Seiberg-Witten equations

ℒ_(Φ_0, A_0): L^1,2(Y-𝒵_0) L^2(Y-𝒵_0)

at a $\Z_2$-harmonic spinor with $\mathcal Z_0\neq \emptyset$ are a degenerate elliptic system whose symbol vanishes along $\mathcal Z_0$. Operators with this type of degeneracy are known as elliptic “edge” operators, and are well-studied in microlocal analysis [27, 20, 21, 8]. For the edge operator $\mathcal L_{(\Phi_0, A_0)}$, there is no natural choice of function spaces on which it is  Fredholm;  in particular, (deglinearization) has an infinite-dimensional cokernel. For any family of model solutions $(\Phi_\e, A_\e)$, the resulting family of linearized equations

ℒ_(Φ_, A_)→0 ℒ_(Φ_0, A_0)

is converging to this limiting operator with inifinite-dimensional cokernel (in no precise sense, as the function spaces change in the limit). As a result,  one cannot expect the linearizations to be uniformly invertible in any reasonable sense.

The consequences of this are two-fold. In the present article, this manifests in the difficulty of proving Theorem <ref>, where the subspace limiting to the infinite-dimensional cokernel ruins any naive approach. The proof unavoidably requires delicate analysis of the degenerating family (deglinearizations2), which is carried out in Sections <ref>-<ref>. The second consequence is for the eventual gluing: even with Theorem <ref> in hand, the gluing problem still appears at first to have an infinite-dimensional obstruction coming from the cokernel of <ref>. This is addressed in [26] by considering deformations of the singular set $\mathcal Z_0$, which requires the study of the infinite-dimensional family of operators $\mathcal L_{\e}(\mathcal Z)$ parameterized by nearby singular sets $\mathcal Z$, though no more is said about this issue here.   1.2.2. Relation to Limiting Configurations

The gluing problem for $\Z_2$-harmonic spinors is effectively a generalization of the gluing problem that arises at the boundary of the moduli space of solutions to Hitchin's equations on a Riemann surface $\Sigma$, and this observation guides parts of our approach.

The boundary objects in the Hitchin moduli space, known as limiting configurations, are singular Higgs fields whose  singular set $\mathcal Z_\Sigma \subset \Sigma$   is a finite collection of points.  Given a limiting configuration $\Phi_0$ and a singular point $z\in\mathcal Z_\Sigma$,  one makes   the ansatz that there are local model solutions which differ from $\Phi_0$ by a complex-gauge transformation $h_\e(r)$ which depends only on the distance $r$ from $z$. That is, are locally  of the form
Φ_= e^h_(r)·Φ_0.

This leads to an $\e$-parameterized family of ODEs for $h_\e(r)$ that can be solved to yield model solutions. These are then spliced onto $\Sigma$ and corrected to true solutions using methods that exploit the
holomorphic structure of Hitchin's equations to circumvent the problem of the degenerating linearization (see [19, 7] for details).

The relation of this case with the gluing problem for $\Z_2$-harmonic spinors is, essentially, that it is a dimensional reduction. More precisely, the gluing problem in the case of Hitchin's equations is the dimensional reduction of the gluing problem for the closely related (though more difficult) Kapustin-Witten equations. For a 3-manifold $Y=S^1\times \Sigma$, the limiting configurations at the boundary of the Hitchin moduli space on $\Sigma$ can be lifted to $\Z_2$-harmonic 1-forms (which are $\Z_2$-harmonic spinors for the Dirac-type operator $(d+d^\star)$) that are invariant in the $S^1$ direction. The singular set is lifted to $\mathcal Z= S^1 \times \mathcal Z_\Sigma$. Up to some minor differences between the equations, the three-dimensional gluing problem for the two-spinor Seiberg-Witten equations can be viewed as a generalizing of the construction for Hitchin's equation to the non-$S^1$-invariant case.

Unfortunately, for the case of a ${\Bbb Z}_2$-harmonic spinor on a general  3-manifold $Y$,  the lack of a holomorphic structure on  $Y$ means virtually none of the techniques used for Hitchin's equations are applicable. First, there is no analogue of the holomorphic structure which can be exploited to circumvent the problem of the degenerating family of linearized operators, and this problem must be confronted. Even disregarding this issue with the linearization, there are several critical issues in extending the 2-dimensional approach to find 3-dimensional model solutions. For one, the holomorphic structure on $\Sigma$ allows one to choose local coordinates putting $\Phi_0$ in a standard form, thus the 2-dimensional model solution is unique up to coordinate change. In contrast, in three-dimensions the local form of $\Phi_0$ lies in an infinite-dimensional space of possibilities. Secondly, one cannot make an effective simplifying ansatz akin to ansatzintro. In addition to having to upgrade the ODE for $h_\e(r)$ to a PDE for $h_\e(r,t)$ depending also on $t$ the tangential coordinate $\mathcal Z$, the lack of a holomorphic structure, means the number of equations also increases. In combination, these features mean there is no analogue of the ansatz ansatzintro that will lead to a system of PDEs near $\mathcal Z$ that is meaningfully simpler than the full Seiberg-Witten equations.

Despite these differences, our approach still relies heavily on a very close analogue of the two-dimensional model ODE solutions, as we now explain.

1.2.3. Our Approach

Given the above, one must abandon the hope of finding explicit model solutions and instead turn to abstract methods. One reliable abstract method is the Implicit Function Theorem (IFT), and in fact, as explained momentarily, any other method would be redundant. Our use of the IFT here relies on the following observation: although the solution to the local PDE near the singular set cannot be found explicitly, its leading order term must be given by the $t$-parameterized family of 2-dimensional model solutions on the normal planes. These are not solutions, and in fact the error from being a solution does not approach 0 in $L^2$ as $\e\to 0$. Yet, surprisingly, it is sufficiently small that with the correctly weighted function spaces the IFT can correct these to true model solutions. Of course, applying the IFT requires analyzing the linearization at these, which has the same shortcoming as described above: this family of linearizations degenerates to an operator which is not Fredholm.

Thus our approach produces model solutions in two steps. The first is to first introduce a $t$-parameterized family of 2-dimensional model solutions which smoothes the $\Z_2$-harmonic spinor to a “de-singularized” pair $(\Phi^{h_\e}, A^{h_\e})$. After this, an application of the IFT corrects them to the desired 3-dimensional model solutions, proving Theorem <ref>. With these model solutions in hand, one forms approximate solutions on the closed manifold by introducing a cut-off function, and the global gluing argument proceeds from there. Schematically, the steps of the gluing are

node distance=3.5cm, auto
(C)[right of=A]$(\Phi^{h_\e}, A^{h_\e})$;
(D)[right of=C]$(\Phi_\e^{\text{mod}}, A_\e^{\text{mod}})$;
(E)[right of=D]$(\Phi_\e^{\text{app}}, A_\e^{\text{app}})$;
(F)[right of=E]$(\Phi_\e, A_\e).$;
[->, decorate] (A) to node $\text{de-sing.}$ (C);
[->, decorate] (C) to node $\text{correct}$ (D);
[->, decorate] (D) to node $\text{cutoff}$ (E);
[->, decorate] (E) to node $\begin{matrix} \text{gluing} \\ \text{iteration}\end{matrix}$ (F);

where the first two steps are accomplished in the present work, and the last two relegated to the sequel [26] as explained in the introduction.

A key advantage of this approach is that it proves Theorems  <ref> and <ref> simultaneously.  The proofs both rely on the study of the degenerating family of linearizations at the de-singularized pair $(\Phi^{h_\e}, A^{h_\e})$. This study extends across Sections <ref>–<ref> and culminates in Theorem <ref>, which describes  the invertibility of this family of operators. Theorems <ref> and <ref> then follow immediately from Theorem <ref>.   While other approaches to Theorem 1.2 (such as finding more explicit local solutions) might be possible, any such approach would be redundant, since Theorem <ref> is needed anyway to establish Theorem <ref>.

§.§ Outline

Section 2 introduces background material and provides an overlay of technical statements on what was said in the introduction. Section 2.1 gives the precise statement of the convergence theorem of Haydys-Walpuski. Sections 2.2 give relevant linear algebra constructions, and Section 2.3 gives a more precise definition of $\Z_2$-harmonic spinors which arises from a version of the Haydys correspondence. Section 2.4 states the Weitzenböck formula for the linearized equations, which is used later.

Section 3 covers some basic properties of the singular Dirac operator Z2harmonicpreliminary. Section 3.1 covers its semi-Fredholm properties, and Section 3.2 establishes local forms for $\Z_2$-harmonic spinors which are of key importance. Relying on these local forms, Section 3.3 is devoted to the topological question of how to reconstruct the $\text{Spin}^c$ structure in  part (i) of Theorem <ref>.

Section 4 constructs the de-singularized configurations and estimates their failure to be true solutions. Section 4.1 reviews the dimensionally reduced problem, which is essentially identical to the corresponding problem for Hitchin's equations found in [19]. Section 4.2 extends these to the parameterized ODE case, and Section 4.3 contains the error calculation.

Section 5 begins the analysis of the linearized equations at the de-singularized configurations. Section 5.1 defines the relevant function spaces. Section 5.2 defines a model operator given by the situation where metric near $\mathcal Z_0$ is Euclidean. In Section 5.3 it is shown that the $\e$-parameterized family of model operators on the planes normal to $\mathcal Z$ are all re-scaling of a single $\e$-invariant operator $\mathcal N$ at the invariant scale $O(\e^{2/3})$.

Section 6 begins the bulk of the technical analysis by studying the scale-invariant normal operator $\mathcal N$. This operator can be understood via complex geometry, and viewing it via this lens makes certain properties manifest. In this section, it is found that $\mathcal N$ naturally has a two (real) dimensional kernel that cannot be perturbed away, despite the fact that the Seiberg-Witten equations on a compact 3-manifold are index 0. This kernel is the first manifestation of the infinite-dimensional cokernel that arises as $\e\to 0$. Section 6.1 and Section 6.2 provide background and review the relevant standard Fredholm theory. Sections 6.3-6.5 study the normal operator $\mathcal N$ in its holomorphic guise, and Section 6.6 provides details on the aforementioned two-dimensional kernel.

Section 7 generalizes the results of the previous section to the 3-dimensional case. This follows essentially from integration by parts and the observation that all the tangential derivatives along $\mathcal Z$ are comparatively mild. In this section, the kernel of the normal operator which is isomorphic to $\C$ is upgraded to a high-dimensional subspace which approaches $L^2(\mathcal Z;\C)$ as $\e\to 0$ to become the infinite-dimensional cokernel in the limit.  To make the integration by parts work involves setting up the quite intricate collection of boundary and projection conditions $\Pi^\mathcal L$ (cf. Theorem <ref>) for the linearized operator, which accounts for this section's length despite the simplicity of the underlying idea. Section 7.1 reviews some standard results about APS boundary conditions for Dirac operators. Section 7.2 discusses the high-dimensional subspace approaching the limiting cokernel, and Section 7.3 sets up the boundary conditions accounting for this. Sections 7.4 and 7.5 then carry out the integration by parts argument, which by that point becomes rather involved. Section 7.6 generalizes to the case of an arbitrary metric near $\mathcal Z$.

Section 8 concludes the proofs of Theorems <ref> and <ref>, which after the analysis of the linearization in Sections 5-7 are essentially immediate. The Appendices cover some calculations that would disrupt the flow of the rest of the article.

§ ACKNOWLEDGEMENTS

This article constitutes a portion of the author's Ph.D. thesis. The author is grateful to his advisors Clifford Taubes and Tomasz Mrowka for their insights and suggestions. The author would also like to thank Aleksander Doan, Andriy Haydys, Rafe Mazzeo, Rohil Prasad, and Thomas Walpuski for helpful discussions. The author is supported by a National Science Foundation Graduate Research Fellowship and by National Science Foundation Grant No. 2105512.

§ $\Z_2$-HARMONIC SPINORS AND COMPACTNESS

Let $(Y, g_0)$ be a closed, oriented Riemannian 3-manifold. Choose a $\text{Spin}^c$ structure $\frak s$ on $Y$, and let $S\to Y$ be the associated spinor bundle. We denote Clifford multiplication by $\gamma: \Omega^1(Y)\to \text{End}(S)$. Since every 3-manifold is spin, we may alternatively specify a $\text{Spin}^c$ structure by choosing a spin structure $\frak s_0$ with spinor bundle $S_0$ and taking $S=S_0\otimes_\C \mathcal L$ where $\mathcal L$ is a complex line bundle. In this second description, the $\text{Spin}^c$ structure obtained depends on the choice of $\frak s_0$.

The two-spinor Seiberg-Witten equations are an extension of the standard Seiberg-Witten equations ([22, 16]) that instead consider spinors valued in two (possibly twisted) copies of $S$. Let $E\to Y$ denote an $SU(2)$-bundle equipped with a fixed smooth background connection $B_0$. Define
$$S_E:= S\otimes_\C E,$$

and denote by $\br \ph, \psi\kt$ the real inner-product on sections of $S_E$ arising from the Hermitian inner-products on $S$ and $E$. Pairs $(\Psi, A) \in \Gamma(S_E)\otimes \mathcal A_{\mathcal L}$ consisting of a spinor in $S_E$ and a $U(1)$ connection on $\mathcal L$ are called configurations.

The Two-Spinor Seiberg-Witten Equations for configurations $(\Psi, A)$ are

\begin{eqnarray}
\slashed D_{A}\Psi&=&0 \label{SW1}\\
\star F_A +\tfrac{1}{2}\mu(\Psi, \Psi)& =& 0
\label{SW2}
\end{eqnarray}

where $\slashed D_{A}$ is the Dirac operator on $S_E$ formed using the Spin connection on $S_0$, the background connection $B_0$ on $E$, and the connection $A$ on $\mathcal L$, and $\tfrac{1}{2}\mu$ is a pointwise quadratic moment map. These equations are invariant under the action of the gauge group $\mathcal G= \text{Maps}(Y; U(1))$.

The moment map $\tfrac{1}{2}\mu: S_E\to \Omega^1(i\R)$ is given in a local orthonormal coframe $e^j$ by
$$\frac{1}{2}\mu(\Psi, \Psi)=\sum_{j=1}^3\frac{i}{2} \br \gamma(ie^j)\Psi, \Psi\kt e^j.$$

In a local trivialization $E|_U\simeq \C^2\times U$, we may write $\Psi=(\Psi_1, \Psi_2)$ as a pair of spinors in $S$ in which case $\mu(\Psi,\Psi)= \mu_\circ(\Psi_1, \Psi_1) + \mu_\circ(\Psi_2, \Psi_2)$, where $\mu_\circ$ is the moment map in the standard Seiberg-Witten equations.

§.§ Compactness Theorem

It is a well-known fact  that the moduli space of solutions to the standard Seiberg-Witten equations modulo the action of the gauge group $\mathcal G$ is compact ([22], Chapter 5). The proof of this relies on the pointwise equality

γ( μ(Ψ, Ψ)) Ψ, Ψ= 14|Ψ|^4,

which via the Weitzenböck formula for $\slashed D_{A}$ leads to an a priori bound
$$\|\Psi\|^2_{L^2}\leq \int_{Y} |s| \ dV$$
for the spinor component of solutions, where $s$ is the scalar curvature of $g$. Starting with this, the proof of compactness is a standard application of elliptic theory ([22], Sections 5.2-5.3).

For the case of the two-spinor Seiberg-Witten equations (SW1)-(SW2), there are non-zero spinors for which $\mu(\Psi,\Psi)=0$, thus no bound akin to (mubound) can hold. The consequence is that for the two-spinor Seiberg-Witten equations

$$\text{\it There may be sequences of solutions $(\Psi_i, A_i)$ such that } \|\Psi_i\|_{L^2}\to \infty.$$

Note that this $L^2$-norm is a gauge-invariant quantity. To understand the behavior of such sequences of solutions, one considers renormalizing by dividing by the $L^2$-norm. Equivalently, we “blow-up” the space of configurations by adding the sphere are infinity in $L^2(Y; S_E)$.

Thus consider re-normalized spinors to replace configurations $(\Psi, A)$ with blown-up configurations $(\Phi, A, \e)$ by setting
$$ \Phi= \e \Psi \hspace{1cm}\text{ where }\hspace{1cm} \e = \frac{1}{\|\Psi\|_{L^2}}. $$

The blown-up Seiberg-Witten Equations for a blown-up configuration $(\Phi, A,\e)\in \Gamma(S_E)\times \mathcal A(\mathcal L) \times [0, \infty)$ are

\begin{eqnarray}
\slashed D_{A}\Phi&=&0 \label{SWBlownup1}\\
\star \e^2F_A +\tfrac{1}{2} \mu(\Phi, \Phi)& =& 0 \\
\label{SWBlownup2}
\|\Phi\|_{L^2}&=&1.
\label{SWBlownup3}
\end{eqnarray}

As before, these equations are invariant under the action $\mathcal G=\text{Maps}(Y;U(1))$. Solutions with $\e\neq 0$ are solutions of the original equations (SW1)-(SW2) where $\|\Psi\|_{L^2}=\tfrac{1}{\e}$.

The upcoming theorem, due to Haydys-Walpuski [13] and building on the work of Taubes in [31], describes the limiting behavior of sequences of solutions for which the $L^2$-norm diverges. Additional regularity results were proved by Taubes [34], and Zhang [43];  a more general approach to the original result was later given by Walpuski-Zhang in [39]. The precise statement of the theorem is rather intricate, and it merits preliminary explanation.

One would naively expect that a sequence of solutions $(\Phi_i, A_i, \e_i)$ with $\e_i\to 0$ would converge to a solution of  (SWBlownup1)-(SWBlownup3) with $\e=0$, i.e. a pair $(\Phi_0, A_0)$ solving

D_A_0Φ_0=0            such that         Φ_0 ∈μ^-1(0)        and       Φ_0_L^2=1.

A version of this statement is true, but there are several caveats.

The first caveat arises from the fact that $\mu^{-1}(0)$ is not fiberwise a manifold; instead, it is singular at  the point $0\in \mu^{-1}(0)$; it is therefore unclear what it means for $\Phi_0$ to solve the equation $\slashed D_{A_0}\Phi_0=0$ at the singular locus $|\Phi|^{-1}(0)$. The second difficulty is describing the limiting process for the connection, since it no longer appears in the limiting $\e=0$ equations  (e=0equation). It turns out that the connection converges to a well-defined limit away from a second singular locus around which the energy density $|F_A|^2$ concentrates and becomes unbounded. The coupling of the equations dictates, however, that this concentration may only occur where the spinor hits the singular point of $\mu^{-1}(0)$ and these two singular loci therefore coincide. Consequently, the statement of the  convergence theorem makes reference to a singular set $\mathcal Z_0$ which plays the dual role of

* The set of $y\in Y$ for which the limiting spinor  hits the singularity, i.e. has $\Phi_0(y)=0 \in \mu^{-1}(0)$.

* The set of $y\in Y$ away from which $|F_{A_i}|^2$ remains bounded.

The following theorem makes these ideas precise. The statement given here combines the result of Haydys-Walpuski, Taubes, and Zhang referenced above.

(Haydys-Walpuski [13], Taubes [34], Zhang [43])  Let $(\Phi_i, A_i, \e_i) \in \Gamma(S_E)\times \mathcal A(\mathcal L)\times (0, \infty)$ denote a sequence of solutions to the blown-up Seiberg-Witten equations
$$\slashed D_{A_i}\Phi_i=0 \hspace{1cm} \star \e_i^2 F_{A_i}+ \tfrac{1}{2}\mu(\Phi_i,\Phi_i)=0 \hspace{1cm} \|\Phi_i\|_{L^2}=1 $$
with respect to a sequence of converging metrics  $g_i\to g_0$ on $Y$ and connections $B_i\to B_0$ on $E$. Then, either

(i) If $\limsup \e_i>0$, then $(\Phi_i, A_i, \e_i)$ converges subsequentially modulo gauge to a solution with $\e>0$.

(ii) If $\limsup \e_i=0$, there exists a triple $(\mathcal Z_0,\Phi_0, A_0)$ where

* $\mathcal Z_0 \subseteq Y$ is a closed rectifiable subset of Haudorff codimension at least 2.

* $\Phi_0$ is a spinor on $Y-\mathcal Z_0$ such that $|\Phi_0|$ extends as a continuous function to $Y$ with $\mathcal Z_0=|\Phi_0|^{-1}(0)$.

* $A_0$ is a flat connection on $\mathcal L|_{Y-\mathcal Z_0}$ with holonomy in $\Z_2$,

such that $(\Phi_0, A_0)$ satisfies the $\e=0$ version of the blown-up Seiberg-Witten equations e=0equation on $Y-\mathcal Z_0$ with respect to the metric $g_0$ and the connection $B_0$ on $E$. Furthermore, there is an $\alpha>0$ such that and after passing to a subsequence and up to gauge transformations defined on $Y-\mathcal Z_0$,

Φ_i L^2,2_locΦ_0  A_i L^1,2_locA_0  |Φ_i| C^0,α|Φ_0|
where local convergence means on compact subsets of $Y-\mathcal Z_0$.

As we will see in the next two subsections, the data of case (ii) is equivalent to that of a $\Z_2$-harmonic spinor. The main result of [24] shows that the convergence (convergencetolim) in this Theorem is $C^\infty_{loc}$ on $Y-\mathcal Z_0$.

Although $\Phi_0$ is a section of a bundle of dimension $>2$, solutions of the equations (e=0equation) are topologically constrained and do not behave generically. The stability of a singular set $\mathcal Z_0$ of Hausdorff codimension 2 follows from the main results of [5, 25, 28].

§.§ The Hyperkähler Quotient

This section explicitly identifies $\mu^{-1}(0)$ in fibers of $S_E$ and gives important linear algebra constructions (see also [5], Appendix A).

Consider the vector space
$$V=\C^2\otimes_\C \mathbb H$$

equipped with its real inner product as a model for the fibers of $S_E$. It carries a pointwise action of $U(1)$ via the first factor, and a Clifford multiplication $\gamma: \Lambda^1(\R^3) \to \End(V)$ given by
$$\gamma(dt)= \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix} \otimes Id\hspace{1cm}\gamma(dx)= \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \otimes Id\hspace{1cm}\gamma(dy)= \begin{pmatrix} 0 & i \\ i & 0  \end{pmatrix}\otimes Id. $$

where $\R^3$ is given coordinates $(t,x,y)$. A pointwise spinor $\Phi\in V$ may be written in the form Φ=  α_1
β_1 ⊗1  +   α_2
β_2 ⊗j.

In this form the pointwise moment map is given by

\begin{eqnarray}
\frac{1}{2}\mu(\Phi,\Phi) & = & \frac{i}{2}(|\beta_1|^2 + |\beta_2|^2 - |\alpha_1|^2 -|\alpha_2|^2) \ dt    \ \  \label{mut}\\
& +&   \frac{i}{2}\text{Re}( -\overline \alpha_1 \beta_1 - \overline \alpha_2 \beta_2) \ dx\label{mux}\\
& +&   \frac{i}{2}\text{Im}( -\overline \alpha_1 \beta_1 - \overline \alpha_2 \beta_2) \ \label{muy}dy.
\end{eqnarray}

Notice that the sign convention here differs from many authors since we have written the Seiberg-Witten equations as $\star F_A + \mu=0$ rather than $\star F_A = \mu$. It is easy to check the under the identification $(\R^3)^* = \text{Im}\mathbb H$ given by $dt \mapsto I, dx\mapsto J, dy\mapsto K$, the map $\tfrac{1}{2}\mu$ is indeed the hyperkähler moment map associated to the $U(1)$ action, justifying the name.

We can identify $V\simeq \text{End}(\C^2; \C^2)$ so that Phiform is written as the matrix Φ_0=  α_1     α_2
β_1     β_2 .

Under the above isomorphism,
$$\mu^{-1}(0) \ \simeq \ \text{Cone}(U(2)).$$

In particular, it is a smooth 5-dimensional manifold away from $0\in V$.

In terms of the matrix (matrixPhi1), the second and third moment map equations  (mux)-(muy) show that the columns are orthogonal in the Hermitian metric, and the first equation (mut) requires that the rows have the same norm. Thus the matrix is a possibly 0 multiple of a unitary matrix.

Next, we establish the form of the hyperkähler quotient orbifold $\mu^{-1}(0)/U(1)$. To do this, we construct slices for the $U(1)$ action. We will show, in fact, that there is a global slice for the action up to a stabilizer of $\Z_2=\{\pm 1\}$. To begin, each factor of $V=\C^2 \otimes_\C \mathbb H$ carries a complex anti-linear involution, denoted by $J: \C^2 \to \C^2$ and $j:\mathbb H\to \mathbb H$ respectively, such that $J^2 = j^2=-1$. Explicitly, these are

β: = -β
α j(q):= qj.

Together these give rise to a real structure $\tau:V\to V$ satisfying $\tau^2=Id$ given by
$$\tau:= J\otimes j.$$

We denote by $V^\text{Re}$ and $V^\text{Im}$ the $+1$ and $-1$ eigenspaces of $\tau$ respectively. So that

\begin{eqnarray}
V^\text{Re}&=& \{ \psi\otimes 1 + J\psi \otimes j \ | \ \psi \in \C^2 \}\label{SRe} \\
V^\text{Im}&=& \{\psi\otimes 1 - J\psi \otimes j \ | \ \psi \in \C^2 \}.
\label{SIm}
\end{eqnarray}

The subspace $V^\text{Re}$ provides a global slice for the $U(1)$ action up to a $\Z_2$-stabilizer. That is,
$$V^\text{Re}\subset \mu^{-1}(0)$$

and each $U(1)$-orbit intersects $V^\text{Re}$ in two point which differ by multiplication by $-1$. Consequently, the hyperkähler quotient is given by
$$\mu^{-1}(0)/U(1) \simeq \mathbb H/\Z_2.$$

A pointwise spinor $\Phi \in V^\text{Re}$ has the form Φ= α_1
β_1 ⊗1 +  -β_1
α_1  ⊗j

which automatically satisfies (mut)-(muy). Moreover, if $e^{i\phi}\Phi\in V^\text{Re}$ is another element in the same $U(1)$ orbit in $V^\text{Re}$ it must also be of the form (Sre). Since
$$e^{i\phi}\Phi= e^{i\phi}\begin{pmatrix}\alpha_1 \\ \beta_1 \end{pmatrix}\otimes 1 +e^{i\phi}  \begin{pmatrix}-\overline \beta_1 \\ \overline \alpha_1  \end{pmatrix}\otimes j =\begin{pmatrix}e^{i\phi}\alpha_1 \\ e^{i\phi}\beta_1 \end{pmatrix}\otimes 1 +e^{2i\phi}  \begin{pmatrix}-\overline {e^{i\phi}\beta_1} \\ \overline {e^{i\phi}\alpha_1}  \end{pmatrix}\otimes j$$

this implies $e^{2i\phi}=1$ hence $e^{i\ph}=\pm 1$.

To complete this section we note one more simple lemma, which is central to many arguments in this paper.  Let V=V^Re⊕V^Im

denote the decomposition from (SRe)-(SIm). The linearization of $\mu$ at $\Phi$ is given by its polarization, which we denote $\mu(-,\Phi)$ . We may extend this to a map $\mu(-,\Phi): V\to (\Lambda^0\oplus \Lambda^1)(i\R)$ by redefining μ(Ψ,Φ) ⇒(-iiΨ,  Φ ,  μ(Ψ, Φ))

where the previous definition now constitutes the 1-form component. Notice polarizing cancels the factor of $\tfrac{1}{2}$. Similarly, we extend Clifford multiplication to $(\Lambda^0\oplus \Lambda^1)(\R^3)$ by scalar multiplication in the first factor.

The following statements hold:

(A)  Clifford multiplication by real and purely imaginary forms
$$\gamma: \Lambda^0(\R)\oplus \Lambda^1(\R) \to \text{End}(V) \hspace{2cm}\gamma: \Lambda(i\R)\oplus \Lambda^1(i\R) \to \text{End}(V)$$

respectively preserve and reverse the splitting (Vsplitting).

(B) If $\Phi \in V^\text{Re}$ is non-zero, then
$$V^\text{Re}=\{\gamma(b)\Phi \ | \ b \in \Lambda^0(\R)\oplus \Lambda^1(\R)\} \hspace{1cm} \text{and}\hspace{1cm} V^\text{Im}=\{\gamma(a)\Phi \ | \ a \in \Lambda^0(i\R)\oplus \Lambda^1(i\R)\}$$

(C) If $\Phi \in V^\text{Re}$ then
$$\ker(\mu(-,\Phi))= V^\text{Im}$$

and the reverse for $\Phi\in V^\text{Im}$.

For (A), simply note that Clifford multiplication by real forms commutes with $J$, hence with $\tau$ and preserves the splitting. For purely-imaginary forms, it anti-commutes by the anti-linearity of $J$. This implies (B) since $\gamma(e^j)\Phi$ is orthogonal to $\gamma(e^k)\Phi$ in the real inner product for $j\neq k$ ranging over $j,k=0,1,2,3$. (C) follows in turn from (B) since $\mu(-,\Phi)$ and $\gamma(-)\Phi$ are adjoints.

§.§ The Haydys Gauge

In the $\e=0$ limit of
the blown-up Seiberg-Witten equations (SWBlownup1)-(SWBlownup2) the variables are $\Phi_0$ and $A_0$, but $A_0$ no longer satisfies an elliptic equation. There is a special choice of gauge, however, that effectively eliminates $A_0$ as a variable. This allows the limiting $\e=0$ equation to be reinterpreted as an equation for only $\Phi_0$ that is elliptic on $Y-\mathcal Z_0$ (though not uniformly so). This gauge is a key part of the Haydys correspondence ([9, 5]) in the case that $\mathcal Z_0=\emptyset$.

Since the limiting connection $A_0$ in Theorem <ref> is flat with holonomy in $\Z_2$, it follows that (ℒ|_Y-𝒵_0)^2 ≃is trivial. Indeed, it carries a flat connection whose holonomy is trivial, this being the one induced by $A_0$. It follows also that $\mathcal L|_{Y-\mathcal Z_0}$ admits a reduction of structure group to a real line bundle. More precisely,

Suppose $\mathcal L|_{Y-\mathcal Z_0}$ admits a flat connection $A_0$ with holonomy in $\Z_2$. Then

(A) The first chern class $c_1(\mathcal L|_{Y-\mathcal Z_0})$ is 2-torsion.

(B) There exists a real line bundle $\ell\to Y-\mathcal Z_0$ such that
$$\mathcal L|_{Y-\mathcal Z_0}\simeq \ell \otimes_\R \underline \C.$$

(C)The set of gauge equivalence classes of connections $A_0$ on $\mathcal L|_{Y-\mathcal Z_0}$ is a torsor on the kernel of the integral Bockstein homomorphism
$$\beta: H^1(Y-\mathcal Z_0; \Z_2)\to H^2(Y-\mathcal Z_0; \Z).$$

The short exact sequence $\Z\overset{\times 2}\to \Z\to \Z_2$ induces the long exact sequence

$$\ldots \lre H^1(Y-\mathcal Z_0;\Z)\lre H^1(Y-\mathcal Z_0;\Z_2)\overset{\beta}\lre H^2(Y-\mathcal Z_0;\Z)\overset{\times 2}\lre H^2(Y-\mathcal Z_0;\Z)\lre \ldots$$

and L^2triv shows that $c_1(\mathcal L|_{Y-\mathcal Z_0})$ is in the kernel of $\times 2$, which is (A). Exactness implies $c_1(\mathcal L|_{Y-\mathcal Z_0})$ the image of a class in $H^1(Y-\mathcal Z_0;\Z_2)$. For (B)-(C), note that flat  connections with holonomy in $\Z_2$ up to gauge (on $Y-\mathcal Z_0$) are in one-to-one correspondence with $\Z_2$-valued representations in $Hom(\pi_1(Y-\mathcal Z_0); \Z_2)\simeq H^1(Y-\mathcal Z_0;\Z_2)$, thus with real line bundles $\ell$ via $w_1(\ell)=\text{hol}_{A_0}$ under this isomorphism.
The complex line bundle whose first chern class is $c_1(\mathcal L|_{Y-\mathcal Z_0})=\beta(w_1(\ell))$ is simply $\mathcal L|_{Y-\mathcal Z_0} =\ell \otimes \C$ (this is the $\check{\text{C}}\text{ech}$ description of $\beta$). (B) therefore holds by exactness, and the set of flat $A_0$ with $\Z_2$-holonomy on a given isomorphism class of complex line bundle is in one-to-one correspondence with the fiber of $\beta$ over its first chern class, which gives (C).

Given the above lemma, we may fix an isomorphism σ: ℒ|_Y-𝒵_0 ≃  ℓ⊗_,

where $\ell$ is the real line bundle specified by the holonomy representation of $A_0$. Such a choice is only determined up to gauge transformations on $Y-\mathcal Z_0$, as composing with a gauge transformation (now thought of as acting on the $\underline \C$ factor) gives another such choice. A choice of $\sigma$ gives a reduction of structure group of the the spinor bundle $S=S_0\otimes_\C(\ell \otimes_\R \C)$ from $\text{Spin}^c$ to $SU(2)$. The auxiliary bundle $E$ has structure group $SU(2)$ by definition, hence there are global versions of the maps J: S|_Y-𝒵_0→S|_Y-𝒵_0  j: E→Efrom the previous subsection (which depend on the choice of $\sigma$). They are given in local trivializations by the same expressions Jdef which commute with the action $SU(2)$. It follows that there is a global splitting S_E|_Y-𝒵_0= S^Re⊕S^Im

which is  determined by $\sigma$. Thus this splitting is specified up to gauge transformations on $Y-\mathcal Z_0$, and the gauge equivalence classes are given by the gauge equivalence classes of $A_0$ as in Lemma <ref>.

The following lemma gives the special gauge choice advertised at the beginning of the section.
Suppose that $(\mathcal Z_0, A_0, \Phi_0)$ is a triple as in Theorem <ref>. There exists a choice of a gauge $u\in \mathcal G|_{Y-\mathcal Z_0}$ such that the following equivalent conditions holds:

(1) After replacing the isomorphism $\sigma$ by $u\circ \sigma$,
$$\sigma: \mathcal L|_{Y-\mathcal Z_0} \ \simeq  \ \ell \otimes_\R \underline \C$$
sends A_0   ↦∇^flat⊗1 + 1⊗dwhere $\nabla^\text{flat}$ is the unique flat connection with holonomy in $\Z_2$ on $\ell$.

(2) In the splitting ReImSplitting determined by the new $\sigma$, one has
$$\Phi_0 \in \Gamma(S^\text{Re}).$$

This choice of gauge is referred to as the Haydys gauge. It is unique up to the action of $\Z_2\subseteq \mathcal G|_{Y-\mathcal Z_0}$.

We construct $u$ so that (2) holds and show this implies (1). Let $\{U_\alpha\}$ be a finite open cover of $Y-\mathcal Z_0$ obtained by restricting a finite open cover on the compact $Y$ to the subspace topology. We may assume that on each $U_\alpha$ there is a trivialization
$$g_\alpha \times f_\alpha: (S_{0}\otimes_\C  \mathcal L \otimes_\C E)|_{U_\alpha}\simeq U_\alpha \times (\C^2 \otimes_\C \C \otimes_\C \mathbb H) $$

with transition functions g_αβ: U_α∩U_β    →    SU(2)×{1}×SU(2))
f_αβ    →    {1}×U(1)×{1}

for $S_0\otimes E$ and $\mathcal L$ respectively. We may assume that $g_{\alpha\beta}$ are induced by trivializations that extend to $Y$.

Let $\nabla$ denote the connection induced by the spin connection and $B$ and the product connection $\text{d}$ on the middle factor. Thus on $U_\alpha$ each we may write
$$\nabla_{A_0}= \nabla + \gamma(i a_\alpha) $$

for a connection form $a_\alpha\in \Omega^1(U_\alpha; \R)$. Locally, in each trivialization there is a real structure given by
$$\tau_\alpha:= J \otimes 1 \otimes j$$

where $J,j$ are as in Jdef. $\tau_\alpha$ do not a priori give rise to a global real structure $\tau$ as the transition functions $g_{\alpha\beta}f_{\alpha\beta}$ do not respect the $SU(2)$ structure.

By Lemma slicelemma, we may choose local gauge transformations $u_\alpha$ on each $U_\alpha$ unique up to a $\Z_2$ factor such that u_αΦ_0|_U_α∈Re(^2⊗_⊗_ℍ)since $\Phi_0\in \mu^{-1}{(0)}$ In the new trivializations $u_{\alpha} f_\alpha: \mathcal L|_{Y-\mathcal Z_0}\to U_\alpha\times \C $, the transition functions
$$f_{\alpha\beta}'= u_\alpha^{-1} u_\beta f_{\alpha\beta}$$
preserve the condition (inslice) thus by Lemma  slicelemma, we must have that $f_{\alpha\beta}'\in \{\pm 1\}.$ Let $\ell$ be the real line bundle determined by $f_{\alpha\beta}': U_\alpha \cap U_{\beta}\to \Z_2$. Since $f_{\alpha\beta}$ and $f_{\alpha\beta}'$ differ by a $\check{\text{C}}\text{ech}$ coboundary, we obtain an isomorphsim $\sigma':  \mathcal L|_{Y-\mathcal Z_0}\simeq \ell\otimes_\R \underline{\C}$, and the gauge transformations $u_\alpha$ patch to form a global gauge transformation $\sigma' \circ u = \sigma$. This yields (2). Moreover, in this gauge $J,j$ are respected by the transition functions, hence $\tau_\alpha= \tau|_{U_\alpha}$ agrees with the global structure defined by (Jdefglobal) using the trivialization $\sigma'$.

Now we show that in this trivialization $\sigma'$, item (1) holds. In the local trivialization on each $U_\alpha$ write $\nabla$ to be connection formed from $\nabla^\text{Spin}$ on $S_0$ and $B_0$ on $E$. We claim that in this trivialization, $u_\alpha A_0= \nabla$, i.e. $u_{\alpha}\cdot  (\text{d}+ia_\alpha)=\text{d}$ is the product connection on the $\C$ factor. To see this, write the Dirac equation
$$\slashed D_{A_0}\Phi_0=\left(\sum_{j=1}^3\gamma(e^j) \nabla_j + \gamma(iu_\alpha\cdot a_\alpha)\right)  u_\alpha\Phi_0=0$$

and $\nabla, \gamma$ preserve $\text{Re}(\C^2\otimes_\C \C\otimes_\C \mathbb H)$, while $\gamma(ia_\alpha)$ exchanges it with $\text{Im}(\C^2\otimes_\C \mathbb H)\otimes \C$ by part (A of Lemma splittinglem and the fact that $\nabla$ is an $SU(2)\times SU(2)$ connection hence respects $J,j$. It follows that

∑_j=1^3γ(e^j) ∇_j   (u_αΦ_0)    ∈    Re(^2⊗_ℍ)⊗

γ(iu_α·a_α)  u_αΦ_0    ∈    Im(^2⊗_ℍ)⊗

must individually vanish, implying $a_\alpha=0$ since $\Phi_0(y)\neq 0$ for $y\in Y-\mathcal Z_0$. Thus in the trivializations $f_{\alpha}'$, $A_0$ is the product connection, so globally it patches to the connection $\nabla^\text{flat}\otimes 1 + 1\otimes \text{d}$ on $\ell\otimes_\R \underline \C$ in the trivialization $\sigma'$. This shows (1). In fact, since there is always a unique gauge transformation up to constants so that (1) holds, the two statements are equivalent up to constant gauge transformations.

From now on, we fix the association  sigmadef to be one of the two determined by the Haydys gauge defined by the previous lemma. This choice subsequently fixes the splitting ReImSplitting.

The splitting $S_E=S^\text{Re} \oplus S^\text{Im}$ determined by the Haydys gauge satisfies the following.

(A) The conclusions of Lemma <ref> hold globally.

(B) The splitting is parallel with respect to $\nabla_{A_0}$. In particular, the Dirac operator splits as
$$\slashed D_{A_0}^\text{Re}: \Gamma(S^\text{Re})\to \Gamma(S^\text{Re}) \hspace{3cm}\slashed D_{A_0}^\text{Im}: \Gamma(S^\text{Im})\to \Gamma(S^\text{Im})$$

(A) is immediate from the pointwise version Lemma <ref>. For (B), note that in the Haydys gauge of Lemma <ref>, the connection formed from $A_0$ and the spin connection respects the $SU(2)$ structure, hence commutes with $J$. The connection $B_0$ on $E$ is an $SU(2)$ connection hence automatically commutes with $j$, thus $\nabla_{A_0}$ commutes with $\tau=J\otimes j$.

With the above preparation, we may give a more precise definition of $\Z_2$-harmonic spinors which refines Definition <ref> in the introduction.

Let $\mathcal Z_0\subseteq Y$ be a smooth, embedded link. Fix a real line bundle $\ell\to Y-\mathcal Z_0$, and set $\mathcal L_0:= \ell \otimes_\R \underline\C$.  Denote by $A_0$ the connection (A_0def) formed from $\nabla^\text{flat}$ on $\ell$ and the product connection as in (A_0def).  An (unoriented) $\Z_2$-harmonic Spinor is a triple $(\mathcal Z_0, A_0, \Phi_0)$ where $\Phi_0 \in \Gamma(S^\text{Re})$ satisfies

(i) $\|\Phi_0\|_{L^2}=1$

(ii)$\slashed D_{A_0}^\text{Re}\Phi_0=0 \hspace{1cm} \text{on }\hspace{.3cm} Y-\mathcal Z_0.$

(iii) $|\Phi_0|$ extends continuously to $Y$ with $\mathcal Z_0= |\Phi_0|^{-1}(0)$.

Such triples are considered up to the action of $\Z_2=\{\pm 1\}$. When $\mathcal Z_0$ is equipped with an orientation, the parenthetical descriptor is removed.

Notice that, although we have reached it in a circuitous way, this definition makes no reference to the Seiberg-Witten equations. The bundle $S=S_0\otimes_\C \mathcal L_0\simeq S_0\otimes_\R \ell$ is simply the spinor bundle associated to another spin structure on $Y-\mathcal Z$, which need not extend over $\mathcal Z_0$. Conversely, given a spinor bundle $S_1\to Y-\mathcal Z_0$ we form $S^\text{Re}\subset S_1\otimes_\C E$ as before. In fact,  it is straightforward to show that since an $SU(2)$ bundle $E$ on a 3-manifold is necessarily trivial, that $S^\text{Re}\simeq S_1$ and the only effect of introducing $E$ is a perturbation to the spin Dirac operator of $S_1$ arising from $B_0$. We also remark that the upcoming Proposition <ref> in Section 3 implies that the continuous extension in item (iii) is equivalent to the integrability condition in <ref>.

Another key point is that Definition <ref> makes no reference to a complex line bundle $\mathcal L$ such that $\mathcal L|_{Y-\mathcal Z_0}=\mathcal L_0$: the information about the isomorphism class of $\mathcal L$ is lost in the limit $\e\to 0$. There are many choices of extensions $\mathcal L\to Y$ whose restriction to $Y-\mathcal Z_0$ is isomorphic to $\mathcal L_0$, and before beginning any analysis of the gluing question one must first answer the topological question of which $\text{Spin}^c$ structure the $\Z_2$-harmonic spinor should be glued into. This is addressed in Section <ref> in the setting where Assumptions <ref>-<ref> hold.

Definition <ref> makes sense if $\mathcal Z_0$ is simply a closed, rectifiable subset of Hausdorff codimension 2. The extension of the definition of an unoriented $\Z_2$-harmonic spinor is trivial; the oriented case requires some geometric measure theory arguments (see [11]). Note that other authors generally do not assume that the definition includes an orientation of $\mathcal Z_0$. The results of [11], however, show that when a $\Z_2$-harmonic spinor arises as a limit of solutions to the Seiberg-Witten equations it carries a preferred orientation.

§.§ The Weitzenböck Formula

This section derives the Weitzenböck formula for the gauge-fixed Seiberg-Witten equations with two spinors linearized at a possibly singular configuration. This formula is the two-spinor version of the one appearing in [32] Equation 5.21.

In dimension 3, it is standard ([16]) to supplement the equations (SW1)-(SW2) by an auxiliary 0-form field $a_0\in \Omega^0(i\R)$; this extends them to an elliptic system modulo gauge. Extend Clifford multiplication to a map $\gamma: (\Omega^0\oplus \Omega^1)\to \text{End}(S)$ denoted by the same letter. The extended (Two-Spinor) Seiberg-Witten Equations for a configuration $(\Psi, A, a_0)\in \Gamma(S_E)\times \mathcal A(\mathcal L)\times \Omega^0(Y;i\R)$ are

\begin{eqnarray}
\slashed D_{A}\Psi + \gamma(a_0)\Psi&=&0\label{SWextended1} \\
\star F_A - da_0 +\tfrac{1}{2}\mu(\Psi, \Psi)& =& 0.
\label{SWextended2}
\end{eqnarray}

This system is again invariant under the action of the gauge group $\mathcal G$ (which acts trivially on $a_0$). For irreducible configurations ($\Psi$ not identically 0), integration by parts shows that $a_0=0$, thus irreducible solutions of (SWextended1)-(SWextended2) are the same as irreducible solutions of the original equations (SW1)-(SW2). For the purposes of the eventual gluing result, it suffices to only consider irreducible solutions. The extended blown-up Seiberg-Witten equations are defined analogously with the addition of the auxiliary $0$-form $a_0$ and the term $\e^2 da_0$ in the second equation.

Let $(\tfrac{\Phi}{\e}, A)\in C^\infty(Y; S_E)\times (\mathcal A(\mathcal L)\otimes \Omega^0(i\R))$ denote a smooth configuration with $\|\Phi\|_{L^2}=1$. Here, we have condensed the notation by replacing $A$ with $A+a_0$.   Differentiating a 1-parameter family of nearby configurations $(\frac{\Phi}{\e},A)+ s(\ph, a)$ shows that the linearization of the equations at $(\tfrac{\Phi}{\e},A)$ acting on the variation $(\ph,a)$ is given by

$$\d{}{s}\Big |_{s=0} SW(\tfrac{\Phi}{\e}+ s\ph, A+ sA)=\begin{pmatrix} \slashed D_A\ph + \gamma(a)\tfrac{\Phi}{\e} \vspace{.15cm}\\ \tfrac{\mu(\ph, \Phi)}{\e}+( \star d -d)a\end{pmatrix}.$$

Supplementing the pair $(\Phi,A)$ with an auxiliary $0$-form $a_0\in \Omega^0(i\R)$ and the gauge-fixing condition

$$-d^\star a -{ i\text{Re}\br i \ph,\tfrac{\Phi}{\e} \kt}=0$$

extends the linearization to the elliptic system

$$\mathcal L_{(\Phi, A,\e)}\begin{pmatrix} \ph \\ a \end{pmatrix}= \begin{pmatrix}\slashed D_A & \gamma(\_)\tfrac{\Phi}{\e} \vspace{.15cm}\\ \tfrac{\mu(\_,\Phi)}{\e} & \bold d \end{pmatrix} \begin{pmatrix} \ph   \vspace{.15cm} \\ a \end{pmatrix} \hspace{2cm}\text{where}\hspace{1cm}\bold da= \begin{pmatrix} 0 & -d^\star \\ -d & \star d \end{pmatrix}\begin{pmatrix}a_0 \\ a_1\end{pmatrix}.$$

and $a=(a_0, a_1)\in (\Omega^0\oplus \Omega^1)(i\R)$. The moment map is extended as in (extendedmu) to $\mu=(-i \br i\ph, \Phi\kt, \mu^1)$ with $\mu^1$ being the 1-form components. The reader is cautioned that because of the singular nature of the connection $A_0$, the linearization at a $\Z_2$-harmonic spinor $(\Phi_0, A_0)$ is not a bounded operator on $L^2(Y-\mathcal Z_0)$.

The Weitzenböck formula is given below. Notice that in this formula something rather miraculous has occurred: a priori one would expect the Weitzenböck formula to contain first order terms in $(\ph, a)$. The fact that these terms cancel is a special property of the Seiberg-Witten equations.

(Weitzenbock Formula)  Let $\mathcal Z_0\subseteq Y$ be a closed subset and $(\Phi_0, A_0)$ a configuration smooth on $Y-\mathcal Z_0$. Then on $Y-\mathcal Z_0$ the operator $\mathcal L_{({\Phi_0}, A_0,\e)}$ satisfies

$$ \mathcal L^\star \mathcal L_{(\Phi_0, A_0,\e)} (\ph, a)= \begin{pmatrix} \slashed D_{A_0}\slashed D_{A_0} \ph  \vspace{.1cm}\\   \bold d\bold d a \end{pmatrix}+\frac{1}{\e^2}\begin{pmatrix} \gamma(\mu(\ph, \Phi_0))\Phi_0 \vspace{.1cm} \\  \mu(\gamma(a)\Phi_0, \Phi_0)\end{pmatrix}+\frac{1}{\e}\mathfrak B (\ph,a)$$

where  the latter is the off-diagonal zeroth-order term

$$\mathfrak B(\ph , a )=\begin{pmatrix} 0 & \gamma(\epsilon \_)\slashed D_{A_0}\Phi_0 - 2\_ \cdot \nabla_{A_0} \Phi_0 \\  \epsilon \mu(\_ ,\slashed D_{A_0}\Phi_0) + 2i \br i \_ , \nabla_{A_0} \Phi_0 \kt  & 0 \end{pmatrix} \begin{pmatrix}\ph \\ a \end{pmatrix}. $$
Here, $a\cdot \nabla\Phi_0$ denotes the contraction of 1-form indices, $2i \br i\ph, \nabla \Phi_0\kt$ is the contraction of spinor components (yielding a 1-form), and $\epsilon$ acts by $(-1)^k$ on $k$-forms.

The operator $\mathcal L$ is formally self-adjoint. Expanding the expression and abbreviating $\slashed D= \slashed D_{A_0}$ and $\nabla=\nabla_{A_0}$,
ℒ^⋆ℒ_(Φ_0, A_0,)(, a)   =      D     γ(_)Φ_0
μ(_, Φ_0)     d D     γ(_)Φ_0
μ(_,Φ_0)     d

=     DD + γ(μ(, Φ_0))Φ_0^2

d d  + μ((γ(a)Φ_0), Φ_0)^2  + 1/    D( γ(a)Φ_0) + γ(d a)Φ_0

d μ(, Φ_0)+ μ(D, Φ_0)

Next, we use the following identities, which are proved in [24]:

\begin{eqnarray}
\slashed D(\gamma(a)\Phi_0) & =& - \gamma(\bold d a)\Phi_0 - \gamma((-1)^{\deg}a)\slashed D\Phi_0 - 2a \cdot \nabla\Phi_0\label{appendix1identities1}\\
\bold d\mu(\ph ,\Phi_0)&=& -\mu((\slashed D\ph), \Phi_0)  +(-1)^{\deg} \mu(\ph,  (\slashed D\Phi_0)) + 2i \br i \ph  \ , \ \nabla \Phi_0 \kt.
\label{appendix1identities}
\end{eqnarray}

Substituting these yields the formula.

In the case that $\mathcal Z_0$ is empty, i.e. if the configuration $(\Phi, A)$ is smooth, integration by parts and the above yields immediately yields the
the following $L^2$ version of the Weitzenböck formula will be used in later sections. This formula does not apply to the linearization at the singular configuration $(\Phi_0, A_0)$.
If $(\Phi, A)$ is a smooth configuration, then $\mathcal L_{({\Phi_0}, A_0,\e)}$ is formally self-adjoint and
ℒ_(Φ,A,)(, a)^2_L^2(Y) =   D_A^2_L^2(Y) + d a^2_L^2(Y) + 1/^2 γ(a)Φ^2_L^2(Y)+1/^2 μ(,Φ)^2_L^2(Y)
+ 1/ (,a )  ,  𝔅 (, a)where the inner product is in $L^2$ and $\mathfrak B$ is as above. An equivalent statement holds in the case that $(Y,\del Y)$ is a manifold with boundary, up to the addition of a boundary term.

The formal self-adjointness follows from the fact that  $\mu,\gamma$ are fiberwise adjoints. This also implies

$$ \br b , \mu(\gamma(b)\Phi, \Phi) ) \kt= \br \gamma(b)\Phi, \gamma(b)\Phi\kt= |b|^2 |\Phi_0|^2$$

for all $b$ and

$$ \br \ph  , \gamma(\mu(\ph, \Phi)\Phi ) \kt= \br \mu(\ph,\Phi),\mu(\ph,\Phi)\kt= |\mu(\ph,\Phi)|^2$$

for all $\Phi$. The expression for the $\tfrac{1}{\e^2}$ terms follows.

Notice that this cancellation of the first-order terms does not hold for the $\e$-version of the blown-up Seiberg equations unless $\e=1$. The unequal renormalization of the spinor and connection components in the blown-up equations disrupts the cancellation. It is for this reason that we prefer to work with the un-renormalized equation wherever possible.

§ THE SINGULAR DIRAC OPERATOR

Locally near a component of $\mathcal Z_0$, the Dirac operator $\slashed D_{A_0}$ takes the form

D_A_0= D   +   O(1/r)

where $\slashed D$ is the Dirac operator on $S_E$ formed using a smooth background connection that extends over $\mathcal Z_0$, and $r$ denotes the distance to $\mathcal Z_0$. In particular, the zeroth order term is unbounded on $L^2$. An equivalent viewpoint is to consider $r\slashed D_{A_0}= r\slashed D+ O(1)$ in which case the zeroth order term is bounded, but the symbol degenerates along $\mathcal Z_0$. Elliptic operators of this type are known as elliptic edge operators, and have been studied extensively in microlocal analysis dating back to the 1980s. Authoritative sources on similar operators include [27, 20, 21, 8] and the references therein.

This section gives some necessary results about the singular Dirac operator. The results are stated here without proof, and the reader is referred to [12, 27, 20] for proofs, as well as [25] which provides more detailed analysis in this particular case.  The final subsection uses these to address the topological problem of reconstructing the $\text{Spin}^c$ structure explained at the end of Section <ref>. It is instructive in understanding the operator (singularDirac) to first consider the following example.

Consider $Y=S^1\times D^2$ with coordinates $(t,x,y)$ and take $\mathcal Z= S^1\times \{0\}$. Consider the trivial bundle $\underline \C^2 \to Y$ of rank 2, and let $\ell \to Y-\mathcal Z$ be the real line bundle that restricts to the mobius bundle on $\{t\}\times \R^2$ equipped with its unique flat connection $A_0$. The spinor bundle $\underline \C^2\otimes_\R \ell$ is globally trivial (its determinant is trivial and $H^2(Y;\Z)$ has no 2-torsion here). In fact,

^2    →    ^2 ⊗_ℓ

ψ   ↦    e^iθ/2ψprovides an explicit trivialization. Indeed, $e^{i\theta/2}$ provides a nowhere-vanishing section of each factor $\underline \C \otimes \ell$ with the proper monodromy condition. In this trivialization, we may write
$$\nabla_{A_0}= \text{d}+ \frac{i}{2} d\theta= \text{d}+ \frac{1}{4}\left(\frac{dz}{z}-\frac{d\overline z}{\overline z}\right) \hspace{2.5cm} \slashed D_{A_0}= \begin{pmatrix}i\del_t & 2\del \\ 2 \delbar & -i\del_t \end{pmatrix}+\frac{1}{4}\gamma\left(\frac{dz}{z}-\frac{d\overline z}{\overline z}\right)$$

where $(r,\theta)$ are polar coordinates on the $\R^2$ factor and $z=x+iy$ and $z=x-iy$ complex ones. After decomposing a spinor in Fourier series
$$\psi= e^{ik\theta}e^{i\ell t} \begin{pmatrix}\psi^+_{k\ell}(r) \\ \psi^-_{k\ell}(r)\end{pmatrix}$$

the Dirac operator becomes a decoupled family of ODEs which may be solved using Bessel functions (see [12, 28]). One finds that ψ_ℓ^Euc= e^-iθ/2 √(|ℓ|) e^iℓt e^-|ℓ| r  1√(z)
sgn(ℓ)√(z)are an $L^2$-orthonormalized set of solutions parameterized by $\ell \in \Z^{\neq 0}$.

This example is the local model of the infinite-dimensional cokernel alluded to in the introduction (section <ref>). Notice that this phenomenon is not an artifact of the non-compactness of $Y$ as $r\to\infty$  —  these solutions concentrate exponentially near $\mathcal Z$ for large $|\ell|$.

The family of solutions (euclideancokernel) display the two following key properties:

(1) $\psi_\ell^\text{Euc} \in L^2$ but $\nabla_{A_0} \psi_\ell^\text{Euc} \notin L^2$, thus these are not $\Z_2$-harmonic spinors as defined in (Z2harmonicpreliminary).

(2) $\psi_\ell^\text{Euc}$ do not extend smoothly across $r=0$; instead they have asymptotic expansions  with half-integer powers of $r$.

In this setting there are no $\Z_2$-harmonic spinors because all solutions whose derivative is $L^2$ along $\mathcal Z$ are not $L^2$ as $r\to \infty$. These are the key properties which generalize to the case of a general 3-manifold.

§.§ (Semi)-Fredholm Theory

Returning to the setting of a general closed 3-manifold $(Y,g_0)$, let $r: Y \to \R^{\geq 0}$ denote a weight function equal to $\text{dist}(-, \mathcal Z_0)$ in a neighborhood of $\mathcal Z_0$ and bounded away from it. Consider the weighted Sobolev spaces defined by:

rH^1_e(Y-𝒵_0 ; S^Re)   :=    {  |   ∫_Y∖𝒵_0 |∇_A_0 |^2 + ||^2/r^2  dV   <∞}

L^2(Y-𝒵_0 ; S^Re)   :=    {ψ  |   ∫_Y∖𝒵_0|ψ|^2  dV  <∞}.

Here, the subscript $e$ stands for “edge”.

The next proposition follows from the general theory of [27]. It is also proved using elementary methods in Section 2 of [25].

The operator
$$\slashed D_{A_0}: rH^1_e(Y-\mathcal Z_0 ; S^\text{Re}) \lre L^2(Y-\mathcal Z_0 ; S^\text{Re}).$$
is (left) Semi-Fredholm, i.e.

* $\ker(\slashed D_{A_0})$ is finite-dimensional, and

* $\text{Range}(\slashed D_{A_0})$ is closed.

Notice that elements $\Phi\in \ker(\slashed D_{A_0})$ satisfy the integrability condition of Definition <ref>, thus this space constitutes the $\Z_2$-harmonic spinors (integrability $|\nabla_{A_0}\Phi|^2$ implies the weighted $L^2$-term is finite as well which gives the reverse inclusion). Assumption <ref> imposes the requirement that this finite-dimensional space is 1-dimensional and spanned by $\Phi_0$.

The next proposition is not explicitly needed, but it is at the heart of the analysis for the question of gluing $\Z_2$-harmonic spinors (recall <ref>). It states that the cokernel of the singular Dirac operator $\slashed D_{A_0}$ is a small perturbation of the case of Example <ref>; in particular, it is infinite-dimensional and concentrates strongly near $\mathcal Z_0$. This proposition is proved in Section 4 of [25].

There is a bounded linear isomorphism

$$ \bigoplus_{\pi_0(\mathcal Z_0)} L^2(S^1;\C) \ \oplus \ \ker(\slashed D_{A_0}|_{rH^1_e}) \ \overset{\simeq} \lre \ \text{Coker}(\slashed D_{A_0}) $$

where the direct sum is over components of $\mathcal Z_0$. It is given by the inclusion on $\ker(\slashed D_{A_0}|_{rH^1_e}) $, and on the summand corresponding to a component $\mathcal Z_j$, by the linear extension of

$$ e^{i \ell t} \ \mapsto \ \psi_{j,\ell}^\text{Re}  \ + \ \xi_{j,\ell}$$

* $\psi_{j,\ell}^\text{Re}$ is given in a local trivialization of $S_E\simeq \underline \C^2 \otimes_\C{\underline{ \mathbb H}}$ extending across $\mathcal Z_j$ by
$$ \psi_{j,\ell}^\text{Re}= \pi^\text{Re} \left(\chi\psi^\text{Euc}_\ell\otimes 1 \right)$$

where $\psi_\ell^\text{Euc}$ is given by Equation (euclideancokernel), $\pi^\text{Re}$ is the projection to $S^\text{Re}$ and $\chi$ is a cut-off function supported on a neighborhood of $\mathcal Z_j$.

* $\xi_\ell$ is a perturbation satisfying $\|\xi_{j,\ell}\|_{L^2}\leq \tfrac{C}{|\ell|}$.

§.§ Local Forms

Because of the effective degeneracy of the symbol, standard elliptic regularity fails for operators of the form ( singularDirac). The proper replacement of elliptic regularity is, as suggested by the form of (euclideancokernel), the existence of asymptotic expansions near $\mathcal Z_0$ that generalize Taylor expansions by allowing non-integral powers. Before writing these local expansions, let us choose local coordinates and express the Dirac operator using these coordinates and an appropriate trivialization.

We endow a tubular neighborhood diffeomorphic to a solid torus $N_{r_0}(\mathcal Z_j)$ of a component $\mathcal Z_j$ of $\mathcal Z_0$ with local coordinates as follows. Let $\gamma: S^1\to \mathcal Z_0$ denote an arclength parameterization of the component $\mathcal Z_j$ whose length is denoted $|\mathcal Z_j|$. When $\mathcal Z_0$ is oriented, it is assumed that $\gamma$ is chosen respecting its orientation. Next, choose a global orthonormal frame $\{n_1, n_2\}$ of the pullback $\gamma^*N\mathcal Z_0$ of the normal bundle to $\mathcal Z_0$. We require that $\{\dot \gamma, n_1, n_2\}$ is an oriented frame with respect to the orientation on $Y$.

A system of geodesic normal coordinates for $r_0 < r_{\text{inj}}$ where $r_{\text{inj}}$ is the injectivity radius of $Y$ is the diffeomorphism $S^1\times D_{r_0}\simeq N_{r_0}(\mathcal Z_j)$ for a chosen component of $\mathcal Z_0$ defined by
$$(t,x,y)\mapsto \text{Exp}_{\gamma(t)}(xn_1 + yn_2).$$

Here $t$ is the coordinate on the $S^1$ factor, which has radius normalized so that $t\in [0,|\mathcal Z_j|)$. In these coordinates the Riemannian metric $g$ can be written
$$g=dt^2 + dx^2 + dy^2 +\  [2x\frak m_x(t)  +2y\frak m_y(t) ] dt^2\  + \ [ \mu(t) y ]  dt dx \  +\  [-\mu(t)x ] dt dy \ + \ O(r^2)$$

where $\mu(t), \frak m_x(t), \frak m_y(t)$ are defined by
μ(t)   =    ∇_γ̇ n_x, n_y= -∇_γ̇ n_y, n_x

m_α(t)    =    ∇_γ̇γ̇, n_αfor $\alpha=x,y$. Given such a coordinate system, $(t,r,\theta)$ are used to denote the corresponding cylindrical coordinates, and $(t,z,\overline z)$ the complex ones on the $D_{r_0}$ factor.

We also have the following trivialization. Recall that as in Definition <ref>, we denote $\mathcal L_0=\ell\otimes_\R \underline \C$.

For each component $\mathcal Z_j$ of $\mathcal Z_0$, there exists a local trivialization
$$\sigma_j: (S_0 \otimes \mathcal L_0 \otimes E)|_{N(\mathcal Z_j)-\mathcal Z_j} \simeq (N_{r_0}(\mathcal Z_j) -\mathcal Z_j ) \ \times \ (\C^2 \otimes_\C \C) \otimes_\C \mathbb H$$

in which

* The connection $A_0$ on the middle factor is given by
$$A_0 \ = \ \text{d} + \frac{i}{2}d\theta +\epsilon_j \tfrac{i}{2}dt$$

where $\epsilon_j=0$ or $1$ depending on $\ell$ and $S_0$.

* The Dirac operator may be written
D_A_0= D + i/2γ(dθ) + d_1 + d_0where $\slashed D$ is the standard Euclidean Dirac operator, and $\frak d_1,\frak d_0$ are respectively a first order and zeroth order term satisfying

$$|\frak d_1 \psi| \leq r |\nabla \psi| \hspace{3cm} |\frak d_2 \psi| \leq C |\psi|.$$

* The anti-linear involution $J$ defined in Jdefglobal is given by $ e^{-i\epsilon_j t/2}e^{-i\theta} J_0 $ where $J_0$ is given by the expression Jdef. Consequently, a spinor $\Phi\in \Gamma(S^\text{Re})$ takes the form Φ_0 =  α
β⊗1 + e^-iθ e^-iϵ_j t  -β

First, we trivialize the middle factor $ \ell \otimes_\R\underline \C$. Fix a vector $s_0 \in (\ell \otimes_\R\underline \C )_{(0,r_0/2,0)}$ in the fiber above the point $(0, r_0/2,0) \in~N(\mathcal Z_j)$. Parallel transport using $A_0$ in the $+\theta$ and $+t$ directions defines a section $s$ with monodromy $-1$ around the meridian of $\mathcal Z_j$ and monodromy $\epsilon_j=0$ or $1$ around the longitude. The latter is determined by the line bundle $\ell$ and by the choice of spin structure $S_0$. The section $e^{i\theta/2} e^{i\epsilon_jt /2}s$ therefore defines a global nowhere-vanishing section of $\underline \C\otimes \ell$. Since $s$ is parallel by construction, in the trivialization

\begin{eqnarray}
\underline \C & \mapsto & \underline \C\otimes_\R \ell\\
f  & \mapsto & e^{i\theta/2} e^{i\epsilon_jt /2}s f\label{trivialization3}
\end{eqnarray}

the connection becomes
$${A_0}=\text{d} + \frac{i}{2}d\theta + \epsilon_j\tfrac{i}{2}dt.$$

The first bullet point follows.

The second bullet point, we extend the above trivialization to $\sigma_j$ by choosing local trivializations of $S_0$ and $E$ that extend across $\mathcal Z_j$. We may additionally specify that in the trivialization of $S_0$, the two factors of $\underline \C$ are the $\pm i$ eigenspaces of $\gamma(e^1)$ where $\{e^1, e^2, e^3\}$ is the orthonormal co-frame on $N(\mathcal Z_j)$ extending $dt,dx,dy$. In this trivialization, the connection on $S_0 \otimes \mathcal L_0 \otimes E$ is given by

$$\nabla_{A_0}=\text{d} + \frac{i}{2}d\theta + \text{b}_0 + \Gamma+ \epsilon_j\tfrac{i}{2}dt$$

where $\text{b}_0$ and $\Gamma$ are zeroth order terms arising from the connection $B_0$ and the Christoffel symbols of $g_0$ respectively. The second bullet point is then immediate from the first, where $\frak d_1$ arises from the $O(r)$ failure of $dt, dx, dy$ to be an orthonormal frame for $g_0$, and $\frak d_1$ arises from a combination of $\text{b}_0, \Gamma$ and $\epsilon_j$.

For the third bullet point, recall that $J$ is given by the local expression Jdef in local trivialization respecting the $SU(2)$ structure on $\underline \C^2 \otimes_\C (\ell \otimes_\R \underline \C)$ (which $\sigma_j$ is not).  In a system of local trivializations on a a contractible open sets $U_\alpha\subset N(\mathcal Z_j)-\mathcal Z_j$ respecting the $SU(2)$ structure (i.e. one in which the transition functions on $\ell\otimes_\R \underline \C$ are simply), $A_0$ is given in each of these trivializations simply by $\text{d}$. These differ from the trivialization trivialization3 by transition functions $e^{i\theta/2}e^{i\epsilon_j t/2}$, hence in the trivialization $\sigma_j$ constructed using  trivialization3, $J$ is given by
$$J=(e^{-i\theta/2} e^{-i\epsilon_jt /2})  J_0(e^{i\theta/2} e^{i\epsilon_jt /2}) $$

and the third bullet point follows from the complex anti-linearity of $J$.

Using these local coordinates and local trivialization, the Dirac operator Diraclocal takes the local form singularDirac and the general regularity theory of [27] applies to give local asymptotic expansions. We consider the following type of asymptotic expansion.

A spinor $\psi\in L^2(Y-\mathcal Z_0; S_E)$ is said to admit a Polyhomogenous Expansion with index set $\Z+\tfrac{1}{2}$ if
$$\psi\sim \sum_{ k\geq 0  }\sum_{m+n=k} r^{1/2} \begin{pmatrix} c_{k,m,n}(t)  z^m \overline {z}^n  \ \ \ \ \  \\ d_{k,m,n}(t)e^{-i\theta} z^m \overline {z}^n\end{pmatrix} $$

where $c_{k,m,n}(t), d_{k,m,n}(t)\in C^\infty(S^1;\mathbb H)$, and where $\sim$ denotes convergence in the following sense: for every $N\in \N$, the partial sums
$$\psi_N=   \sum_{ k\leq N  }\sum_{m+n=k} r^{1/2}\begin{pmatrix} c_{k,m,n}(t)  z^m \overline {z}^n  \ \ \ \ \  \\ d_{k,m,n}(t)e^{-i\theta} z^m \overline {z}^n\end{pmatrix}$$
satisfies the pointwise bounds

\begin{eqnarray} |\psi-\psi_N| &\leq& C_{N} r^{N+1}\label{polyhom1} \\ |\nabla_t^{\alpha}\nabla^{\beta}_{A_0}(\psi-\psi_N) |&\leq& C_{N,\alpha,\beta} r^{N+1-|\beta|}\label{polyhom2}\end{eqnarray}

for constants $C_{N,\alpha,\beta}$ determined by the background data and choice of local coordinates and trivialization. Here, $\beta$ is a multi-index of derivatives in the normal directions.

The appropriate version of elliptic regularity for $\Z_2$-harmonic spinors, which follows from [27], is the following.

Suppose that $\Phi_0\in rH^1_e(Y-\mathcal Z_0; S^\text{Re})$ is a $\Z_2$-harmonic spinor. Then $\Phi_0$ admits a polyhomogenous expansion. Thus in the trivialization of Lemma localtrivialization, $\Phi_0$ has a local expression

\begin{eqnarray} \Phi_0&\sim &\begin{pmatrix} c(t) r^{1/2}  \ \ \ \ \  \\ d(t)r^{1/2}e^{-i\theta} \end{pmatrix}\otimes 1+\begin{pmatrix} -\overline d(t) r^{1/2}  \ \ \ \ \  \\ \overline c(t)r^{1/2}e^{-i\theta} \end{pmatrix} \otimes j \label{polyhomogeneous1} \\ & &  +   \sum_{ \underset{m+n=k}{k\geq 1}  }r^{1/2}\begin{pmatrix} c_{m,n}(t)  z^m \overline {z}^n  \ \ \ \ \  \\ d_{m,n}(t)e^{-i\theta} z^m \overline {z}^n\end{pmatrix}  \otimes 1  +r^{1/2}\begin{pmatrix}-\overline d_{m,n}(t)  z^m \overline {z}^n  \ \ \ \ \  \\ \overline c_{m,n}(t)e^{-i\theta} z^m \overline {z}^n\end{pmatrix} \otimes j  \label{polyhomogeneous2}\end{eqnarray}

where $c(t), d(t),c_{k,m,n}(t), d_{k,m,n}(t) \in C^\infty(S^1;\C)$. The terms on the second line of the expression will often be abbreviated by adding $O(r^{3/2})$ to the leading order term, with the understanding that this notation refers to a collection of bounds as in (polyhom1)-(<ref>).

The existence of such an expansion is a consequence of the regularity theory in [27, 12]. The relation between the two components follows from the form of $S^\text{Re}$ Lemma localtrivialization.

The smooth functions $c(t), d(t), c_{m,n}(t), d_{m,n}(t)$ depend on the choice of frame $\{n_x, n_y\}$ made to define local coordinates in Definition <ref>. More invariantly, these are sections of $(N\mathcal Z_0)^{\otimes(1/2+m-n)}$ where $m=n=0$ for $c(t),d(t)$.

§.§ Reconstructing $Spin^c$ Structures

In this subsection, the above local forms are used to show that the $\text{Spin}^c$ structure arising from the line bundle $\mathcal L_0\to Y-\mathcal Z_0$ can be extended to one on $Y$ satisfying the conclusion of Theorem <ref>. This section therefore finishes the topological portion of Theorems <ref>-<ref>.

Since $\Phi_0$ behaves like $r^{1/2}$ along $\mathcal Z_0$, it does not extend to the closed manifold as a smooth section. A version of $\Phi_0$ squared does, however. For any section $\Phi$, let $\det \Phi \in \Gamma(Y-\mathcal Z_0; \mathcal L_0^2)$ be the section defined as follows.
The identification used in (matrixPhi1) induces a bundle isomorphism $S_E\simeq \text{Hom}(E^*, S)$ so that $\Phi$ can be regarded as a bundle map $E^*\to S$; $\det(\Phi)$ is then a section of $\det(E^*)^{-1}\otimes \det(S)=\mathcal L_0^{2}$. As in (matrixPhi1), there are local trivialization in which $\det(\Phi)$ is given by the determinant of the matrix

Φ_0 =  α_1     α_2
β_1     β_2 .

The following inequality holds pointwise for any $\Phi$:

$$\tfrac{1}{4}|\Phi|^4 \leq |\mu(\Phi,\Phi)|^2 + |\det\Phi|^2.$$

In particular, $\det(\Phi_0)$ vanishes nowhere on $Y-\mathcal Z_0$.

Suppose that $\Phi$ has the above form, and write  $\alpha=(\alpha_1, \alpha_2)$ and $\beta=(\beta_1, \beta_2)$. Using the expressions for the moment map (mut-muy) denote the $dt$ component by $\mu_\R$ and the $dx+idy$ component by $\mu_\C$. Using the Hermitian inner product on $\mu_\C$ and $\det(\Phi)$ shows:

|μ_(Φ,Φ)|^2   =    (|α_1|^2 +|α_2|^2 - |β_1|^2 +|β_2|^2)^2

=    |α|^4 +|β|^4 - 2|α|^2 |β|^2

|μ_(Φ,Φ)|^2    =     |α_1 β_1 +α_2β_2|^2

=    |α_1|^2 |β_1|^2 +|α_2|^2 |β_2|^2 + 2 Re(α_1 β_1 α_2 β_2)

|Φ|^2    =   |α_1 β_2 - α_2 β_1|^2

=     |α_1|^2 |β_2|^2 + |α_2|^2|β_1|^2 - 2 Re(α_1 β_2 α_2 β_1)

and the two real parts are negatives after conjugating. Adding these yields
$$|\Phi|^4= |\alpha|^4 + |\beta|^4 + 2|\alpha|^2 |\beta|^2 \leq |\mu_\R(\Phi,\Phi)|^2 + 4 |\alpha|^2|\beta|^2 \leq  |\mu_\R(\Phi,\Phi)|^2 + 4|\mu_\C(\Phi,\Phi)|^2 + 4|\det(\Phi)|^2.$$

The second statement follows directly from this inequality applied to $\Phi_0$; one has $\mu(\Phi_0, \Phi_0)=0$ by definition of a $\Z_2$-harmonic spinor, hence $\det(\Phi_0)$ vanishing on $Y-\mathcal Z_0$ would imply that $\Phi_0$ vanished there as well, which is forbidden by Assumption <ref>.

Using $\det(\Phi_0)$, we can now construct the $\text{Spin}^c$ structure used in the gluing.

There exists a unique $\text{Spin}^c$ structure with spinor bundle $S=S_0\otimes_\C \mathcal L$ where $\mathcal L \to Y$ is a complex line bundle such that

(i) $\mathcal L|_{Y-\mathcal Z_0}\simeq (\ell \otimes_\R \underline \C)$.

(ii) The trivialization of Lemma <ref> extend to local trivialization of $\mathcal L$ where the conclusion of Lemma <ref> continue to hold.

(iii) The determinant $\det(S)=\mathcal L^2$ satisfies c_1(ℒ^2)= - PD[𝒵_0].

Define $\mathcal L$ by extending the trivializations of Lemma <ref> on $N_{r_0}(\mathcal Z_j)$ across $\mathcal Z_j$ for every component of $\mathcal Z_0$. Thus items (i) and (ii) hold automatically. Note, however, that while the expressions for the connection form $id\theta$ and the involution $J=e^{-i\theta}J_0$ they are not smooth across the origin.

Thus it suffices to show (iii). The local trivializations for $\mathcal L$ as defined induce local trivializations $\mathcal L^2|_{N(\mathcal Z_j)}\simeq N(\mathcal Z_j)\times \C$ of $\mathcal L^2$. In these, the local expression (polyhomogeneous1)-(polyhomogeneous2) show that locally
$$\det(\Phi_0)=(|c(t)|^2 + |d(t)|^2) \overline z  \ + \ \overline zf(z,\overline z).$$

Assumption <ref> implies that $|c(t)|^2 + |d(t)|^2>0$, thus using Lemma <ref> it follows that $\det(\Phi_0)$ is a smooth section of $\mathcal L^2$ vanishing transversely along $|\det \Phi_0|^{-1}(0)= \mathcal Z_0$. Since $\overline z$ is orientation reversing, and the coordinates for the local expressions respect the orientation of $\mathcal Z_0$ by construction, (iii) follows.

Notice that the orientation convention here is opposite that in [11]. As observed in [11], the condition (c1constraint) places topological restrictions on the homology class represented by $\mathcal Z_0$. We emphasize that these restrictions follow from the existence of a $\Z_2$-harmonic spinor and do not require an additional assumption.

§.§ Restatement of Assumptions

The above proposition completes any global topological statements required for the proofs of Theorems <ref>-<ref>. The remainder of the article is analytic in nature and works exclusively in the local coordinates and trivializations constructed using Lemma <ref> and Proposition <ref>. To summarize briefly, the starting point of the local analysis is the following local expressions for the pair $(\Phi_0, A_0)$:

\begin{eqnarray}
\Phi_0&=& \begin{pmatrix} c(t) r^{1/2}  \ \ \ \ \  \\ d(t)r^{1/2}e^{-i\theta} \end{pmatrix}\otimes 1+\begin{pmatrix} -\overline d(t) r^{1/2}  \ \ \ \ \  \\ \ \overline c(t)r^{1/2}e^{-i\theta} \end{pmatrix} \otimes j+O(r^{3/2}) \label{finallocalforms1})\\
A_0 &=& \frac{i}{2}d\theta\label{finallocalforms2}
\end{eqnarray}

and Assumption <ref> in its pragmatic form is the statement that the quantity $K(t)=\tfrac{2}{3}(|c(t)|^2+ |d(t)|^2)$ is nowhere vanishing. Additionally, in a slight abuse of notation we have switched from letting $A_0$ denote the connection to letting it denote the connection form in the local trivialization. Additionally, for notational simplicity we will assume that $\mathcal Z_0$ consists of a single component and that $\epsilon_j=0$ and $B_0$ is the product connection in the trivialization for this component. It is a trivial matter at the end of Section <ref> to eliminate these restrictions.

§ DE-SINGULARIZED CONFIGURATIONS

In this section we begin main portion of the analysis required for the proof of Theorem <ref> and Theorem <ref>. As explained in the introduction, the construction requires several steps, the first of which is the “de-singularization” step

node distance=3.6cm, auto
(C)[right of=A]$(\Phi^{h_\e}, A^{h_\e})$;
[->, decorate] (A) to node $\text{de-sing.}$ (C);

of the limiting $\Z_2$-harmonic spinor to an $\e$-parameterized family of nearby smooth configuration, which we undertake in the current section.

The de-singularized configurations are a  family of solutions to an ODE parameterized by $(t,\e)\in S^1\times [0,\e_0)$. More specifically, they are the $S^1$-parameterized family of the two-dimensional fiducial solutions on planes normal to $\mathcal Z_0$ for the leading order term of $\Phi_0$. These two-dimensional fiducial solutions are exact solutions in the Euclidean metric, and first appeared in the context of Hitchin's Equations in [19, 7], though their existence may have been known to physicists before that. They are obtained from the limiting $\Z_2$-harmonic spinor by applying a singular complex gauge transformation which solves a degenerate second order ODE. Section 4.1 gives the construction of these two-dimensional fiducial solutions, and follows [19] quite closely. Section 4.2 departs from the approach of [19] and from the holomorphic setting to introduce the parameterized version which yields the de-singularized configurations. Section 4.3 calculates the size of the error term by which these fail to be true solutions.

As explained at the end of the previous section, we now fix local coordinates on $N_{r_0}(\mathcal Z_0)$ and a trivialization of $S_E|_{N_{r_0}(\mathcal Z_0)}$ in which the local expressions are given by (finallocalforms1-(finallocalforms2).

§.§ Dimensional Reducation

This subsection constructs fiducial solutions on the complex plane. Let $(\C,g_0)$ denote the complex plane equipped with the flat Euclidean metric. The spinor bundle $S_{g_0}\simeq \C \times \C^2$ is identified with the trivial $\C^2$-bundle, and $E \simeq \C \times  {\mathbb H}$ with the trivial quaternionic line bundle. Assume in this case that $B_0$ is the product connection.  We may write a configuration $(\Phi, A)\in \Gamma(S_E)\times \mathcal A_{U(1)}$ as
Φ   =    α
β=  α_1
β_1 ⊗1+  α_2
β_2 ⊗j

A    =    12(a dz - a dz  )
where $\alpha,\beta$ are $\mathbb H$-valued functions, and $\alpha_i, \beta_i, a$ are complex-valued functions, and where we associate a connection form $A$ with the connection $d+A$ in the given trivialization. To convert from the complex coordinates to the real ones, we use the isomorphism
Ω^1(i)   ≃    Ω^1,0()

i(a_x dx + a_y dy)    ↦    (a_y + i a_x)dz
on $1$-forms. Under this association (see also Section 6.1 and Section 3 of [2]), the dimensionally-reduced blown-up Seiberg-Witten equations become

\begin{eqnarray}
\begin{pmatrix} 0 & -2\del_A \\ 2\delbar_A & 0 \end{pmatrix}\begin{pmatrix}\alpha \\ \beta \end{pmatrix}&=& 0 \label{1stSWdr}\\
\mu_\C(\Phi)&=&0\label{2ndSWdr} \\
F_A + \frac{\mu_\R(\Phi)}{\e^2}&=& 0\label{3rdSWdr}
\end{eqnarray}

where $(\mu_\R, \mu_\C)=\star_3 \mu$ under the isomorphism $\R\omega \oplus \Omega^{0,1}\simeq i T^* (S^1\times \C)$ so that, explicitly,

μ_(Φ)   =   (- 1/2 ∑_j=1,2 α_j β_j) dz

μ_(Φ)   =   (- 1/2 ∑_j=1,2 |α_i|^2-  |β_i|^2) i dx∧dy.

Note also that we do not assume that $\|\Phi\|_{L^2}=1$ (in fact we won't even assume it is in $L^2$). As we are looking for local solutions which will later be transferred to the closed manifold $Y$ where the normalization is global, this is irrelevant for our immediate purposes.

The configuration space of pairs $(\Phi, A) \in \Gamma(C; S_E \oplus \Omega^{1}({i\R}))$ carries an action of the complex gauge group
$$\mathcal G^\C:= \{ e^h \ | \ h: \C \to \C\}$$

e^h\cdot (\alpha,\beta,A)\mapsto (e^h \alpha, e^{-\overline h}\beta, A+ \del \overline h -\delbar h).

The first two Seiberg-Witten equations (1stSWdr - 2ndSWdr) are invariant under the action of $\mathcal G^\C$, while the third (3rdSWdr) is invariant only under the action of the real gauge group ($h\in i\R$).

Consider a $\Z_2$-harmonic spinor which is equal to the leading order term in its asymptotic expansion, so that
Φ_0   =    cr^1/2
d r^1/2e^-iθ⊗1+  -dr^1/2

A_0    =    1/4(dz/z-dz/z)
where, $|c|^2 + |d|^2>0$. Later we will take $c=c(t)$ and $d=d(t)$ for each fixed $t\in S^1$. We look for an $\e$-parameterized family of solutions satisfying the following ansatz:

Assume that there is an $\e$-parameterized family of solutions is complex gauge equivalent to $(\Phi_0, A_0)$ via a complex gauge transformation $h_\e$ for every $\e$. Assume additionally that $h_\e=h_\e(r)$ is rotationally invariant, so that
$$(\Phi^{h_\e}, A^{h_\e}):= e^{h_\e(r)}\cdot (\Phi_0, A_0).$$

We do not assume that $h_\e(r)$ is smooth and allow the possibility that $h_\e(r)\to \infty$ as $r\to 0$.

Let $(\Phi^{h_\e}, A^{h_\e})$ be configurations satisfying Ansatz <ref> above. Then $(\Phi^{h_\e}, A^{h_\e})$ satisfy the dimensionally reduced blown-up Seiberg-Witten equations (1stSWdr-3rdSWdr) on $\C$  if and only if the gauge transformations $h_\e(r)$ satisfy the $\e$-parameterized family of ODEs

r^2 Δ_r h_(r)= 9/4K^2 r^3/2^2sinh(2h_(r)) where $\Delta_r$ is the radial part of the Laplacian $\Delta_r=\tfrac{1}{r}\del_r(r\del_r)$ and $K^2=\tfrac{2}{3}(|c|^2+ |d|^2)$.

To begin, recall the polar coordinate expressions

\begin{equation}
\delbar_z = \f{1}{2}e^{i\theta}(\del_r+\f{i}{r}\del_\theta)\hspace{1cm}
\del_z = \f{1}{2}e^{-i\theta}(\del_r-\f{i}{r}\del_\theta).
\label{delbarpolar}
\end{equation}

The gauge-transformed configurations may be written as follows, where the second expression is the definition of the function $f_\e(r)$.

Φ^h_   =     e^h_(r) c r^1/2
e^-h_(r)dr^1/2e^-iθ ⊗1+  -e^h_(r) d r^1/2
e^-h_(r)cr^1/2e^-iθ ⊗j
A^h_   =   A_0+ h_(r) -h_(r):=  f_(r) (dz/z-dz/z).

We may now substitute these expressions into the Seiberg-Witten equations (3rdSWdr). By complex gauge invariance, $\Phi^{h_\e}$ automatically satisfies the $\mu_\C=0$ equation. For the Dirac equation,  the $\otimes 1$ and $\otimes j$ components behave identically so it suffices to calculate the first. This equation becomes

D_A^h_Φ^h_   =    0     -2(+ (A^h_)_z)
2(+ (A^h_)_z)    0  ce^h_(r)r^1/2
de^-h_(r)r^1/2e^-iθ ⊗1 + …⊗j.

Focusing on the first component, this becomes

-[e^-iθ(_r-ir_θ)+2f_(r)e^-iθ/r]de^-h_(r)r^1/2 e^-iθ    =    e^iθe^h_(r)/2r^1/2( -2f_(r)+(r_r)h_(r)+12)

and the first factor is non-zero, hence $f_\e(r)$ must satisfy
$$f_\e(r)=\tfrac{1}{4} +\tfrac{1}{2}r\del_r h_\e(r).$$

and the second component and the $\otimes j$ term give the same equation.

For the third equation (3rdSWdr), we compute
μ_   =   (-i/2∑_j |α_j|^2-|β_j|^2) dx ∧dy

=    -i/2(r|c|^2 e^2h_(r)-r|d|^2 e^-2h_(r) +r |d|^2 e^2h_(r)-r|c|^2e^-2h_(r))dx ∧dy

=    -i9/4K^2 r/ 2  sinh(2h_(r)) dx ∧dy

F_A^h_   =    ((f) dz∧dzz- (f) d z∧dzz )

=    (- 12e^iθ1z_r f  -12e^-iθ1z_r f ) dz∧dz

=    (-1r_r f) dz∧dz=  (1r_r f) 2i dx∧dy.
Combining these we obtain the system of ODEs

\begin{eqnarray}
f_\e(r)&=&\frac{1}{4} +\frac{1}{2}r\del_r h_\e(r)\\
\tfrac{2}{r}\del_r f_\e(r) &=&\frac{9}{4}\frac{K^2 r }{2\e^2}\sinh(2h_\e(r))
\end{eqnarray}

and substitution the first into the second then multiplying by $r^2$ yields the proposition.

Up to a constant factor, Equation (ODEfamily) is the same equation obtained for the corresponding situation using Hitchin's equations. It is of Painlevé type and it is solved in [19], Section 3 via the following substitution. It turns out that all the $\e$-parameterized family of solutions are all re-scalings of a single invariant solution.

Let $\tau= \frac{K}{\e} r^{3/2}$ and $h_\e(r)=G(\tau)$ so that
$$r\del_r = \frac{3}{2} \tau \del_\tau$$
and Equation (ODEfamily) becomes

(τ_τ)^2 G= 1/2τ^2 sinh(2G).

This equation admits a distinguished solution which is defined by the two conditions that $h$ has an asymptote at $\tau=0$ and decays to $0$ as $\tau\to \infty$. The next below proposition collects the essential properties of this solution, which we do not prove and instead refer the reader to [19] (see Equation (25) and the accompanying discussion).

First, we change variables once more. It turns out that it is quite confusing to use the variable $\tau$ which depends non-linearly on $r$. We will instead opt for a linear scaling by replacing $\tau$ by $\rho=\tau^{2/3}$.

Define the $\e$-invariant length by
$$\rho:= \left(\frac{K}{\e}\right)^{2/3}r.$$

Then define $H(\rho):=G(\rho^{3/2})$ so that the $\e$-parameterized family of solutions are given by
$$h_\e(r)= H(\rho).$$

The essential properties of the solution are now expressed in terms of $H(\rho)$. For the proof of these properties, see Lemma 3.3 of [19] (performing the above substitution for $\rho$ into their results).

There exists a unique $\e$-parameterized family of solutions $h_\e(r)$ to (ODEfamily) such that $h_\e(r)$ has an asymptote at $r=0$ and decays to $0$ as $r\to \infty$. This family $h_\e(r)= H(\rho)$ are all dilations of a single $\e$-independent function such that $H(\rho)=G(\tau)$ solves  (scaledODE). It satisfies the following properties

* $H(\rho)$ is strictly positive and monotonically decreasing.

* $H(\rho)$ decays faster than exponentially as $\rho\to \infty$. More specifically, there are constant $C,c, \rho_0$ such that for $\rho \geq \rho_0$
$$H(\rho)\leq C \text{Exp} (-c\rho^{3/2})$$

and similarly for the derivatives of $H(\rho)$.

* At $\rho=0$ there is an asymptotic expansion of the form H(ρ)=-log(ρ^1/2∑_j=0^∞a_j ρ^2j)in which $a_0\neq 0$. In particular, up to leading order $H(\rho)= \frac{}{}\log(\rho^{-1/2})$ so that
$$ e^{H(\rho)}= \rho^{-1/2}+ O(1). $$

* The function $f_\e(r)=\tfrac{1}{4} + \tfrac{1}{2}\rho \del_\rho H(\rho)$ vanishes to second order at $r=0$, and increases monotonically to its limiting value of $\tfrac{1}{4}$. Additionally, $|\tfrac{f_\e(r)}{r} |\leq C \e^{-2/3}$ for all $r$, and the difference of $f_\e(r)$ and $f_\e'(r)$ from their limiting values decreases exponentially, i.e.
|f_(r)-14|   ≤    CρExp(-cρ^3/2)

|f_'(r)|     ≤    C ρExp(-cρ^3/2)

and similarly for the higher derivatives.

The first statement is immediate from the re-scaling above. The first through third items follow immediately from Equation (27) in [19]. The first two statement of the fourth bullet point follow from parts (a)-(c) of Lemma 3.3 in [19]. The exponential bound on $f'_\e(r)$ follows from that on $H(\rho)$ by the equation $\tfrac{2}{r}\del_r f_\e(r) =\tfrac{9}{4}\tfrac{K^2 r }{2\e^2}\sinh(2h_\e(r))$, and the one on $f'_\e(r)$ by the fundamental theorem of calculus.

The properties of the function $h_\e(r), f_\e(r)$ above translate into similar properties of the two-dimensional fiducial solutions $(\Phi^{h_\e}, A^{h_\e})$. We state these momentarily for the 3-dimensional case. The radial profiles of the two-dimensional fiducial solutions are plotted below with those of the limiting $\Z_2$-harmonic spinor:

(-45,8)$\large O(\e^{2/3})$
(190,8)$\large O(\e^{2/3})$
(-125,58)$\large O(\e^{\tfrac{1}{3}})$
(112,90)$\large O(\e^{-\tfrac{2}{3}})$
(112,132)$\large O(\e^{-\tfrac{4}{3}})$
(70,12)$r\to \infty$
(300,12)$r\to \infty$

The radial profiles of the de-singularized configurations and limiting $\Z_2$-harmonic spinor.

§.§ De-Singularization on $Y$

This subsection introduces the de-singularized configurations on the closed manifold $Y$, which are a $t$-parameterized of the 2-dimensional fiducial solutions of the previous subsection on each plane normal to $\mathcal Z_0$. Here, we work in local coordinates on a tubular neighborhood $N_{\lambda}(\mathcal Z_0)$ of radius $\lambda$ possibly depending on $\e$.

Returning to the case of full generality when $\Phi_0$ may have higher order terms, write

Φ_0   =    c(t)r^1/2
d(t) r^1/2e^-iθ⊗1+  -d(t)r^1/2
c(t) r^1/2e^-iθ⊗j   +  O(r^3/2)

A_0    =    i/2dθ= 1/4(dz/z-dz/z)

as before, and let
$$K^2(t):=\frac{2}{3}( |c(t)|^2+ |d(t)|^2).$$

Assumption <ref> requires that $K(t)$ is bounded below by a constant greater than 0 depending only on $\Phi_0$.

Define the $t$-dependent  $\e$-invariant length by
$$\rho_t:= \left(\frac{K(t)}{\e}\right)^{2/3}r,$$

and the de-singularized configurations by
$$(\Phi^{h_\e}, A^{h_\e}):= e^{\chi_\e(r)h_\e(r,t)}\cdot (\Phi_0, A_0)$$

where $h_\e(r,t)= H(\rho_t)$ and where $\chi_\e(r)$ is a cutoff function equal to 1 on a neighborhood of $r\leq \lambda(\e)$ around $\mathcal Z_0$. In the right hand side, $\cdot$ still denotes the action of the complex gauge group on the normal planes.

Notice that  since for $r\geq \lambda$, the function $h_\e(r,t)$ is exponentially small in $\e$ (provided $\lambda\geq \e^{2/3}$), the cutoff function changes it in a very minor way. With the cutoff, the de-singularized configurations extend to all of $Y$ by setting them equal to $(\Phi_0, A_0)$ outside the tubular neighborhood $N_\lambda(\mathcal Z_0)$.

The following properties are retained from the 2-dimensional version:

The de-singularized configurations satisfy the following properties:

* The configuration $(\Phi^{h_\e}, A^{h_\e})$ is smooth.

* The pair converges to the limiting configuration $(\Phi_0, A_0)$ in $C^\infty_{loc}(N_\lambda(\mathcal Z_0))$ exponentially quickly in the sense that there is a constant $c_0$ such that for $r\geq c_0 \e^{2/3}$, one has
$$\|\Phi^{h_\e}-\Phi_0\|_{C^k} \leq C_k \e^{-2k/3}\text{Exp}(-\frac{cr^{3/2}}{\e}) \hspace{1cm}\| A^{h_\e}- A_0\|_{C^k}\leq C_k r\e^{-2(k+1)/3} \text{Exp}(-\frac{cr^{3/2}}{\e})$$

* There are pointwise bounds
$$\tfrac{|\Phi^{h_\e}|}{\e}\geq c \e^{-2/3} \hspace{1cm}  |A^{h_\e}| \leq C \e^{-2/3} \hspace{1cm}|\nabla_{A_{h_\e}}\Phi^{h_\e}|\leq \frac{C}{r^{1/2}}.$$

and $|\Phi^{h_\e}|$ is monotonically increasing in $r$ for small $r$.

For smoothness in the normal directions, notice the expansion of (<ref>) from Proposition <ref> shows $H(\rho)=\log(a_0 \rho^{1/2}(1+O(\rho)^2))$ where $O(\rho^2)$ contains only even powers and is therefore smooth. It follows that $e^{h_\e}= r^{1/2}(1+O(r^2))$ where the $O(r^2)$ is also smooth, thus the leading order term of $\Phi^{h_\e}$ is
$$\begin{pmatrix}e^{h_\e(r)} c(t) r^{1/2} \\ e^{-h_\e(r)}d(t)r^{1/2}e^{-i\theta}  \end{pmatrix}\otimes 1 + \begin{pmatrix}-e^{h_\e(r)} \overline d(t) r^{1/2} \\ e^{-h_\e(r)}\overline c(t)r^{1/2}e^{-i\theta}  \end{pmatrix}\otimes j $$

is smooth and constant at $r=0$ in the top component, and vanishes like $\overline z$ at $r=0$ in the second. The same applies to the higher order terms from Proposition <ref>  which only contain additional factors of $z^m\overline z^n$.  Similar considerations show that $\rho \sinh(2H(\rho))$ is smooth and vanishes to second order at the origin, which implies the same for $f_\e(r)$ thus $A^{h_\e}=f_\e(r) \left(\frac{dz}{z}- \frac{d\overline z}{\overline z}\right)$ is smooth and vanishes to first order. For smoothness in the $t$-directions, notice that

t e^H(ρ_t)= e^H(ρ_t) Ḥρ_t ρ̣_̣ṭt= e^H(ρ_t) Ḥρ_t 2K'(t)/3K(t)ρ_t

is again smooth across the origin since $\rho_t\tfrac{\del H}{\del \rho_t}\sim \text{const}$ is also smooth across the origin.

The second and third bullet points follow directly from rescaling the corresponding properties of $H(\rho)$ from Proposition <ref>, and using the expression (<ref>) to bound the $t$-derivatives.

§.§ Calculation of Error

Denote by $E^{(0)}_\e$ the error by which the de-singularized configurations fail to solve the (un-renormalized) Seiberg-Witten equations. That is,

$$SW(\tfrac{\Phi^{h_\e}}{\e}, A^{h_\e})= E^{(0)}_\e.$$

The superscript is present to indicate that this is initial error in an eventual iteration process.

Let $\gamma<<1$ be a small positive constant. There is an $\e_0$ such that for $\e<\e_0$, the error $E^{(0)}_\e$ satisfies
$$\|E^{(0)}_\e\|_{L^2(Y)}\leq C\e^{-\gamma},$$

where $C$ is constant independent of $\e$. Moreover, the error is exponentially concentrated along $\mathcal Z_0$ in the sense that there is a constant $c_0$ such that for $r\geq c_0\e^{2/3}$ there is a pointwise bound
$$\|E^{(0)}_\e\|_{C^0}\leq \frac{C}{\e^2}\text{Exp}(-\frac{cr^{3/2}}{\e}).$$

One should think of, say, $\gamma=10^{-6}$. The purpose of this small constant is so that in the region where $r\geq \e^{2/3-\gamma}$, the difference of the de-singularized configurations from the limiting $\Z_2$-harmonic spinor $(\Phi_0, A_0)$ is exponentially small. It could just as easily be a power of $\log(\e)$, and is in fact probably not necessary at all, but the proof is more intricate.

The second statement is immediate from the exponential decay properties in the second bullet point of (desingularizedproperties) and the fact that $(\Phi_0, A_0)$ solves the Seiberg-Witten equations in the region $r\geq c_0 \e^{2/3}$.

Write $E_\e^{(0)}= (E_\e', E_\e'')$ for the spinor and form components of the error respectively, so that

D_A^h_ Φ^h_    =    E_'

⋆F_A^h_ + μ(Φ^h_, Φ^h_)/^2   =    E_”.

We calculate the error in two regions, the “interior” region $\text{Int}=\{r\leq \e^{2/3-\gamma'}\}$ where $\gamma'=\gamma/10$ for $\gamma$ as in the statement of the lemma, and the “exterior” region  $Y-\text{Int}$ where $r\geq \e^{2/3-\gamma'}$, so that
$$\|E_{\e}^{(0)}\|_{L^2}\leq \|E_{\e}^{(0)}\|_{L^2(\text{Int})}+ \|E_{\e}^{(0)}\|_{L^2(Y-\text{Int})}.$$

The boundedness in the exterior region is immediate given the exponential decay from the first sentence of the proof. For the interior region, the triangle inequality implies that:

E_'_L^2(Int)     =    D_A^h_Φ^h__L^2(Int)
≤     D^0_A^h_Φ^h__L^2(Int)+   (D-D^0)Φ^h__L^2(Int)+ cl(B_0)Φ^h__L^2(Int) +   (cl-cl^0)A^h_Φ^h__L^2(Int)

E_”_L^2(Int)    =     ⋆F_A^h_ + μ(Φ^h_, Φ^h_)^2_L^2(Int)
≤     ⋆_0 F_A^h_ + μ^0(Φ^h_, Φ^h_)^2_L^2(Int)+    (⋆-⋆_0) F_A^h_ _L^2(Int) +  (μ-μ^0)(Φ^h_, Φ^h_)^2_L^2(Int)
where $\slashed D^0, \star_0, \mu^0, \text{cl}^0$ denote the Dirac operator, hodge star, moment map, and Clifford multiplication in the product metric.

Investing the first term, recall that by definition of the de-singularized solutions these solve the leading order term in the product metric for the 2-dimensional Dirac operator, hence

D^0_A^h_Φ^h__L^2(Int)    ≤    σ_t _t (Φ^h_)_L^2(Int)+  (D^0_A^h_)^(Φ^h_)^h.o._L^2(Int)
where $(\Phi^{h_\e})^{h.o.}$ denotes the desingularization of the $O(r^{3/2})$ terms, and $\slashed D^\C$ denotes the two-dimensional Dirac operator (1stSWdr).  Since there are bounds $|\Phi|, |\del_t \Phi_0|< C r^{1/2}$,  the first of these terms is bounded by
≲    1/(∫_S^1(∫(|_t e^H(ρ_t)| + |_t e^-H(ρ_t)|)^2 |Φ_0|^2  + (e^H(ρ_t)+ e^-H(ρ_t))^2  |_t Φ_0|^2 ) r dr dθ) dt )^1/2

≲    1/(∫_S^1(∫(|_t e^H(ρ_t)| + |_t e^-H(ρ_t)|)^2r  + (e^H(ρ_t)+ e^-H(ρ_t))^2  r ) r dr dθ) dt )^1/2

≲     1/(∫_S^1 (/K)^2(∫_ρ_t≤C^-γ'  (e^H(ρ_t)+ e^-H(ρ_t))^2   | Ḥρ_t2K̇(t)/3K(t)ρ_t|^2ρ_t^2 dρ_t dθ) dt )^1/2

≲    Const
since the integrand is a bounded function of $\rho_t$ at the origin (by the same reasoning as in the proof of Proposition <ref>) and decays exponentially as $\rho_t\to \infty$. For the second term, using complex gauge-invariance, one has

≲   1/( ∫_S^1 (∫_r ≤^2/3-γ' | e^H(ρ_t) +e^-H(ρ_t)|^2 |∇_A_0 (Φ_0)^h.o.|^2 rdr dθ)dt )^1/2

≲    1/( ∫_S^1 (/K(t))^2 (∫_ρ_t≤^-γ' | e^H(ρ_t) +e^-H(ρ_t)|^2 ρ_t^2  d ρ_t dθ)dt )^1/2

≲    C ^-γ
since $|\nabla_{A_0}(\Phi_0)^{h.o.}|\leq C r^{1/2}$.

The other terms are similar using the fact that $g-g^0=O(r)$ where $g^0$ is the product metric: for each, one rescales to the $\rho_t$ coordinates, collects powers of $\e$ and observes that the rescaled integrand is a fixed integrable function of $\rho_t$.

§ RE-SCALING

To prove Theorem <ref> we must correct the de-singularized configurations to fiducial solutions on a tubular neighborhood $N_{\lambda(\e)}(\mathcal Z_0)$ we must solve the non-linear equation  (ℒ_(Φ^h_, A^h_, ) + Q)(_, a_) = E^(0)_.

The solvability of this equation—and therefore the conclusion of Theorem <ref>—follows from a standard application of the Implicit Function Theorem after showing a version of the following statement:

Theorem: In the proper context, the linearization of the Seiberg-Witten Equations at the de-singularized configurations $\mathcal L_{(\Phi^{h_\e}, A^{h_\e}, \e)}$ is invertible, and the norm of the inverse is suitably controlled as $\e\to 0$.

The precise version of this theorem, wherein the meaning of “proper context” and “suitably controlled” are clarified, is given in Theorem <ref> in Section 7.

The remainder of the paper is devoted to the set-up and proof of Theorem <ref>. In the present section, we define weighted Sobolev spaces which provide the functional-analytic setting for the theorem, and begin the study of the the linearization in the model case that the metric on $Y$ is a product near $\mathcal Z_0$.

§.§ Function Spaces

Let $N_{\lambda}(\mathcal Z_0)$ be the tubular neighborhood of $\mathcal Z_0$ of radius $\lambda$. Eventually, $\lambda=\lambda(\e)$ will depend on the parameter $\e$ as in the statements of Theorem <ref>-<ref>. .

We now define a family of weighted Sobolev spaces, which naturally arise from the Weitzenböck formula in Section <ref>. To this end, let $R_\e$ denote a weight function given by R_= √(κ^2 ^4/3 + r^2)

where $r=\text{dist}(-, \mathcal Z_0)$ and
$$\kappa= \frac{1}{\underset{t\in S^1}{\min}  \  K(t)^{2/3}}$$

on a tubular neighborhood $N_{\lambda_0}(\mathcal Z_0)$ for some $\e$-independent constant $\lambda_0$, and smoothing off so that $R_\e= \text{constant }$ outside this neighborhood. This weight function is approximately equal to $r$ for $r>O(\e^{2/3})$, and for $r\sim \e^{2/3}$ it levels off so that it is globally bounded below by a constant times $\e^{2/3}$.   In the invariant scale, this leveling off occurs at $\rho_t=O(1)$. Furthermore, taking the minimum over $S^1$ ensures the weights satisfy
$$\left(\frac{\e}{K(t)}\right)^{4/3}\frac{1}{R_\e^2}\leq \frac{1}{{1+\rho_t^2}}$$
for every $t$.

Consider the norms on sections of $(\ph,a)\in \Gamma(S_E) \oplus (\Omega^0\oplus \Omega^1)(i\R)$ given by

\begin{eqnarray}
\|(\ph,a)\|_{H^1_{\e}}&:=& \left(\int_{N_\lambda(\mathcal Z_0)} | \nabla \ph|^2 + |\nabla a|^2 + \frac{|\ph|^2}{R_\e^2}+ \frac{|\mu(\ph, \Phi^{h_\e})|^2}{\e^2} + \frac{|a|^2 |\Phi^{h_\e}|^2}{\e^2} \ dV \right)^{1/2}\label{H1norm}\\
\|(\ph,a)\|_{L^2}&:=& \left(\int_{N_\lambda(\mathcal Z_0)} |\ph|^2 + |a|^2 \ dV \right)^{1/2}\label{L^2norm}
\end{eqnarray}

in which $\nabla$ and $dV$ denote the connection formed from the Levi-Civita connection, the spin connection, and the connection $B$ on $E$, and $dV$ the Riemannian volume form.

We then define

Let H^1_(N_λ(𝒵_0))   =    {(,a)  |  (,a)_H^1_(N_λ(𝒵_0))<∞}

L^2(N_λ(𝒵_0))   =    {(,a)  |  (,a)_L^2(N_λ(𝒵_0))<∞} denote the Hilbert spaces of sections on which the above norms are finite, equipped with the inner products arising from the polarizations of the respective norms (H1norm-L^2norm). When it is clear from the context, we will omit the reference to the domain $N_\lambda(\mathcal Z_0)$ from the notation.

Note that the operator $\mathcal L_{(\Phi^{h_\e}, A^{h_\e},\e)}$ is uniformly bounded (in $\e$) on these spaces. These norms are natural in the sense all but the middle term of the norm arise from the positive terms of the Weitzenböck formula for $\mathcal L_{(\Phi^{h_\e}, A^{h_\e}, \e)}$ by omitting the cross term $\tfrac{1}{\e}\mathfrak B$. Since $\Phi^{h_\e}\sim \Phi_0$ outside a $\rho_t\sim 1$ neighborhood of $\mathcal Z_0$, the fourth term gives the $\ph^{\text{Im}}$ component a stronger weight that $\ph^\text{Re}$.

We also have the following weighted versions of the above spaces: let $\nu \in \R$ be a real number. Then we define the $(\e,\nu)$-weighted norms by

(,a)_H^1_,ν   :=    (∫_N_λ(𝒵_0) (| ∇|^2 + |∇a|^2 + ||^2/R_^2+ |μ(, Φ^h_)|^2/^2 + |a|^2 |Φ^h_|^2/^2) R_^2ν  dV )^1/2

(,a)_L^2_ν   :=    (∫_N_λ(𝒵_0)( ||^2 + |a|^2) R_^2ν  dV )^1/2

H^1_,ν(N_λ(𝒵_0))   =    {(,a)  |  (,a)_H^1_,ν(N_λ(𝒵_0))<∞}

L^2_ν(N_λ(𝒵_0))   =    {(,a)  |  (,a)_L^2_ν(N_λ(𝒵_0))<∞} to be the spaces of sections on on which these norms are finite.

Since $N_\lambda(\mathcal Z_0)$ is compact, there is an equivalence of norms showing $H^1_\e= L^{1,2}(N_\lambda(\mathcal Z_0))$ for every $\e$, just not uniformly in $\e$.

§.§ The Model Operator

The operator $\mathcal L_{(\Phi^{h_\e}, A^{h_\e},\e)}$ can be treated as a small perturbation of the the operator in the case that the metric near $\mathcal Z_0$ is a product and $\Phi_0$ has only leading order terms. In this section we begin the study of this model case. The general case is deduced easily from this one in Section  <ref>.

Thus assume from here until Section <ref>, that for some $\e$-independent constant $\lambda_0$, the pair $(g,B)$ and $\Z_2$-harmonic spinor $(\Phi_0, A_0)$ are given by

$$g= dt^2 + dx^2 + dy^2 \hspace{2cm} B_0=\text{d} \text{ is the product connection on $\mathbb H$}$$
$$\Phi_0=\begin{pmatrix} c(t)r^{1/2} \\ d(t)r^{1/2}e^{-\theta}\end{pmatrix}\otimes 1+ \begin{pmatrix}- \overline d(t)r^{1/2} \\ \overline c(t)r^{1/2}e^{-\theta}\end{pmatrix}\otimes j \hspace{2.4cm} A_0 = \frac{i}{2}d\theta$$

in the geodesic normal coordinates and trivialization of $S_E$ on $N_{\lambda_0}(\mathcal Z_0)$. Moreover, we assume that $\chi_\e(r)h_\e(r)=h_\e(r)$ on $N_{\lambda_0}(\mathcal Z_0)$.

The operator

$$\mathcal L^{h_\e}=\begin{pmatrix} \slashed D_{A^{h_\e}}  & \mathcal \gamma(\_)\tfrac{\Phi^{h_\e}}{\e} \\ \tfrac{\mu (\_, \Phi^{h_\e})}{\e} & \bold d\end{pmatrix} $$

defined using the above data is referred to as the  model Linearization at the de-singularized configurations. It may be written as
$$\mathcal L^{h_\e}= \sigma(dt) \del_t + \mathcal N_t$$

where $\sigma$ is the principal symbol and $\mathcal N_t$ is the Normal Operator  defined by

\begin{equation}
\mathcal N_t=\begin{pmatrix} \slashed D^\C_{A^{h_\e}}  & \mathcal \gamma(\_)\tfrac{\Phi^{h_\e}}{\e} \\ \tfrac{\mu(\_, \Phi^{h_\e})}{\e} & \bold d^\C\end{pmatrix}
\label{normaloperator}
\end{equation}

where $\slashed D^\C, \bold d^\C$ are the operators on the normal disks $\{t\}\times D_{\lambda_{0}}$. Explicitly, writing $a=(a_0, a_tdt + a_x dx + a_ydy)$,

$$\slashed D_{A^{h_\e}}^\C=\begin{pmatrix} 0 & -2\del_{A^{h_\e}} \\ 2\delbar_{A^{h_\e}} & 0 \end{pmatrix}  \hspace{1cm} \bold d^{\C}=\begin{pmatrix} 0 &0 & -d^{\star}  \\ 0 &0 & d \\  -d & -\star d &  0
\end{pmatrix}\begin{pmatrix} a_0 \\ a_t  \\   a_xdx+ a_ydy \end{pmatrix}\label{dsigma}$$

where $d, \star$ now denote the operators on $D_{\lambda_0}$.

Without changing notation, we continue to use $H^1_{\e,\nu}$ and $L^2_\nu$ to refer to the Hilbert spaces defined using the data in the model case.

§.§ Re-Scaling

Since the $\e$-paramterized family of de-singularizing complex gauge transformations $h_\e(r)=H(\rho)$ for fixed $t$ are all dilations of a single $\e$-invariant function, the $\e$-parameterized family of normal operators $\mathcal N_t$ are likewise all scalings of a single $\e$-independent operator. In this subsection, we rescale the normal disks to $\mathcal Z_0$ to express $\mathcal N_t$ in terms of the scale-invariant coordinate $\rho_t$ of Definition <ref>. Throughout this subsection, we omit the $t$-dependence from the notation, and use $\widehat{\mathcal N}$ to denote the rescaled version of $\mathcal N$ and similarly for other structures.

Let r:= ( K/)^2/3λ

so that the scaling Υ_: D_r    →   D_λ
ρ   ↦     r =(/K)^2/3ρdefines an isomorphism between the disk in the rescaled coordinate $\rho$ and that in the original coordinate $r$. We consider the re-scaled disk $D_\bold r$ equipped with polar coordinates $(\rho,\theta)$, with accompanying euclidean and complex coordinates $(x', y')$ and $(w, \overline w)$ respectively. It is considered with the Euclidean metric $(dx')^2 + (dy')^2$ (as opposed to the pullback of the Euclidean metric on $D_\lambda$). Since only the normal coordinates are scaled, the $dx, dy$ components of $a$ scale as $1$-forms, while the remaining components and spinor scale as functions. Explicitly, the pullback of forms and spinors from $D_\lambda$ are related by

Υ^*_(dx)= (/K)^2/3dx'   Υ^*_((r))   =    (ρ(r)).
Υ^*_(dy)= (/K)^2/3 dy'   Υ^*_(a_0(r))   =    a_0(ρ(r))

Υ^*_(a_t(r))   =    a_t(ρ(r)).

We define the Scale-invariant Configurations by

Φ^H:= 1/K  (K/)^1/3 Υ_^* Φ^h_     =   1/K( e^Hc ρ^1/2
e^-Hd ρ^1/2e^-iθ⊗1+ -e^Hd ρ^1/2
e^-Hc ρ^1/2e^-iθ⊗j)

A^H= Υ^*(A^h_)    =    (1/4+1/2ρ_ρH(ρ))(dw/w-dw/w),

And the Scale-invariant Normal Operator by

\begin{equation}
\widehat {\mathcal N}_t:= \begin{pmatrix}\begin{pmatrix} 0 & -2\del_{A^H} \\ 2\delbar_{A^H}&0 \end{pmatrix} & \gamma(\_)\Phi^H \\ \mu(\_, \Phi^H) &
\bold d^\C \end{pmatrix}
\end{equation}

where $\del_{A^H}, \gamma, \mu, \bold d^\C$ are defined using the Euclidean metric in $(x',y')$ coordinates.

The rescaling $\Upsilon_\e$ extends to a map of sections, which is weighted to be a pointwise isometry:

$$\overline \Upsilon_\e: \Gamma( D_\lambda;  S_E \oplus (\Lambda^0 \oplus \Lambda^0\oplus \Lambda^1)(i\R))\to \Gamma( D_\bold r;  S_E \oplus (\Lambda^0 \oplus \Lambda^0\oplus \Lambda^1)(i\R))$$

$$\overline \Upsilon_\e(\ph, a_0, a_t, a_1):= \left(\Upsilon_\e^* \ph, \Upsilon_\e^* (a_0, a_t), \left( \frac{K}{\e}\right)^{2/3} \Upsilon^*_\e a_1\right)$$

so that, explicitly
(r)    ↦    (ρ)

(a_0(r), a_t(r))    ↦    (a_0(ρ), a_t(ρ) )
a_x dx + a_y dy    ↦    a_x(ρ) dx' + a_y(ρ) dy'.
This map preserves the pointwise norms since $dx, dx'$ are unit norm in the Euclidean metrics on $D_\lambda, D_\bold r$ respectively.

It's equivalent to use $\Upsilon_\e^*$ in place of $\overline \Upsilon_\e^*$ and define the Scale-invariant Normal Operator using the pullback metric in place of the Euclidean metric $g_\rho$. Re-scaling the 1-form components by hand, however, makes the operator manifestly $\e$-independent.

The Normal operator and the Scale-invariant Normal operator are related by

$$\overline \Upsilon_\e\mathcal N_t(\ph, a)= \left(\frac{K(t)}{\e}\right)^{2/3} \widehat{\mathcal N}_t(\overline \Upsilon_\e(\ph,a)).$$

Changing $\nabla_{r}$ derivatives to $\nabla_{\rho}$ one has
$$\nabla_x= \left(\frac{K}{\e}\right)^{2/3}\nabla_{x'}$$

and similarly for $y$. Likewise, for the connection, Clifford multiplication by the unit forms $dw$ and $dz$ is identical in the two Euclidean metrics, and
$$\frac{1}{z}= \left(\frac{K}{\e}\right)^{2/3} \frac{1}{w}$$
and identically for $\overline w$. Thus the diagonal terms scale as claimed. For the off-diagonal terms, one has

Φ^h_/   =    1/( e^Hc r^1/2
e^-Hd r^1/2e^-iθ⊗1+ -e^Hd r^1/2
e^-Hc r^1/2e^-iθ⊗j)
=    1/ (/K)^1/3( e^Hc ρ^1/2
e^-Hd ρ^1/2e^-iθ⊗1+ -e^Hd r^1/2
e^-Hc ρ^1/2e^-iθ⊗j)

=    (K/)^2/3Φ^H by definition of $\Phi^H$, hence the off-diagonal terms scale in the same way.

§.§.§ Scale-Invariant Hilbert Spaces

We also define a scale-invariant version of the Hilbert space $ H^1_{\e}$. Let $\bold r>>0$ continue denote the re-scaled radius as in (rescaledradius) (eventually, we will take $\bold r=O(\e^{-1/6}))$. Let $R=\sqrt{1+\rho^2}$ denote a weight function

The scale-invariant norm on sections of $S_E\oplus (\Omega^0\oplus \Omega^1)$ on $D_\bold r$  is given by

\begin{equation}
\|(\ph, a)\|_{\widehat{ H}^1(D_\bold r)}:=\left( \int_{D_\bold r} |\nabla \ph|^2 +|\nabla a|^2 + \frac{|\ph|^2}{R^2}+ |\mu(\ph,\Phi^H)|^2 + |a|^2 |\Phi^H|^2  \ dV \right)^{1/2}
\label{Hhatnorm}
\end{equation}

and the scale-invariant $L^2$ norm by

\begin{equation}
\|(\ph, a)\|_{L^2(D_\bold r)}:=\left( \int_{D_\bold r} |\ph|^2 + |a|^2  \ dV \right)^{1/2}
\label{L2hatnorm}
\end{equation}

where, $dV$ denotes the Euclidean volume form and $\nabla$ the product connection induced by the chosen trivialization defined using structures defined by the scale invariant coordinate $\rho$. As in the unscaled case, there is an equivalence of norms so that $H^1(D_\bold r)= L^{1,2}(D_\bold r)$ for every $\bold r$, just not uniformly so.

The scale-invariant norm is the two-dimensional and  scale-invariant version of the $\e$-weighted norm of $ H^1_{\e}$ in the following sense. The two-dimensional version of the $H^1_\e$-norm, i.e. assuming that all configurations and $\Phi_0$ are $t$-invariant, is given by the positive square root of

\begin{equation}\int_{D_\lambda}  |\nabla\ph|^2 + |\nabla a|^2 + \frac{|\ph|^2}{R_\e^2} + \frac{|\mu(\ph,\Phi^{h_\e})|^2}{\e^2} + \frac{|a|^2|\Phi^{h_\e}|^2}{\e^2} \ dV.\label{2dimhnorm}\end{equation}
Then $\overline \Upsilon_\e$ provides an equivalence (of $t$-invariant configurations)

$$\|\overline \Upsilon_\e (\ph,a)\|_{\widehat { H}^1}\simeq \|(\ph,a)\|_{ { H}^1_\e}.$$

i.e. the two are bounded by a universal constant times the other once $\e<<1$. The equivalence is only not an isometry because $R_\e$ was defined in Equation (Redef) to be a minimum over $t$: if we had defined $R_\e$ to be a $t$-dependent weight then the above expression for the norms is an equality.

The $L^2$ norm is not scale-invariant: instead one has
$$\|\overline \Upsilon_\e(\ph,a)\|_{L^2}= \left(\tfrac{K(t)}{\e}\right)^{2/3} \|(\ph,a)\|_{L^2}. $$

Combining this with the relation

$$ \overline{\Upsilon}_\e \mathcal N(\ph,a)=\left(\tfrac{K(t)}{\e}\right)^{2/3} \widehat{\mathcal N}(\overline\Upsilon_\e(\ph,a)) $$

from Proposition <ref>, we see the diagram

node distance=2.5cm, auto

(A)$ H^1_\e(D_\lambda)$;
(B)[below of=A]$\widehat{ H}^1(D_\bold r)$;
(C)[right of=A]$L^2(D_\lambda, dV_r)$;
(D)[right of=B]$L^2(D_\bold r, dV_\rho)$;
[->] (A) to node $\overline{\Upsilon}_\e$ (B);
[->] (A) to node $\mathcal N$ (C);
[->] (B) to node $\widehat{\mathcal N}$ (D);
[->] (C) to node $\overline{\Upsilon}_\e$ (D);

§ THE NORMAL OPERATOR

This section analyzes the scale-invariant normal operator $\widehat{\mathcal N_t}$ as a boundary value problem on disks in $\R^2$. Explicitly, writing a pair $(\ph,a)$ as $\ph=(\alpha,\beta)$ and $a=(a_0, a_tdt + a_x d\hat x + a_y d\hat y)$, the scale invariant Normal operator is given by

\begin{equation}\widehat {\mathcal N_t}(\alpha,\beta,a)= \begin{pmatrix} \begin{pmatrix} 0 & -2\del_{A^H} \\ 2\overline \del_{A^H} & 0 \end{pmatrix} & \gamma( \_ \ )\Phi^H  \\ \begin{pmatrix}\mu_\R( \_ , \Phi^H) \\ \mu_\C( \_ , \Phi^H)\end{pmatrix} & \begin{pmatrix} 0 &  -d^\star +d  \\  -d -\star d   & 0  \end{pmatrix}
\end{pmatrix} \begin{pmatrix}\alpha \\ \beta \\ a_0 + a_t dt \\ a_x d\widehat x + a_y d\widehat y \end{pmatrix}   \end{equation}

where all structures are defined using the Euclidean metric on $\R^2$.
The main result is the below Proposition <ref>, which identifies the kernel of $\widehat{\mathcal N}$ and shows that on its orthogonal complement the inverse is bounded uniformly in the scale-invariant norms. The proof of Proposition <ref> requires several steps, and parts of the proof are somewhat subtle.

The two key ingredients of the proof are a holomorphic description of $\widehat {\mathcal N}$ in the case that $a_0=a_t=0$, which gives control of the operator on disks of fixed radius, and the Weitzenböck formula, which gives control of the operator for large radii. The subtlety of the proof lies in making these ideas work in congress.
This Section is organized as follows. Section 6.1 provides some brief set-up and gives the precise statement for the properties of $\widehat{\mathcal N}$. Section 6.2 is devoted to a review of the relevant Fredholm theory for first order boundary value problems with Atiyah-Patodi-Singer boundary conditions and for polynomially weighted Sobolev spaces, and Section 6.3 gives the precise boundary conditions. Section 6.4 contains the proof in the case that $a_0=a_t=0$, and Section 6.5 completes the general case. Finally, in Section 6.6, different projection operators for the kernel of $\widehat{\mathcal N}$ are discussed to be used in the next section.

§.§ Set-up

We will impose Atiyah-Patodi-Singer boundary conditions on pairs $(\ph,a)$ on $D_\bold r$ to make $\widehat{\mathcal N}$ a Fredholm operator. The boundary conditions are that pairs $(\ph,a)$ lie in the kernel of a certain boundary projection

\begin{equation}
\widehat{\Pi}^{H^1}(D_\bold r): L^{1,2}(D_\bold r)\lre H^+\subseteq L^{1/2,2}(S^1_\bold r \ ; \ S_E \oplus (\Omega^0\oplus \Omega^1))
\end{equation}

where $H^+$ is a “half-dimensional” spectral subspace of boundary values. The precise definition of $\widehat{\Pi}^{H^1}$ will be given at the end of Section <ref> in Definition <ref>. Define the following Hilbert Spaces:

\begin{eqnarray}
\widehat{H^1}(D_\bold r)&:=&\{(\ph,a) \ | \  \ \|(\ph,a)\|_{H^1(D_\bold r)}< \infty \ \text{ and } \ \widehat{\Pi}^{H^1} (\ph,a)=0\}\label{H^12ddef} \\
L(D_\bold r) &:=& L^2 (D_\bold r \ ; \ S_E \oplus (\Omega^0\oplus \Omega^1))
\end{eqnarray}

where the first is equipped with the inner product resulting arising from the polarization of the norm (Hhatnorm), and the latter with the standard $L^2$ product. We may now state the main result of Section 6.

The operator

\begin{equation}\widehat{\mathcal N_t}: \widehat{ H}^1(D_\bold r)\lre  L^2(D_\bold r)\end{equation}

is a bounded Fredholm operator of (real) Index 2. For $\bold r$ sufficiently large, it is surjective with a kernel of real dimension 2 and the inverse on the complement of the kernel is uniformly bounded. That is,

\begin{equation}
\|(\ph,a)\|_{\widehat { H}^1} \leq C( \|\widehat{\mathcal N_t}(\ph,a)\|_{L^2}  + \|p^{\ker} (\ph,a)\|_2)
\end{equation}

holds for $C$ independent of $\bold r, t$ and $p^{\ker}$ is a projection operator to $\C$.

The presence of a non-trivial kernel merits explanation. Indeed, its appearance may, at first, be surprising since the Seiberg-Witten equations are self-adjoint in 3-dimensions, and thus in most contexts have index 0. However, since the boundary conditions imposed here are somewhat immaterial (when pasting the fiducial solutions onto a 3-manifold, they are cut off near the boundary), we are free to choose any boundary conditions we wish without affecting any eventual gluing construction, and we could have, of course, selected boundary conditions of index zero. The subtelty is that such a choice will never result in a uniform bound on the inverse.

This is an essential consequence of the geometry, and is the first manifestation of the convergence of the linearization to a non-Fredholm limit discussed in the introduction. The limiting operator in the normal planes $\slashed D_{A_0}^\C$ has two-dimensional kernel on $L^2$ which consists of elements that decay like $O(r^{-1/2})$ away from $\mathcal Z$. Since $\slashed D_{A^{h_\e}}\to \slashed D_{A_0}$ (in no precise sense, since the difference in not bounded in $L^2$), there is a two-dimensional space of configurations approaching this limiting kernel, which have similar asymptotics. But because these elements decay toward the boundary, they cannot be excluded by disallowing their boundary values; cuttong off these elements with $r^{-1/2}$ decay towards the boundary will necessarily lead to a violation of any uniform bound on the inverse. Thus we cannot use the naive index 0 boundary condition, and must instead allow boundary modes capturing these kernel elements, and project to their orthogonal complement in the correct norm. This problem becomes quite subtle in the 3-dimensional case, when the limiting operator has $\slashed D_{A_0}$ has an infinite-dimensional kernel in $L^2$ as is discussed in Section 7. The two dimensional kernel of $\widehat{\mathcal N}_t$ therefore plays an essential role in this and the following section. It is identified explicitly over the course of the proof.

Proposition <ref> combined with the relation <ref> from Proposition <ref> immediately implies the following result for the un-rescaled operator $\mathcal N_t$:

For every fixed $t_0\in \mathcal Z$, on smooth configurations $(\ph,a)$ satisfying the un-rescaled version of the boundary conditions $\Pi^{H_\e^1}(\ph,a)=0$,  the following estimate on the normal disk $\{t\}\times D_\lambda$ holds uniformly in $t,\e$.
(,a)_H^1_({t}×D_λ)≤C  𝒩_t (,a)_L^2({t}×D_λ)+ π^_t(,a)_2

Here, the left side denotes (2dimhnorm) (and does not include $\nabla_t$ terms) formed using the Euclidean norm.

The remainder of Section 6 is devoted to the proof of Proposition <ref>. Before beginning the proof,   it is convenient place the form components $( a_xd\widehat x+ a_yd \widehat y \ , \ a_0 + a_tdt)$ in a holomorphic context. There are isomorphisms

\begin{eqnarray}
\Omega^{1}(i\R)&\overset{(1)}\to &\Omega^{0,1}(\C) \hspace{2.75cm}  \Omega^0\oplus \Omega^0(i\R)\overset{(3)} \to \Omega^{1,0}(\C)\\
i(a_x d\widehat x+ a_y d\widehat y)& \mapsto &   (a_y - ia_x)d\overline w\hspace{2cm} i(a_0+a_tdt)\mapsto (a_0+ia_t)dw \label{isomorphisms}\end{eqnarray}

on the domain, and
Ω^0⊕Ω^2(i)   (2)→   Ω^1,1()   Ω^1(i) (4)→Ω^1,1()

(ih_2, ih_2 dx∧dy)    ↦     (h_1-ih_2)dw∧dw i(pdx + qdy)↦(p+iq)dw∧dw on the codomain. Setting
ζ   :=    (a_0+ia_t)dw

ω   :=    (a_y -ia_x)dw
$\widehat{\mathcal N}_t$ may be considered as an operator $\Gamma( \C^2 \oplus \C^2 \oplus \Omega^{1,0}\oplus \Omega^{0,1}) \to \Gamma(\C^2 \oplus \C^2 \oplus \Omega^{1,1}\oplus \Omega^{1,1})$
now given by

$$\widehat {\mathcal N}_t(\alpha,\beta,\zeta  ,\omega )= \begin{pmatrix} \begin{pmatrix} 0 & -2\del_{A^H} \\ 2\overline \del_{A^H} &0 \end{pmatrix} & \gamma( \ \ )\Phi^H  \\ \begin{pmatrix}\mu_\R( \ , \Phi^H) \\ \mu_\C( \ , \Phi^H)\end{pmatrix} & \begin{pmatrix} 0 & 2\del  \\  -2\delbar & 0  \end{pmatrix}
\end{pmatrix} \begin{pmatrix}\alpha \\ \beta \\ \zeta \\ \omega \end{pmatrix}   $$

where Clifford multiplication becomes
\begin{equation}\gamma(p dw,q d\overline w)= \begin{pmatrix} ip & -\overline q \\ -q &  i\overline p \end{pmatrix}\label{cliffordmult}\end{equation}
and the moment map on $\psi=(\alpha,\beta)$ is
\begin{eqnarray} \mu_\R(\psi,\Phi^H)&=& -\alpha_1\overline{\alpha}_1^H + \overline \beta_1 \beta_1^H -\alpha_2\overline{\alpha}_2^H + \overline \beta_2 \beta_2^H  \\ \mu_\C(\psi, \Phi^H)&=& -\overline \alpha_1 \beta^H_1 - \beta_1 \overline {\alpha}_1^H -\overline \alpha_2 \beta^H_2 - \beta_2 \overline {\alpha}_2^H \label{mumaps}  \end{eqnarray}
its adjoint as before. We view $\widehat{\mathcal N}_t$ in this guise for the remainder of Section 6. We also leave the $t$-dependence implicit for the remainder of the section.

§.§ Fredholm Theory

This subsection discusses Fredholm theory for Dirac operators in two different contexts: 1) as boundary-value problems with Atiyah-Patodi-Singer boundary conditions, and 2) on non-compact domains with polynomially weighted Sobolev spaces. The proof of Proposition <ref> will require both of these perspectives, as the weighted spaces are needed for estimates to be uniform in the radius $\bold r$. Since it suffices for our purposes, the discussion here is limited to the relevant cases of the operators $\del,\delbar$ on the disk; the reader is referred to [16, 18, 23] for more general discussions.

§.§.§ APS Boundary Conditions

First, we consider boundary-value problems for $\del,\delbar$. Let $D\subseteq \C$ denote the unit disk, and $L^{k,2}(D; \C)$ the standard Sobolev spaces of complex-valued functions. The continuous restriction or trace map
$$\Tr: L^{k,2}(D;\C)\to L^{k-1/2, 2}(\del D; \C)$$

gives functions well-defined boundary values for $k\geq 1$. Within the space of boundary values for $k=1$, we have the half-dimensional subspaces
H^+_[m]    =     { u∈L^1/2,2   |  u=∑_k ≥m a_k e^ik θ } ⊆L^1/2,2(D, )

H^-_[m]    =     { u∈L^1/2,2   |  u=∑_k ≤m a_k e^ik θ } ⊆L^1/2,2(D, )

of functions whose Fourier series have non-vanishing components only on the positive and negative sides of $m\in \Z$ (inclusive) respectively. We denote the projections to these spaces  by Π_[m]^±: L^1/2,2(D; )→H^±_[m]respectively.

Now consider $\delbar$ on $D$. Its (infinite dimensional) kernel consists of holomorphic functions on the disk, whose boundary values lie in $H^+_{[0]}$. The following two propositions are standard results, whose proofs can be found in [23] page 85.

The operator

\begin{equation}
(\delbar, \Pi^+_{[0]}): L^{1,2}(D;\C)\to L^{2}(D;\C)\oplus H^{+}_{[0]}
\end{equation}

is invertible, and, a fortiori, Fredholm of Index 0.

More generally,
The operator

\begin{equation}
(\delbar, \Pi^+_{[m]}): L^{1,2}(D;\C)\to L^{2}(D;\C)\oplus H^{+}_{[m]}
\label{delbarm}
\end{equation}

* (if $m>0$) empty cokernel and kernel of dimension $m$ spanned by $\{1, z,\ldots, z^{m-1}\}$.

* (if $m<0$) empty kernel and cokernel of dimension $-m$ spanned by $ \{(0, e^{-i\theta}), \ldots, (0, e^{-im\theta})\}. $