\stackMath

Gluing $\mathbb{Z}_{2}$ -Harmonic Spinors and Seiberg–Witten Monopoles on 3-Manifolds

Gregory J. Parker Department of Mathematics, Stanford University [email protected]

Abstract.

Given a $\mathbb{Z}_{2}$ -harmonic spinor satisfying some genericity assumptions, this article constructs a 1-parameter family of two-spinor Seiberg–Witten monopoles converging to it after renormalization. The proof is a gluing construction beginning with the model solutions from [41]. The gluing is complicated by the presence of an infinite-dimensional obstruction bundle for the singular limiting linearized operator. This difficulty is overcome by introducing a generalization of Donaldson’s alternating method in which a deformation of the $\mathbb{Z}_{2}$ -harmonic spinor’s singular set is chosen at each stage of the alternating iteration to cancel the obstruction components.

1. Introduction

The Uhlenbeck compactification of the moduli space of anti-self-dual (ASD) Yang-Mills instantons on a compact 4-manifold exemplifies a philosophy for constructing natural compactifications of moduli spaces that is ubiquitous in modern differential geometry. The construction has two main steps: first, K. Uhlenbeck’s compactness theorem shows that any sequence $A_{n}$ of ASD instantons subconverges either to another instanton, or to a limiting object consisting of a background instanton $A_{\infty}$ of lower charge and bubbling data $B$ [57, 58]. Second, C. Taubes’s gluing results reverse this process, showing that each pair $(A_{\infty},B)$ is the limit of smooth ASD instantons [47]. Together, these results allow the construction of boundary charts which endow the moduli space with the structure of a smoothly stratified manifold. This structure of the moduli space is the basis for the celebrated applications of Yang-Mills theory to 3 and 4-dimensional topology [7].

C. Taubes’s more recent extension of Uhlenbeck’s compactness theorem to $\text{PSL}(2,\mathbb{C})$ connections introduced a new type of non-compactness [50], which has since been shown to be quite general in 3 and 4 dimensional gauge theories. It is exhibited by most generalized Seiberg–Witten (SW) equations [15, 61], a class of equations that includes the Kapuastin–Witten equations [54, 55], the Vafa–Witten equations [53], the complex ASD equations [49], the Seiberg–Witten equations with multiple spinors [32, 52], and the ADHM Seiberg–Witten equations [65]. For these equations, a sequence of solutions need not converge but, after renormalization, subconverges either to another solution or to the limiting data of a Fueter section – a solution of a different elliptic PDE that is usually degenerate, and in many cases non-linear [8, 20, 48].

It is natural to ask whether this more subtle limiting process can be reversed by a gluing construction. An affirmative answer would provide an essential step in constructing compactifications of the moduli spaces of solutions to generalized Seiberg–Witten equations, which are expected to be necessary to study the conjectured relations of these equations to the geometry of manifolds cf. [59, 13, 63, 64, 22, 33, 23, 14]. Such a gluing result would produce, from a given Fueter section $\Phi$ , a family of solutions to the corresponding generalized Seiberg–Witten equation that converges to $\Phi$ after renormalization.

The purpose of this article is to prove a gluing result of this form in the case of the two-spinor Seiberg–Witten equations on a compact 3-manifold, where the corresponding Fueter sections are $\mathbb{Z}_{2}$ -harmonic spinors. In most cases, $\mathbb{Z}_{2}$ -harmonic spinors possess a singular set which is stable under perturbations, along which the relevant linearized operator degenerates. This degenerate operator carries an infinite-dimensional obstruction bundle, making the gluing problem considerably more challenging than most gluing problems in the literature.

Despite the presence of the infinite-dimensional obstruction bundle, the gluing can still be accomplished for a set of parameters with finite codimension. Geometrically, the infinite-dimensional obstruction arises because the location of the singular set may vary during the limiting process. This freedom plays an important role in previous work of the author [43], R. Takahashi [46], and S. Donaldson [12] on the deformation theory of $\mathbb{Z}_{2}$ -harmonic spinors. To account for it here, deformations of the $\mathbb{Z}_{2}$ -harmonic spinor’s singular set are included as an infinite-dimensional gluing parameter. This leads to an infinite-dimensional family of Seiberg–Witten equations coupled to embeddings of the singular set; the first-order effect of deforming the singular set may then be calculated by differentiating this family with respect to the embedding. The crucial result that allows the gluing to succeed is that the linearized deformations of the singular set perfectly pair with the infinite-dimensional obstruction, allowing it to be cancelled.

Once this analytic set-up is in place, the gluing is accomplished by adapting Donaldson’s alternating method [9] to the semi-Fredholm setting. The starting point is an approximate solution constructed by splicing the model solution constructed in [41] onto a neighborhood of the singular set. The gluing iteration is then a three-step cycle, which adds corrections to the approximate solution near the singular set, then away from it, and at the start of each new cycle, deforms the singular set to cancel the obstruction. The iteration is complicated by the fact that the linearized deformation operator displays a loss of regularity, which necessitates refining the approach to deforming the singular set in [43, 46, 12] by introducing specially adapted families of smoothing operators. The end result of the iteration is a family of Seiberg–Witten solutions converging to a given $\mathbb{Z}_{2}$ -harmonic spinor after renormalization. The framework developed here may also be useful for addressing other geometric problems requiring the deformation of a singular set [12, 26, 62, 11].

1.1. The Seiberg–Witten Equations

Let $(Y,g)$ be a closed, oriented, Riemannian 3-manifold, and fix a $\text{Spin}^{c}$ -structure with spinor bundle $S\to Y$ . Choose a rank 2 complex vector bundle $E\to Y$ with trivial determinant, and fix a smooth background $SU(2)$ -connection $B$ on $E$ . The two-spinor Seiberg–Witten equations are the following equations for pairs $(\Psi,A)\in\Gamma(S\otimes_{\mathbb{C}}E)\times\mathcal{A}_{U(1)}$ of an $E$ -valued spinor and a $U(1)$ connection on $\text{det}(S)$ :

	$\displaystyle\not{D}_{AB}\Psi$	$\displaystyle=$	$\displaystyle 0$		(1.1)
	$\displaystyle\star F_{A}+\tfrac{1}{2}\mu(\Psi,\Psi)$	$\displaystyle=$	$\displaystyle 0$		(1.2)

where $\not{D}_{AB}$ is the twisted Dirac operator formed using $B$ on $E$ and the Spin^c connection induced by $A$ on $S$ , $F_{A}$ is the curvature of $A$ , and $\mu:S\otimes E\to\Omega^{1}(i\mathbb{R})$ is a pointwise-quadratic map. The equations are invariant under $U(1)$ -gauge transformations. Solutions of (1.1–1.2) are called monopoles.

Unlike for the standard (one-spinor) Seiberg–Witten equations, sequences of solutions to (1.1–1.2) may lack subsequences where $\|\Psi\|_{L^{2}}$ remains bounded, thus no subsequences can converge. The renormalization procedure alluded to above simply scales this $L^{2}$ -norm to unity by setting $\Phi=\varepsilon\Psi$ where $\varepsilon=\tfrac{1}{\|\Psi\|_{L^{2}}}$ . The re-normalized equations become

$\displaystyle\not{D}_{AB}\Phi$	$\displaystyle=$	$\displaystyle 0$	(1.3)
$\displaystyle\star\varepsilon^{2}F_{A}+\tfrac{1}{2}\mu(\Phi,\Phi)$	$\displaystyle=$	$\displaystyle 0$	(1.4)
$\displaystyle\\|\Phi\\|_{L^{2}}$	$\displaystyle=$	$\displaystyle 1$	(1.5)

and diverging sequences are now described by the degenerating family of equations with parameter $\varepsilon\to 0$ . A theorem of Haydys–Walpuski [32] (Theorem 3.2 in Section 3.1) shows that sequences of solutions for which $\varepsilon\to 0$ must converge, in an appropriate sense, to a solution of the $\varepsilon=0$ equations, i.e. to a pair $(\Phi,A)$ where $\Phi$ is a normalized harmonic spinor with pointwise values in $\mu^{-1}(0)$ , which is a 5-dimensional cone. Up to gauge, such solutions are equivalent (via the Haydys Correspondence, Section 3.2) to harmonic spinors valued in a vector bundle up to a sign ambiguity. The latter are $\mathbb{Z}_{2}$ -harmonic spinors, which are the simplest non-trivial type of Fueter section.

A key feature of convergence for a sequence $(\Phi_{i},A_{i},\varepsilon_{i})$ is the concentration of curvature, which gives rise to a singular set. As $\varepsilon_{i}\to 0$ , curvature may concentrate along a closed subset $\mathcal{Z}$ of Hausdorff codimension 2, so that the $L^{p}$ -norm of $F_{A_{i}}$ diverges on any neighborhood of $\mathcal{Z}$ for $p>1$ . In fact, [24] shows that $\mathcal{Z}$ represents the Poincaré dual of $-c_{1}(S)$ in $H_{1}(Y;\mathbb{Z})$ , hence it is necessarily non-empty if the $\text{Spin}^{c}$ structure is non-trivial. Away from $\mathcal{Z}$ , the connections converge to a flat connection with holonomy in $\mathbb{Z}_{2}$ , the data of which is equivalent to that of a real Euclidean line bundle $\ell\to Y-\mathcal{Z}$ . This limiting connection and line bundle may have holonomy around $\mathcal{Z}$ that is the remnant of the curvature that has “bubbled” away. If this holonomy is non-trivial, the Dirac equation twisted by such a limiting connection is singular along $\mathcal{Z}$ ; a $\mathbb{Z}_{2}$ -harmonic spinor is, more accurately, a solution of such a singular equation on the complement of $\mathcal{Z}$ .

1.2. $\mathbb{Z}_{2}$ -Harmonic spinors

Let $(Y,g)$ be as in Section 1.1, and now fix a spin structure with spinor bundle $S_{0}\to Y$ . Let $B_{0}$ be a zeroth order, $\mathbb{R}$ -linear perturbation to the spin connection that commutes with Clifford multiplication. Given a closed submanifold $\mathcal{Z}\subset Y$ of codimension 2, choose a real Euclidean line bundle $\ell\to Y-\mathcal{Z}$ and let $A$ be the unique flat connection with $\mathbb{Z}_{2}$ -holonomy on $\ell$ . The twisted spinor bundle $S_{0}\otimes_{\mathbb{R}}\ell$ carries a Dirac operator $\not{D}_{A}$ twisted by $A$ on $\ell$ , and perturbed using $B_{0}$ . A $\mathbb{Z}_{2}$ -harmonic spinor is a solution $\Phi\in\Gamma(S_{0}\otimes_{\mathbb{R}}\ell)$ of the twisted Dirac equation on $Y-\mathcal{Z}$ satisfying

\not{D}_{A}\Phi=0\hskip 28.45274pt\text{and}\hskip 28.45274pt\nabla_{A}\Phi\in L^{2}.

(1.6)

$\mathcal{Z}$ is called the singular set of the $\mathbb{Z}_{2}$ -harmonic spinor. A $\mathbb{Z}_{2}$ -harmonic spinor is denoted by the triple $(\mathcal{Z},A,\Phi)$ where $\mathcal{Z}$ is the singular set, $A$ is the unique flat connection determined by the line bundle $\ell$ , and $\Phi$ is the spinor itself.

When $\mathcal{Z}=\emptyset$ is empty, solutions of (1.6) are classical harmonic spinors whose study goes back to the work of Lichnerowicz [35], Atiyah-Singer [3], and Hitchin [27]. When the singular set is non-empty, the twisted Dirac operator is a type of degenerate elliptic operator known as an elliptic edge operator. This class of operators has little precedent in gauge theory, but extensive tools for their study have been developed in microlocal analysis [36, 39, 18]. Doan–Walpuski established the existence and abundance of solutions with $\mathcal{Z}\neq\emptyset$ on compact 3-manifolds in [16], and a stronger version of their result will appear in [31]. Additional examples have been constructed in [56, 25, 29].

The equations (1.6) do not carry an action of $U(1)$ -gauge transformations; there is only a residual action of $\mathbb{Z}_{2}$ by sign. In particular, (1.6) is not a gauge theory. On the other hand (1.6) is $\mathbb{R}$ -linear, so admits a scaling action by $\mathbb{R}^{+}$ (note that the assumption that the perturbation $B_{0}$ is $\mathbb{R}$ -linear means that the Dirac operator is also only $\mathbb{R}$ -linear). $\mathbb{Z}_{2}$ -harmonic spinors are considered as equivalence classes modulo these two actions; the scaling action is eliminated by fixing the normalization condition (1.5, leaving a residual $\mathbb{Z}_{2}$ action. Notice that (1.6) defines $\mathbb{Z}_{2}$ -harmonic spinors without reference to the Seiberg–Witten equations; it is therefore not a priori clear whether an arbitrary $\mathbb{Z}_{2}$ -harmonic spinor (1.6) should arise as a limit of (1.3)–(1.5).

1.3. The Gluing Problem

The gluing problem may now be stated more precisely. Fix a $\mathbb{Z}_{2}$ -harmonic spinor $(\mathcal{Z}_{0},A_{0},\Phi_{0})$ . The goal is to construct a family of solutions $(\Phi_{\varepsilon},A_{\varepsilon})$ to (1.3–1.5) for sufficiently small $\varepsilon>0$ such that

(\Phi_{\varepsilon},A_{\varepsilon})\longrightarrow(\mathcal{Z}_{0},A_{0},\Phi_{0})

(1.7)

in the sense of Haydys–Walpuski’s compactness theorem (Theorem 3.2). In particular, this requires reconstructing a smooth $E$ -valued spinor $\Phi_{\varepsilon}$ (note $\Phi_{0}$ is a section of a different bundle of real rank 4 over $Y-\mathcal{Z}$ ), and re-introducing the highly concentrated curvature by smoothing the singular connection $A_{0}$ . The latter implicitly requires recovering the $\text{Spin}^{c}$ -structure, which is lost in the limit as $\varepsilon\to 0$ .

In the simplest case, when $\mathcal{Z}_{0}=\emptyset$ and standard elliptic theory applies, Doan–Walpuski [15] showed that all classical harmonic spinors arise as limits of a family as in (1.7). Reversing the convergence by a gluing in the singular case $\mathcal{Z}_{0}\neq\emptyset$ is far more challenging, and requires new analytic tools for elliptic edge operators and their desingularizations. The concentration of curvature along the singular set $\mathcal{Z}_{0}$ as $\varepsilon\to 0$ manifests by making the linearization of the $\varepsilon=0$ version of (1.3)–(1.5) a singular elliptic edge operator with an infinite-dimensional obstruction to solving. This prevents the application of the standard Fredholm approaches that have historically been used in gluing problems [7, 34, 38, 10].

The presence of the infinite-dimensional obstruction bundle does not mean that the gluing can only be accomplished for a subset of parameters of infinite codimension. Rather, this obstruction is an artifact arising from inadvertently fixing the singular set $\mathcal{Z}_{0}$ . In fact, the location of the singular set is a degree of freedom that may also vary as $\varepsilon\to 0$ . Indeed, work of the author [43] and R. Takahashi [46] has shown that constructing families of $\mathbb{Z}_{2}$ -harmonic spinors with respect to families of metrics $g_{s}$ for $s\in\mathcal{S}$ requires allowing the singular set to depend on $s$ . Because the gluing problem is a de-singularization of the same situation, one anticipates the same phenomenon will occur. It is therefore necessary to include space of all possible singular sets nearby $\mathcal{Z}_{0}$ as a parameter in the gluing construction. This approach has some precedent in the work of Pacard–Ritoré [44] on gluing problems in minimal surface theory arising from the Allen-Cahn and Yang-Mills-Higgs equations [5], though the obstruction in these situations is more tractable.

As explained in the introduction, the main idea of this paper is to show that the deformations of the singular set pair with the infinite-dimensional obstruction to create a Fredholm gluing theory. Key aspects of this approach rely on the theory developed in [41], and [43], and this article is in some sense the sequel to and culmination of these. The first, [43] develops the deformation theory for the singular set for $\mathbb{Z}_{2}$ -harmonic spinors alone, without reference to Seiberg–Witten theory. The second, [41] constructs model solutions near the singular set $\mathcal{Z}$ , which are the starting point of the gluing construction. To keep this article self-contained, the relevant parts of [43] and [41] are reviewed in detail in Sections 4–6 and Section 7, respectively.

1.4. Main Results

To state the main result, we first describe several necessary assumptions on the starting data.

Let $Y$ be a compact, oriented three-manifold. The two-spinor Seiberg–Witten equations (1.1–1.2) depend on a smooth background parameter pair $p=(g,B)$ of a Riemannian metric and an auxiliary $SU(2)$ -connection on $E$ in the space

\mathcal{P}=\{(g,B)\ |\ g\in\text{Met}(Y)\ ,\ B\in\mathcal{A}_{SU(2)}(E)\}

(1.8)

of all such choices. The definition of a $\mathbb{Z}_{2}$ -harmonic spinor makes reference to a similar parameter pair $(g,B)$ where $B$ is now a perturbation to the spin connection on $S_{0}$ that is inherited from the $SU(2)$ connection denoted by the same symbol. The gluing construction can be carried out beginning from a $\mathbb{Z}_{2}$ -harmonic spinor satisfying several conditions; these conditions constrain the parameters to lie in the complement of the locus $\mathcal{P}^{\prime}\subset\mathcal{P}$ admitting $\mathbb{Z}_{2}$ -harmonic spinors with worse singular behavior.

Definition 1.1.

A $\mathbb{Z}_{2}$ -harmonic spinor $(\mathcal{Z}_{0},A_{0},\Phi_{0})$ with respect to a parameter pair $p_{0}=(g_{0},B_{0})$ is said to be regular if it satisfies the following three conditions:

(i)

(Smooth) the singular set $\mathcal{Z}_{0}\subset Y$ is a smooth, embedded link, and the holonomy of $A_{0}$ is equal to $-1$ around the meridian of each component.
(ii)

(Isolated) $\Phi_{0}$ is the unique $\mathbb{Z}_{2}$ -harmonic spinor for the pair $(\mathcal{Z}_{0},A_{0})$ with respect to $p_{0}=(g_{0},B_{0})$ up to normalization and sign.
(iii)

(Non-degenerate) $\Phi_{0}$ has non-vanishing leading-order, i.e. there is a constant $c>0$ such that

$|\Phi_{0}|\geq c\cdot\text{dist}(-,\mathcal{Z}_{0})^{1/2}.$

For a fixed singular set $\mathcal{Z}_{0}$ , the twisted Dirac operator,

\not{D}_{A_{0}}:H^{1}(S_{0}\otimes_{\mathbb{R}}\ell)\longrightarrow L^{2}(S_{0}\otimes_{\mathbb{R}}\ell)

(1.9)

is semi-Fredholm, where $H^{1}$ denotes the Sobolev space of sections whose covariant derivative is $L^{2}$ with appropriate weights, and $S_{0}$ is the spinor bundle of the spin structure hosting $\Phi_{0}$ (see Section 4 for details). For weights requiring that a solution satisfy the integrability requirement in (1.6), this operator is left semi-Fredholm and has infinite-dimensional cokernel. This cokernel gives rise to the infinite-dimensional obstruction of the linearized Seiberg–Witten equations at $\varepsilon=0$ .

As explained above, cancelling the obstruction requires deforming the singular set. Any singular set $\mathcal{Z}$ nearby $\mathcal{Z}_{0}$ defines a flat connection $A_{\mathcal{Z}}$ whose holonomy is homotopic to $A_{0}$ , thus it defines an accompanying real line bundle isomorphic to the original. Allowing the singular set to vary over the space of embedded links $\mathcal{Z}$ gives an infinite-dimensional family of Dirac operators parameterized by embeddings, which we combine into a universal Dirac operator

\not{\mathbb{D}}(\mathcal{Z},\Phi)=\not{D}_{A_{\mathcal{Z}}}\Phi,

(1.10)

which is fully non-linear in the first argument and linear in the second. The theory of the universal Dirac operator was developed in detail in [43] and is reviewed in Section 6.

The idea that the derivative of (1.10) with respect to the embedding should cancel the cokernel was investigated for the twisted Dirac operator alone (as opposed to the Seiberg–Witten equations) in [43] (see also the work of Takahashi [46] and Donaldson [12], which take a different approach). The first main result of [43] is the following theorem, which makes precise the idea that the linearized deformations of the singular set at a $\mathbb{Z}_{2}$ -harmonic spinor cancel the obstruction.

Theorem 1.2.

([43, Thm 1.3]) Let $(\mathcal{Z}_{0},A_{0},\Phi_{0})$ be a regular $\mathbb{Z}_{2}$ -harmonic spinor, and let $\Pi_{0}$ denote the projection onto the cokernel of (1.9). Then the cokernel component of the linearization of the universal Dirac operator with respect to deformations of $\mathcal{Z}_{0}$

\Pi_{0}\circ(\text{d}_{\mathcal{Z}_{0}}\not{\mathbb{D}})_{(\mathcal{Z}_{0},\Phi_{0})}:L^{2,2}(\mathcal{Z}_{0};N\mathcal{Z}_{0})\longrightarrow\text{Coker}(\not{D}_{\mathcal{Z}_{0}})

(1.11)

is an elliptic pseudo-differential operator of order $\tfrac{1}{2}$ and its Fredholm extension has index $-1$ . ∎

Here, the domain is thought of as the tangent space at $\mathcal{Z}_{0}$ to the space of embedded singular sets of Sobolev regularity $(2,2)$ . Section 5 gives a precise characterization of the cokernel, which results in an equivalence $\text{Coker}(\not{D}_{\mathcal{Z}_{0}})\simeq\Gamma(\mathcal{Z}_{0};\mathcal{S})$ with the space of sections of a vector bundle $\mathcal{S}\to\mathcal{Z}_{0}$ . Viewed in this guise, (1.11) is a map between spaces of sections of vectors bundles on $\mathcal{Z}_{0}$ , and the assertion that it is an elliptic pseudo-differential operator then has the standard meaning.

Definition 1.3.

A $\mathbb{Z}_{2}$ -harmonic spinor is unobstructed if the Fredholm extension of (1.11) has trivial kernel.

Since the Seiberg-Witten equations in dimension 3 have index 0, one does not expect 1-parameter families of solutions $(\Phi_{\varepsilon},A_{\varepsilon})$ converging to a $\mathbb{Z}_{2}$ -harmonic spinor for the fixed parameter $p_{0}=(g_{0},B_{0})$ , but rather for a 1-dimensional family of parameters $p_{\tau}=(g_{\tau},B_{\tau})$ which coincides with $p_{0}$ at $\tau=0$ . The third condition necessary for the gluing result is a requirement on such a 1-parameter family, and says that $(\mathcal{Z}_{0},A_{0},\Phi_{0})$ arises as a transverse spectral crossing in this family.

The following theorem, proved in [43], ensures that this notion makes sense in an analogous way to classical harmonic spinors in three dimensions. The added complexity here is that the singular set $\mathcal{Z}_{\tau}$ of the spinor is defined implicitly as a function of the parameter $\tau$ . The proof of the theorem uses the linearized result of Theorem 1.2 and the Nash-Moser Implicit Function Theorem.

Theorem 1.4.

([43] Corollary 1.5) Suppose that $p_{\tau}=(g_{\tau},B_{\tau})$ is a smooth path of parameters, and that $(\mathcal{Z}_{0},A_{0},\Phi_{0})$ is a regular, unobstructed $\mathbb{Z}_{2}$ -harmonic spinor with respect to $p_{0}$ . Then, there is a $\tau_{0}>0$ such that for $\tau\in(-\tau_{0},\tau_{0})$ , there exist $\mathbb{Z}_{2}$ -eigenvectors $(\mathcal{Z}_{\tau},A_{\tau},\Phi_{\tau})$ with eigenvalues $\Lambda_{\tau}\in\mathbb{R}$ satisfying

\not{D}_{A_{\tau}}\Phi_{\tau}=\Lambda_{\tau}\Phi_{\tau},

(1.12)

where each member of the tuples is defined implicitly as a smooth function of $\tau$ . Moreover, $(\mathcal{Z}_{\tau},A_{\tau},\Phi_{\tau})$ are regular and unobstructed for each $\tau\in(-\tau_{0},\tau_{0})$ .

Definition 1.5.

The family of Dirac operators $\not{D}_{A_{\tau}}$ is said to have transverse spectral crossing at $\tau=0$ if the family of $\mathbb{Z}_{2}$ -eigenvectors $(\mathcal{Z}_{\tau},A_{\tau},\Phi_{\tau},\Lambda_{\tau})$ has

\dot{\Lambda}(0)\neq 0,

where $\dot{\ }$ denotes the derivative with respect to $\tau$ .

We now state the main result, which establishes the existence of two-spinor Seiberg-Witten solutions converging to a $\mathbb{Z}_{2}$ -harmonic spinor satisfying the above conditions. Here, $S_{0}$ denotes the spinor bundle of a Spin structure that hosts a $\mathbb{Z}_{2}$ -harmonic spinor as in (1.6), while $S$ is reserved for the spinor bundle of the $\text{Spin}^{c}$ -structure in the Seiberg–Witten equations.

Theorem 1.6.

Suppose that $(\mathcal{Z}_{0},A_{0},\Phi_{0})$ is a regular, unobstructed $\mathbb{Z}_{2}$ -harmonic spinor with respect to a parameter $p_{0}=(g_{0},B_{0})$ , and that $p_{\tau}=(g_{\tau},B_{\tau})$ is a path of parameters such that the corresponding family (1.12) has transverse spectral crossing.

Then, for each orientation of $\mathcal{Z}_{0}$ , there is a unique $\text{Spin}^{c}$ structure with spinor bundle $S\to Y$ , an $\varepsilon_{0}>0$ , and a family of configurations $(\Psi_{\varepsilon},A_{\varepsilon})$ for $\varepsilon\in(0,\varepsilon_{0})$ satisfying the following.

(1)

The $\text{Spin}^{c}$ structure is characterized by

c_{1}(S)=-\text{PD}[\mathcal{Z}_{0}]\hskip 28.45274pt\text{and }\hskip 28.45274ptS|_{Y-\mathcal{Z}_{0}}\simeq S_{0}\otimes_{\mathbb{R}}\ell.

(2)

The configurations $(\Psi_{\varepsilon},A_{\varepsilon})\in\Gamma(S_{E})\times\mathcal{A}_{U(1)}$ solve two-spinor Seiberg-Witten equations

	$\displaystyle\not{D}_{A_{\varepsilon}}\Psi_{\varepsilon}$	$\displaystyle=$	$\displaystyle 0$
	$\displaystyle\star F_{A_{\varepsilon}}+\tfrac{1}{2}{\mu(\Psi_{\varepsilon},\Psi_{\varepsilon})}$	$\displaystyle=$	$\displaystyle 0$

on $Y$ with respect to $(g_{\tau},B_{\tau})$ where $\tau=\tau(\varepsilon)$ is defined implicitly as a function of $\varepsilon\in(0,\varepsilon_{0})$ and satisfies either $\tau(\varepsilon)>0$ or $\tau(\varepsilon)<0$ .

(3)

The spinor has $L^{2}$ -norm

$\|\Psi_{\varepsilon}\|_{L^{2}(Y)}=\frac{1}{\varepsilon},$ (1.13)

and after renormalizing by setting $\Phi_{\varepsilon}=\varepsilon\Psi_{\varepsilon}$ , the solutions converge to $(\mathcal{Z}_{0},A_{0},\Phi_{0})$ , i.e.

$\Phi_{\varepsilon}\to\Phi_{0}\hskip 28.45274pt\text{and}\hskip 28.45274ptA_{\varepsilon}\to A_{0}$

in $C^{\infty}_{loc}(Y-\mathcal{Z}_{0})$ after applying gauge transformations defined on $Y-\mathcal{Z}_{0}$ , and $|\Phi_{\varepsilon}|\to|\Phi_{0}|$ in $C^{0,\alpha}(Y)$ for some $\alpha>0$ .

Remark 1.7.

It is expected that all $\mathbb{Z}_{2}$ -harmonic spinors are regular and unobstructed for generic parameters among those admitting $\mathbb{Z}_{2}$ -harmonic spinors (see Section 1.5). We do not undertake the task of establishing the genericity results here. A partial result for the genericity of the non-degeneracy condition in Definition 1.1 is proved in [25] in the situation of $\mathbb{Z}_{2}$ -harmonic 1-forms. It is straightforward to show (see Remark 1.11) that a generic path $(g_{\tau},B_{\tau})$ has transverse spectral crossing.

Remark 1.8.

By a simple diagonalization argument, Theorem 1.6 may be extended to the case of a $\mathbb{Z}_{2}$ -harmonic spinor that is instead a limit of $\mathbb{Z}_{2}$ -harmonic spinors satisfying the hypotheses of the theorem. In particular, the singular set of such a limiting $\mathbb{Z}_{2}$ -harmonic spinor need not be embedded.

The discussion in the upcoming Section 1.5 suggests it is likely that every isolated $\mathbb{Z}_{2}$ -harmonic spinor arises as such a limit. If this were the case, Theorem 1.6 would imply every isolated $\mathbb{Z}_{2}$ -harmonic spinor on a compact 3-manifold arises as the limit of Seiberg–Witten monopoles (in fact in multiple ways—one for each $\text{Spin}^{c}$ structure whose first chern class is Poincaré dual to some orientation of the singular set).

Remark 1.9.

The solutions $(\Psi_{\varepsilon},A_{\varepsilon})$ have curvature that is highly concentrated in a $O(\varepsilon^{2/3})$ neighborhood around a smooth embedded curve $\mathcal{Z}_{\varepsilon}$ that lies in a neighborhood $U$ of the singular set $\mathcal{Z}_{\tau(\varepsilon)}$ . Slightly surprisingly, the proof shows that the $C^{0}$ -distance from $\mathcal{Z}_{\varepsilon}$ to $\mathcal{Z}_{\tau(\varepsilon)}$ is smaller than the concentration scale of the curvature, while $C^{1}$ -distance is larger than the concentration scale. For the specific parameters used in the proof of Theorem 1.6, one finds that $\mathcal{Z}_{\varepsilon}$ lies in a $O(\varepsilon^{23/24-\gamma})$ neighborhood of $\mathcal{Z}_{\tau(\varepsilon)}$ in $C^{0}$ and a $O(\varepsilon^{11/24-\gamma})$ neighborhood in $C^{1}$ for some $\gamma<<1$ .

1.5. Wall-Crossing Formulas

The non-compactness of the moduli space $\mathcal{M}_{SW}$ of solutions to (1.3)–(1.5) prevents the (signed) count of two-spinor Seiberg–Witten monopoles from being a topological invariant. In particular, the compactness theorem (Theorem 3.2) suggests that along a path of parameters $p_{\tau}$ such that $p_{0}$ admits a $\mathbb{Z}_{2}$ -harmonic spinor, a family of monopoles may diverge so that the signed count of solutions changes; in fact, Theorem 1.6 shows that this necessarily happens if the $\mathbb{Z}_{2}$ -harmonic spinor is regular and unobstructed.

Rather than being a topological invariant, it is conjectured that signed count $\#\mathcal{M}_{SW}$ is a chambered invariant with wall-crossing formulas. That is, it is conjectured that the subset $\mathcal{W}_{\mathbb{Z}_{2}}\subseteq\mathcal{P}$ of parameters admitting $\mathbb{Z}_{2}$ -harmonic spinors has codimension 1, and divides its complement in $\mathcal{P}$ into a collection of open chambers inside which the count is invariant, with a well-defined formula for how the count changes as it crosses the “wall” $\mathcal{W}_{\mathbb{Z}_{2}}$ . This chambered invariant is conjectured to fit into a larger scheme of constructing invariants by summing chambered invariants with cancelling wall-crossing formulas (see [13, 33, 23, 14] for details and examples).

The main result of [43] provides a step towards confirming this picture by proving that $\mathcal{W}_{\mathbb{Z}_{2}}$ indeed forms a “wall” near regular, unobstructed $\mathbb{Z}_{2}$ -harmonic spinors.

Theorem 1.10.

( [43, Thm 1.4]) Suppose that $(\mathcal{Z}_{0},A_{0},\Phi_{0})$ is a regular, unobstructed $\mathbb{Z}_{2}$ -harmonic spinor with respect to $p_{0}\in\mathcal{P}$ . Then there is an open neighborhood $\mathcal{U}$ of $p_{0}$ such that ,

\mathcal{W}_{\mathbb{Z}_{2}}\cap\mathcal{U}\subseteq\mathcal{P}

is locally a smooth Fréchet submanifold of codimension 1.

More generally, it is expected that $\mathcal{W}_{\mathbb{Z}_{2}}$ is a stratified space with the following global structure. The top stratum $\mathcal{W}{{}^{\text{reg}}}_{\mathbb{Z}_{2}}$ should consist of a disconnected Fréchet submanifold of codimension 1 whose components are labeled by isotopy classes of embedded links in $Y$ , and where each parameter $p\in\mathcal{W}_{\mathbb{Z}_{2}}^{\text{reg}}$ admits a (unique) regular, unobstructed $\mathbb{Z}_{2}$ -harmonic spinor. Confirming this expectation would, in particular, require confirming the prediction of Taubes that the singular set of a $\mathbb{Z}_{2}$ -harmonic spinor is smooth for generic choices of smooth parameters [50, pg. 9]. Deeper strata of higher finite codimension are expected to consist of the locus where the singular set has the structure of an embedded graph with increasingly complicated self-intersections, and the loci where the regular or unobstructed condition fails. There should also be strata of infinite codimension where wilder singular behavior can occur. The work of [56, 16, 29, 28] support this picture.

Theorem 1.6 shows that the count $\#\mathcal{M}_{SW}$ changes along a path of parameters crossing $\mathcal{W}^{\text{reg}}_{\mathbb{Z}_{2}}$ transversely, which provides a key step in confirming the conjectured wall-crossing formula. In particular, Theorem 1.6 constructs a subset of the parameterized moduli space over $p_{\tau}$ for either $\tau\geq 0$ or $\tau\leq 0$ that is homeomorphic to a half-open interval $[0,\varepsilon_{0})$ . A complete proof of a wall-crossing formula would additionally require investigating orientations to determine the sign of the crossing as in [16], and showing that the homeomorphism to $[0,\varepsilon_{0})$ determines a boundary chart for the moduli space. The latter involves showing the surjectivity of gluing, i.e. that the family of monopoles in Theorem 1.6 is the unique family converging to $(\mathcal{Z}_{0},A_{0},\Phi_{0})$ , and will be the subject of future work.

Remark 1.11.

A path $p_{\tau}=(g_{\tau},B_{\tau})$ has transverse spectral crossing (Definition 1.5) at a regular, unobstructed $\mathbb{Z}_{2}$ -harmonic spinor $(\mathcal{Z}_{0},A_{0},\Phi_{0})$ if and only if $p_{\tau}\pitchfork\mathcal{W}_{\mathbb{Z}_{2}}^{\text{reg}}$ is a transverse intersection. The genericity of this condition follows easily from the latter characterization.

1.6. Outline

The article has 12 sections which build towards the proof of Theorem 1.6.

The proof is accomplished by generalizing Donaldson’s alternating method to the semi-Fredholm setting. Section 2 is a self-contained introduction to gluing using the alternating method and describes this generalization. Briefly, the alternating method decomposes the manifold into two regions

Y=Y^{+}\cup Y^{-}

each of which admits a model solution. These model solutions are spliced into a global approximate solution, which is then corrected to a true solution by alternating making corrections localized on $Y^{+}$ and $Y^{-}$ using the linearized equations. We take $Y^{+}=N(\mathcal{Z}_{0})$ to be a tubular neighborhood of the singular set, and $Y^{-}$ to be the complement of a slightly smaller tubular neighborhood.

In our semi-Fredholm setting, the linearized equations on $Y^{-}$ have an infinite-dimensional obstruction coming from the cokernel of (1.9), and an addition step cancelling this obstruction by deforming the singular set is required. Thus the alternation becomes a three-step cyclic iteration:

(1.14)

Sections 3–12 are devoted to setting up and carrying out this cyclic iteration scheme. In this, there are several key technical difficulties (described in detail in Section 2.5) that conspire to make naive attempts at the iteration (1.14) fail to converge. The most important of these is that the deformation operator (1.11) displays a loss of regularity [19]. Overcoming this loss of regularity requires developing a more sophisticated way of deforming singular sets than was used in [43, 46, 12]. These new deformations depend on the spectral decomposition of a linearized deformation. Somewhat surprisingly, this approach eliminates the need to use tame Fréchet spaces and Nash-Moser theory, which are the standard tools used to deal with a loss of regularity as in [43, 46, 12]. Section 2 ends with a glossary of notation.

Section 3 covers background necessary to set up the gluing argument, beginning by reviewing the compactness theorem for (1.3)–(1.5) and the Haydys Correspondence.

Sections 4–7 analyze the linearized Seiberg–Witten equations on $Y^{\pm}$ . Section 4 begins by studying the singular Dirac operator (1.9) for a fixed singular set, which appears as a direct summand of the linearized Seiberg–Witten equations on $Y^{-}$ .

Section 5 considers the infinite-dimensional obstruction of (1.9). In particular, it is shown that there is a natural basis of the obstruction indexed by the eigenvalues of a 1-dimensional Dirac operator on $\mathcal{Z}$ , whose pointwise norms increasingly concentrate near $\mathcal{Z}$ as the eigenvalue increases.

Section 6 deals with the universal Dirac operator (1.10). The main goal is to calculate the derivative with respect to the embedding of the singular set to give a refined version of Theorem 1.2. This is done by defining a family of diffeomorphisms $F_{\eta}:Y\to Y$ parameterized by linearized deformations $\eta\in T\text{Emb}(\mathcal{Z}_{0};Y)$ such that the curves $\mathcal{Z}_{\eta}=F_{\eta}(\mathcal{Z}_{0})$ form an open neighborhood of $\mathcal{Z}_{0}$ in a sufficient space of embeddings. By the naturality of the Dirac operator, varying the singular set over curves $\mathcal{Z}_{\eta}$ while fixing the metric $g_{0}$ is equivalent to varying the metric over the family of pullback metrics $g_{\eta}=F_{\eta}^{*}(g_{0})$ while keeping $\mathcal{Z}_{0}$ fixed:

\begin{pmatrix}\text{varying }\mathcal{Z}_{\eta}\\ \text{fixed }g_{0}\end{pmatrix}\hskip 28.45274pt\frac{\partial}{\partial\mathcal{Z}}\not{D}_{\mathcal{Z}}\ \ \ \overset{F_{\eta}^{*}}{\Longleftrightarrow}\ \ \ \frac{\partial}{\partial g}\not{D}_{\mathcal{Z}_{0}}^{g}\hskip 28.45274pt\begin{pmatrix}\text{varying }g_{\eta}\\ \text{fixed }\mathcal{Z}_{0}\end{pmatrix}.

(1.15)

The derivative with respect to the family of pullback metrics on the right is then calculated using a result of Bourguignon-Gauduchon [6] (see Theorem 6.4).

Section 7 describes the model solutions on $Y^{+}$ constructed in [41] and summarizes results about the linearized Seiberg–Witten equations at these.

Section 8 constructs the infinite-dimensional family of model solutions parameterized by deformations $\mathcal{Z}_{0}$ alluded to in the introduction. This family is used to define a universal version of the Seiberg–Witten equations analogous to (1.10), which is differentiated using the same trick as in (1.15). Section 9 shows that this derivative of the universal Seiberg–Witten equations with respect to deformations of $\mathcal{Z}_{0}$ is simply a small perturbation of the derivative of the universal Dirac operator (1.10). This allows the deformations to be used to cancel the obstruction on $Y^{-}$ , just as in Theorem 1.2.

Sections 10–11 use the results of Sections 4–9 to carry out the cyclic iteration scheme (1.14). The iteration is shown to converge to a smooth two-parameter family of Seiberg–Witten “eigenvectors” solving

\text{SW}(\Phi,A)=\Big{(}\Lambda(\tau)+\mu({\varepsilon,\tau})\Big{)}\chi\frac{\Phi_{\tau}}{\varepsilon}

(1.16)

for every pair $(\varepsilon,\tau)$ , where $\mu(\varepsilon,\tau)\in\mathbb{R}$ , $\Lambda(\tau)$ is as in Theorem 1.4, and $\chi$ is a smooth function on $Y$ . Thus (1.16) are solutions of the Seiberg–Witten equations precisely when the one-dimensional obstruction $\Lambda(\tau)+\mu(\varepsilon,\tau)$ vanishes. Finally, Section 12 uses a trick due to T. Walpuski [60] to show the condition that this one-dimensional obstruction vanishes defines $\tau(\varepsilon)$ implicitly as a function of $\varepsilon$ , completing the proof of Theorem 1.6.

Acknowledgements

The author gratefully acknowledges the support and guidance of his Ph.D. advisor, Clifford Taubes throughout this project. This work also benefitted from the interest and expertise of many other people, including Aleksander Doan, Siqi He, Rafe Mazzeo, Tom Mrowka, and Thomas Walpuski. The author is supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship (Award No. 2303102).

2. Gluing by the Alternating Method

This section reviews Donaldson’s Alternating method for gluing [9, Sec. 4], and introduces the semi-Fredholm generalization that is used to prove Theorem 1.6. The reader is referred to [10] for additional exposition of general approaches to gluing, and to [34, Ch. 19], and [7, 38, 60] for expositions of other particular instances of gluing results.

2.1. The Structure of Gluing Problems

Let $Y$ be a compact manifold, and let $\mathfrak{F}:\mathcal{X}\to\mathcal{Y}$ be a non-linear elliptic PDE viewed as a continuous map between Banach spaces $\mathcal{X},\mathcal{Y}$ of sections of vector bundles. Suppose that $Y$ is the union of two overlapping open regions

Y=Y^{+}_{\circ}\cup Y^{-}_{\circ},

each of which hosts a solution $\Phi^{\pm}$ (called model solutions) of $\mathfrak{F}(\Phi^{\pm})=0$ on $Y^{\pm}_{\circ}$ (or more generally, a near solution, $\mathfrak{F}(\Phi^{\pm})\approx 0$ ).

The associated gluing problem is to produce a global solution $\Phi$ of $\mathfrak{F}(\Phi)=0$ on $Y$ beginning with the model solutions. This is done by splicing the model solutions together to form an approximate solution

\Phi_{1}:=\Phi^{+}\ \#_{\varepsilon}\ \Phi^{-}

on $Y$ , which is then corrected to a true solution. Here, $\#_{\varepsilon}$ denotes a splicing procedure, usually performed using a cut-off function, which is allowed to depend on a parameter $\varepsilon$ called the gluing parameter(s). The true solution is obtained by a particular gluing procedure, which is method for correcting approximate solutions,

that is applied iteratively starting from the initial approximation above to construct a sequence $\Phi_{1},\Phi_{2},\ldots,$ $\Phi_{N}$ of successively improving approximations. Usually, the gluing procedure is a variation of Newton’s method which repeatedly solves some version of the linearized equations to produce corrections $\delta\Phi_{N}$ .

If the resulting sequence $\Phi_{N}\to\Phi_{\varepsilon}$ converges in a sufficient function space for appropriate choices of the gluing parameter $\varepsilon$ , the limit is a global solution of $\mathfrak{F}(\Phi_{\varepsilon})=0$ on $Y$ (or more generally, a family of global solutions parameterized by appropriate choices of $\varepsilon$ ). Oftentimes, the gluing procedure and convergence are packaged into a suitable version of the Implicit Function Theorem or related contraction mapping.

The following example explains how the above framework applies in the context of gluing ASD instantons. This framework has been applied in dozens of other well-known gluing problems in geometric analysis, and many of these could be substituted as equally instructive examples.

Example 2.1.

(Gluing ASD Instantons, [47]) In the context of Uhlenbeck compactness on a compact $4$ -manifold $X^{4}$ , one seeks to construct an ASD instanton of charge $k+1$ by gluing two connections $A^{\pm}$ with $A^{+}$ being a “bubble”. In this case, $A^{+}$ is the standard instanton of charge $1$ and dilation parameter $\lambda$ on a ball $X^{+}_{\circ}=B_{\sqrt{\lambda}}(x_{0})$ of radius $\sqrt{\lambda}$ around the bubbling point, and $A^{-}$ is an instanton of charge $k$ on the complement $X^{-}_{\circ}=X^{4}-B_{\lambda}(x_{0})$ of a smaller ball. Here, the dilation $\lambda$ of the standard instanton is the gluing parameter. See [7, 17, 40] for details.

Notice the following two features of Example 2.1 that will be pertinent for the upcoming case of $\mathbb{Z}_{2}$ -harmonic spinors. (1) the decomposition $X^{4}=X^{+}_{\circ}\cup_{\lambda}X^{-}_{\circ}$ depends on the gluing parameter $\lambda$ , with $X^{+}_{\circ}$ shrinking as $\lambda\to 0$ . (2) While the problem has a natural “invariant scale” of radius $\lambda$ , the two regions $X^{\pm}_{\circ}$ overlap in a “neck region” which extends from radius $\lambda$ to $\sqrt{\lambda}$ , and the much of the gluing analysis occurs there.

Returning now to the case of $\mathbb{Z}_{2}$ -harmonic spinors, let $(Y,g_{0})$ be a compact 3-manifold and $(\mathcal{Z}_{0},A_{0},\Phi_{0})$ a $\mathbb{Z}_{2}$ -harmonic spinor satisfying the hypotheses of Theorem 1.6. In this case the regions $Y^{\pm}_{\circ}$ are a tubular neighborhood of the singular set $\mathcal{Z}_{0}$ whose radius depends on the parameter $\varepsilon$ , and its complement. More specifically, let $\lambda=\lambda(\varepsilon)$ be a radius to be specified later, and set

Y^{+}_{\circ}=N_{\lambda}(\mathcal{Z}_{0})\hskip 56.9055ptY^{-}_{\circ}=Y-N_{\lambda^{4/3}}(\mathcal{Z}_{0}).

(2.1)

The model solutions $(\Phi^{+}_{\varepsilon},A^{+}_{\varepsilon})$ on $Y^{+}_{\circ}$ are the concentrating local family of model solutions constructed in [41] and reviewed in Section 7; the model solutions on $Y^{-}_{\circ}$ are simply the limiting $\mathbb{Z}_{2}$ -harmonic spinor $(\Phi_{0},A_{0})$ . In this case, we tacitly refer to the region $Y^{+}$ including the singular set as the “inside” region, and $Y^{-}$ as the “outside” region; the overlap $Y^{+}_{\circ}\cap Y^{-}_{\circ}$ is referred to as the “neck” region as in Example 2.1. The gluing parameters in our situation are triples $(\varepsilon,\tau,\mathcal{Z})$ of the $L^{2}$ -norm, the metric parameter, and a deformation of the singular set, and there is, more precisely, a decomposition (2.1) for each such triple.

2.2. The Alternating Method

The gluing procedure employed to prove Theorem 1.6 is a generalization of the alternating method.

The alternating method iteratively corrects approximate solutions by alternating making corrections localized to the two regions $Y^{\pm}_{\circ}$ . This method was first used by S. Donaldson [9] to give a different perspective on C. Taubes’s gluing theorem for ASD instantons (Example 2.1). More specifically, the successively approximations at the $N^{\text{th}}$ stage have the form

\Phi_{N}=\Phi_{1}+\chi^{+}\varphi_{N}+\chi^{-}\psi_{N}

(2.2)

where $\varphi_{N},\psi_{N}$ are corrections supported in the two regions respectively, and $\chi^{\pm}$ are cut-off function restricting them to their respective regions. The iteration may be summarized by the following schematic:

where $\Psi_{N}=\Phi_{1}+\chi^{+}\varphi_{N}+\chi^{-}\psi_{N+1}$ . Thus a single stage of the overall iteration constitutes one alternation back and forth. In the notation of Section 2.1, one has $\delta\Phi_{N}=\chi^{+}\delta\varphi_{N}+\chi^{-}\delta\psi_{N}$ where $\delta\varphi_{N}=\varphi_{N}-\varphi_{N-1}$ and likewise for $\psi$ . The iteration converges to a solution if the errors $\mathfrak{e}_{N}\to 0$ .

To explain further, let us give a more precise meaning to the steps of “solve on $Y^{\pm}$ ”. The equations at a small perturbation $\Phi+\varphi$ of an approximate solution $\Phi$ may be written

\mathfrak{F}(\Phi+\varphi)=\mathfrak{F}(\Phi)+\mathcal{L}_{\Phi}(\varphi)+Q(\varphi)

(2.3)

where $\mathcal{L}_{\Phi}=\text{d}_{\Phi}\mathfrak{F}$ is the linearization of $\mathfrak{F}$ at $\Phi$ , and $Q$ the higher-order terms. The solving steps on $Y^{\pm}_{\circ}$ involve solving the linearized equation. Since $Y^{\pm}_{\circ}$ are open regions, it is often necessary to choose extensions $Y^{\pm}_{\circ}\subseteq Y^{\pm}$ on which $\mathcal{L}_{\Phi}$ has sufficient invertibility properties. This is typically done by attaching a tubular ends to $\partial Y^{\pm}_{\circ}$ or imposing boundary conditions (see also Section 2.3). Write $\mathcal{L}^{\pm}$ for the linearization acting on sections over sufficiently chosen extensions $Y^{\pm}$ . We assume that $\mathcal{L}^{\pm}$ naturally extends the restriction of the linearization on $Y^{\pm}_{\circ}$ , and that the vector bundles and model solutions likewise admit natural extensions to $Y^{\pm}$ .

The alternating method, in its standard guise, requires the following two assumptions.

(I)

The linearizations $\mathcal{L}^{\pm}$ at the sequence of approximate solutions are uniformly invertible in the gluing parameter $\varepsilon$ .
(II)

The solution $\varphi$ of $\mathcal{L}^{\pm}\varphi=g$ decays away from the support of $g$ in both regions in the following sense. If $\text{supp}(g)\subseteq\text{supp}(d\chi^{\pm})$ , then

$\|\sigma_{\mathcal{L}}(d\chi^{\mp})g\|\leq C\delta\|g\|$ (2.4)

for some $\delta<1$ , where $\sigma_{\mathcal{L}}$ is the principal symbol.

Given that (I)–(II) hold, the starting point of the alternating method is an initial approximation $\Phi_{1}$ so that the error $\mathfrak{F}(\Phi_{1})=\mathfrak{e}_{1}$ of this initial approximation is sufficiently small and supported where $d\chi^{\text{+}}\neq 0$ . The iteration then proceeds inductively, with a single step given as follows.

(1)

Let $\psi$ be the unique solution of

$(\mathcal{L}^{-}+Q)\psi=-\mathfrak{e}_{N}$ (2.5)

on $Y^{-}$ given by the Inverse Function Theorem.
(2)

Set $\psi_{N+1}=\psi_{N}+\psi$ , and $\Psi_{N}=\Phi_{1}+\chi^{+}\varphi_{N}+\chi^{-}\psi_{N+1}$ . This intermediate approximation satisfies

$\mathfrak{F}(\Psi_{N})=\cancel{\mathfrak{e}_{N}-\mathfrak{e}_{N}}+d\chi^{-}\psi+g_{N}:=f_{N}$

where $g_{N}=\chi^{-}Q(\psi)-Q(\chi^{-}\psi)$ .
(3)

Since $f_{N}$ is supported where $d\chi^{-}\neq 0$ , the solution $\psi$ decays across the neck region by condition (II) so that $\|f_{N}\|\leq C\delta\|\mathfrak{e}_{N}\|$ for some fixed factor $\delta<1$ .
(4)

Repeat steps (1)–(3) on $Y^{+}$ to obtain a correction $\varphi$ so $\varphi_{N+1}=\varphi_{N}+\varphi$ , then set

$\Phi_{N+1}=\Phi_{1}+\chi^{+}\varphi_{N+1}+\chi^{-}\psi_{N+1}.$

The resulting error, $\mathfrak{e}_{N+1}$ , then satisfies $\|\mathfrak{e}_{N+1}\|\leq(C\delta)^{2}\|\mathfrak{e}_{N}\|$ .

If $\delta$ is sufficiently small that $(C\delta)<1$ , the sequence $\Phi_{N}$ constructed by iterating Steps (1)–(4) converges to a limit $\Phi_{0}$ which solves $\mathfrak{F}(\Phi_{0})=0$ .

Refer to caption — Figure 1. An illustration of the alternating iteration procedure in Steps (1)–(4) above. (Top) The cut-off functions $\chi^{\pm}$ , (red) the error terms with alternating support and decreasing norm, (blue/green) the decay of solutions across the neck region.

The content of proving a gluing result using the alternating method is showing a sufficient version of the hypotheses (I) and (II) are met. Verifying the statement (I) requires identifying suitable extensions $Y^{\pm}_{\circ}\subseteq Y^{\pm}$ and proving that the restricted operator can be sufficiently inverted. The decay requirements in condition (II) can be concluded from knowledge of the linearization’s Green’s function, or from using weighted function spaces.

The main advantage of the alternating method over other gluing procedures, and indeed the reason it is suitable in our setting, is that it can effectively treat asymmetry between the two regions $Y^{\pm}$ . In particular, the method only requires analysis of $\mathcal{L}^{\pm}$ in the two distinct regions separately, and never the analysis of a global linearization whose properties are an (in our case quite opaque) combination of those inherited from the two regions. This allows the asymmetric character of the equation in the two regions to be isolated and analyzed separately; indeed, [41] and [43] are most effectively viewed for the present purposes as manuals for the Seiberg–Witten theory on $Y^{+}$ and $Y^{-}$ respectively.

Remark 2.2.

A slight variation on Steps (1)–(4) above is to solve the strictly linear equation at each step. Thus Step (1) is replaced by solving

\mathcal{L}^{-}\psi=-e_{N}+Q(\psi_{N}+\varphi_{N})

where $Q$ denotes the higher-order error from the correction at the previous stage, and likewise for the inside. This formulation is equivalent, though it comes at the cost of disrupting the fact that the error terms are entirely supported where $d\chi^{\pm}\neq 0$ . In our case, the higher-order terms are sufficiently mild that the linearized version suffices.

2.3. Gluing as Non-linear Excision

The alternating method can be rephrased using the language of parametrix patching and contraction mappings. This rephrasing is helpful for addressing questions the regarding uniqueness of gluing and smooth dependence on parameters. It also clarifies the relationship of the alternating method to other gluing methods that use parametrices.

At first glance, the alternating method does not result in a contraction mapping or a Fredholm problem in an obvious way. Indeed, there is an infinite-dimensional ambiguity in the construction, coming from the freedom to alter (2.2) by $\varphi_{N},\psi_{N}\mapsto\varphi_{N}+\xi,\psi_{N}-\xi$ on the overlap region. This ambiguity can be resolved by framing the alternating method as a non-linear analogue of the excision problem for the index of elliptic operators. With this perspective, it becomes clear that the redundancy in the description can be eliminated by considering the “virtual” gluing problem solved by $\varphi_{N}-\psi_{N}$ on the neck region. The author learned this perspective on gluing from [40].

To explain further, let us briefly review the excision principle for elliptic operators. For convenience, assume here that $\mathfrak{F}$ is a first-order elliptic operator. Let $S_{1},S_{2}\to Y$ be vector bundles whose sections form the domain and codomain of the linearization, so that $\mathcal{L}_{\Phi_{1}}:H^{1}(S_{1})\to L^{2}(S_{2})$ is Fredholm. The closed manifold $Y$ can be reconstructed from $Y^{\pm}$ by the following cut-and-paste procedure. Let $U=Y^{+}_{\circ}\cap Y^{-}_{\circ}\subseteq Y^{\pm}_{\circ}$ be the overlap. Assume that the inclusion $U\hookrightarrow Y^{\pm}$ separates $Y^{\pm}-U$ into two connected components, so that we may write the extended domain $Y^{\pm}=Y^{\pm}_{\circ}\cup_{U}N^{\pm}_{\circ}$ as the union of the original domain and the extension (this being a tubular end or collar neighborhood of $\partial Y^{\pm}_{\circ}$ ). Cutting the domains $Y^{\pm}$ along $U$ and pasting them together again exchanging the second component,

\displaystyle\begin{matrix}Y^{+}&=&Y^{+}_{\circ}\cup_{U}N^{+}_{\circ}\\ Y^{-}&=&N^{-}_{\circ}\cup_{U}Y^{-}_{\circ}\end{matrix}\hskip 42.67912pt\Leftrightarrow\hskip 42.67912pt\begin{matrix}Y=Y^{+}_{\circ}\cup_{U}Y^{-}_{\circ}\\ N=N^{-}_{\circ}\cup_{U}N^{+}_{\circ}\end{matrix}

(2.6)

gives rise to the closed manifold $Y$ and an extra copy of the neck region $N$ . As in Section 3.2, assume that the restrictions of $S_{1},S_{2}$ extend so that these bundles are well-defined on any of the four manifolds above; let $\mathcal{L}^{+},\mathcal{L}^{-},\mathcal{L}_{\Phi_{1}},\mathcal{L}_{N}$ denote the four manifestations of the linearized equations.

The cut-off functions $\chi^{\pm}$ extend by $0$ or $1$ to each piece. Using these, define maps

\alpha:H^{1}(Y^{+})\oplus H^{1}(Y^{-})\to H^{1}(Y)\oplus H^{1}(N)\hskip 42.67912pt\beta:L^{2}(Y)\oplus L^{2}(N)\to L^{2}(Y^{+})\oplus L^{2}(Y^{-})

\alpha=\begin{pmatrix}\chi^{+}&\chi^{-}\\ -\chi^{-}&\chi^{+}\end{pmatrix}\hskip 85.35826pt\beta=\begin{pmatrix}\chi^{+}&-\chi^{-}\\ \chi^{-}&\chi^{+}\end{pmatrix}

(2.7)

which cut-and paste sections of $S_{1},S_{2}$ along with (2.6). These satisfy $w:=\alpha\circ\beta=(\chi^{+})^{2}+(\chi^{-})^{2}\geq 1$ .

Proposition 2.3 (Excision).

Assume that $\mathcal{L}^{+},\mathcal{L}^{-},\mathcal{L}_{\Phi_{1}}$ are Fredholm. Then $\mathcal{L}_{N}$ is also Fredholm, and the indices satisfy

\text{ind}(\mathcal{L}^{+})+\text{ind}(\mathcal{L}^{-})=\text{ind}(\mathcal{L}_{\Phi_{1}})+\text{ind}(\mathcal{L}_{N}).

Proof.

Let $P^{\pm}:L^{2}(Y^{\pm})\to H^{1}(Y^{\pm})$ denote parametrices for $\mathcal{L}^{\pm}$ , and define

P=\alpha\circ(P^{+},P^{-})\circ\frac{\beta}{w}:L^{2}(Y)\oplus L^{2}(N)\to H^{1}(Y)\oplus H^{1}(N).

(2.8)

$P$ is Fredholm because $P^{\pm}$ are and $\alpha,\beta$ are invertible. Setting $L=(\mathcal{L}_{\Phi_{1}},\mathcal{L}_{N})$ , a quick calculation (see [34, Page 245]) shows that $LP-\text{Id}$ and $PL-\text{Id}$ are compact operators consisting of terms involving $d\chi^{\pm}$ , hence $L$ is Fredholm. It follows that $\mathcal{L}_{N}$ is also Fredholm, and the indices satisfy $\text{ind}(L)=-\text{ind}(P)=-\text{ind}(P^{+})-\text{ind}(P^{-})=\text{ind}(\mathcal{L}^{+})+\text{ind}(\mathcal{L}^{-})$ . ∎

The correction to the approximate solution (2.2) constructed by the alternating method, is the $H^{1}(Y)$ -component of $\alpha(\varphi_{N},\psi_{N})$ with $\alpha$ as in (2.7). The relevance of excision to the alternating method is that it shows resolving the ambiguity on $Y^{+}_{\circ}\cap Y^{-}_{\circ}$ requires dictating the fate of the $H^{1}(N)$ -component of $\alpha(\varphi_{N},\psi_{N})$ . The alternation described in Section 2.2 implicitly requires that this component is zero. To see this, let $\zeta^{\pm}:Y\to\mathbb{R}$ be a partition of unity so that $\zeta^{\mp}=1$ on $\text{supp}(d\chi^{\pm})$ , and set

\mathcal{H}=\left\{\begin{pmatrix}P^{+}\zeta^{+}g\\ P^{-}\zeta^{-}g\end{pmatrix}\ \Big{|}\ (g,0)\in L^{2}(Y)\oplus L^{2}(N)\right\}\subseteq H^{1}(Y^{+})\oplus H^{1}(Y^{-})

(2.9)

i.e. $\mathcal{H}$ is the image of $(P^{+},P^{-})\circ\beta$ where $\beta$ is as in (2.7) but now formed using $\zeta^{\pm}$ in place of $\chi^{\pm}$ . A quick inspection of Steps (1)–(4) in Section 2.2 shows that the correction $(\varphi_{N},\psi_{N})$ lands in $\mathcal{H}$ at each stage of the iteration.

At a functional analytic level, the alternating method constructs a sequence of approximate inverses to $\mathfrak{F}$ on the space $\mathcal{H}$ . This is most easily understood as a slight extension of the linear setting (i.e. $Q=0$ in 2.3), which we now explain in detail. In the linear setting, Steps (1)–(4) produce a sequence of approximate inverses to $\mathcal{L}_{\Phi_{1}}\circ\alpha$ by a nested parametrix patching. (2.8) is an unsophisticated version of parametrix patching; indeed, it patches together the two parametrices $P^{\pm}$ on their respective regions using the cut-off functions $\chi^{\pm}$ . When the $L^{2}(N)$ -component in the codomain vanishes, one may replace the $L^{2}(Y)\to\mathcal{H}$ component of (2.8) by

P_{1}=\chi^{+}P^{+}\zeta^{+}+\chi^{-}P^{-}\zeta^{-}.

(2.10)

In this instances $\mathcal{L}_{\Phi_{1}}P_{1}$ and $P_{1}\mathcal{L}_{\Phi_{1}}$ fail to be the identity by compact terms involving $d\chi^{\pm},d\zeta^{\pm}$ . The alternating method corrects $P_{1}$ to recursively cancel these errors, as the next proposition demonstrates.

Proposition 2.4.

Suppose that assumptions (I)–(II) from Section 2.2 hold. Then

\mathcal{L}_{\Phi_{1}}\circ\alpha:\mathcal{H}\to L^{2}(Y)

is invertible.

Proof.

Since $\mathcal{L}^{\pm}$ are surjective, it may be assumed that $P^{\pm}$ are injective; thus $P_{1}:L^{2}(Y)\to\mathcal{H}$ is an isomorphism, and in particular has index 0. Set

e^{\pm}_{1}:=d\chi^{\pm}.P^{\pm}\zeta^{\pm}

(2.11)

so that $\mathcal{L}_{\Phi_{1}}P_{1}=\text{Id}+e^{+}_{1}+e^{-}_{1}$ . Then, with $P_{1}$ as in (2.10) with $P_{1}^{\pm}$ being the two terms, recursively define

P_{N+1}^{\pm}:=P_{N}^{\pm}\ -\ \chi^{\mp}P^{\mp}(\underbrace{d\chi^{\pm}.P^{\pm}e_{N}^{\pm}}_{f^{\mp}_{N}})\ +\ \chi^{\pm}P^{\pm}(\underbrace{d\chi^{\mp}.P^{\mp}f_{N}^{\mp}}_{e^{\pm}_{N+1}})

(2.12)

and take $P_{N}=P_{N}^{+}+P_{N}^{-}$ . Expanding shows

\mathcal{L}_{\Phi_{1}}P_{N}=\text{Id}+e_{N}^{+}+e_{N}^{-}

is a telescoping series. Moreover, using (II) repeatedly shows that $e_{N}^{+}+e_{N}^{-}=O(\delta^{N})$ , where $\delta$ is as in (2.4) because both $e_{N}^{\pm}$ are a nested composition of $N$ terms each of which gains a decay factor of $\delta$ from condition (II). Provided $C\delta<1$ for $C$ depending on $P^{\pm}$ , it follows that $\mathcal{L}_{\Phi_{1}}$ is surjective and $P_{N}$ injective. Since $P_{N}$ differs from $P_{1}$ by compact terms involving $d\chi^{\pm}$ , $P_{N}$ is index 0, hence it is an isomorphism thus so is $\mathcal{L}_{\Phi_{1}}$ . ∎

The nested parametrix patching in the proof of Proposition 2.4 may be also be phrased as a contraction finding the true inverse of $\mathcal{L}_{\Phi_{1}}$ . Indeed, for $g\in L^{2}(Y)$ , the map $\text{d}T_{g}:\mathcal{H}\to\mathcal{H}$ given by

\text{d}T_{g}:=\text{Id}-P_{1}(\mathcal{L}_{\Phi_{1}}(\_)-g)

(2.13)

applied iteratively starting from $0\in\mathcal{H}$ reconstructs the sequence of approximations $P_{N}g$ . For $\delta$ sufficiently small, $\text{d}T^{2}_{g}$ is a contraction. (In fact, on the subset of $\mathcal{H}$ for which the error is supported where $d\chi\pm\neq 0$ , $\text{d}T_{g}$ itself behaves as a contraction, and the first application imposes this condition).

The non-linear version of the alternating method is essentially dentical, where $T:\mathcal{H}\to\mathcal{H}$ is now given by

T:=\text{Id}-P_{1}\mathfrak{F}(\Phi_{1}+\_)

(2.14)

and where $P_{1}$ is as in (2.10), with $P^{\pm}$ now being inverses of the non-linear equations (2.5). In this case, $\mathfrak{F}(\Phi_{N})$ on the sequence of approximate solutions constructed by Steps (1)–(4) in Section 2.2 expands to mirror the telescoping cancellation of Proposition 2.4, but (2.11) now also includes non-linear error terms. Again, $T^{2}$ is a contraction, provided sufficient control on the non-linearity.

Remark 2.5.

In many gluing problems (including all previous gluing problems in gauge theory known to the author), a single iteration of the nested parametrix construction (2.10) resulting in $P_{1}$ is sufficient. In these situations, the geometry (e.g. stretching a neck so that $|d\chi|\to 0$ ) or function spaces (appropriate weights) is such that in (2.11), $e_{1}^{\pm}\to 0$ in the appropriate norm as the gluing parameter $\varepsilon\to 0$ . When this is not the case (and indeed it is not for gluing $\mathbb{Z}_{2}$ -harmonic spinors), the nested construction of Proposition 2.4 provides a more refined approximate inverse. The alternating method described in Sections 2.2 and 2.3 therefore subsumes the methods used in many standard gluing constructions with appropriate reorganization of the proof. More relevant here, it allows the upcoming generalization.

2.4. The Semi-Fredholm Alternating Method

As explained in the introduction, condition (I) in Section 2.2 fails for $\mathcal{L}^{-}$ in the context of gluing $\mathbb{Z}_{2}$ -harmonic spinors because the linearized Seiberg–Witten equation at a $\mathbb{Z}_{2}$ -harmonic spinor are only semi-Fredholm and therefore not invertible. In fact, for effectively the same reason, condition (I) fails for $\mathcal{L}^{+}$ as well, but in this case what fails is the uniformity of invertibility, rather than invertibility itself. The gluing problem for $\mathbb{Z}_{2}$ -harmonic spinors therefore departs the setting of standard gluing problems in several key ways, and requires the following more sophisticated generalization of the alternating method.

Replace assumptions (I)–(II) from Section 2.2 by the following weaker hypotheses:

(I’)

The linearizations $\mathcal{L}^{+}$ at the sequence of approximate solutions are invertible.
(II’)

Condition (II) from Section 2.2 continues to hold for constants $\delta^{\pm}$ .
(III’)

The linearizations $\mathcal{L}^{-}$ at the sequence of approximate solutions are left semi-Fredholm with uniformly bounded left-inverses.
(IV’)

There is a family of operators $F_{\xi}$ on configurations, parameterized by an infinite-dimensional parameter $\xi\in\mathfrak{W}$ which generate a corresponding family of approximate solutions $F_{\xi}(\Phi_{N})$ . Moreover, the projection $T_{\Phi_{N}}:=\Pi_{0}\circ\text{d}_{\xi}\mathfrak{F}_{\Phi_{N}}$ of the derivative

$T_{\Phi_{N}}:\mathfrak{W}\longrightarrow\text{Coker}(\mathcal{L}^{-})$ (2.15)

is uniformly invertible at the family of approximate solutions, where $\mathfrak{F}(\xi,\Phi):=\mathfrak{F}(F_{\xi}(\Phi))$ , and $\Pi_{0}$ denotes the projection to the cokernel.

Notice that Item (I’) is identical to Item (I) from Section 2.2 for the inside except that it omits the requirement of uniformity. Note also the similarity of the map in Item (IV’) to that of (1.11) in Theorem 1.2 in the introduction.

The approximate solutions in the sequence (2.2) now also depends on the parameter $\xi$ that is adjusted in each step of the iteration, and have the form

\Phi_{N}=F_{\xi_{N}}(\Phi_{1})+\chi^{+}\varphi_{N}+\chi^{-}\psi_{N}.

(2.16)

The alternating iteration procedure is similarly adjusted to become the three-stage cycle (1.14), which may now be written as follows:

(2.17)

Commensurately, Steps (1)–(4) in the previous subsection are augmented by a Step (0) corresponding to the horizontal arrow in (2.17), and Step (1) is altered accordingly. Here, we retain the notation of Steps (1)–(4) from the previous subsection. Assuming that the codomain of $\mathcal{L}^{-}$ is a Hilbert space, $\Pi_{0}$ may be viewed as the orthogonal projection to cokernel viewed as the orthogonal complement of the range, thus $(1-\Pi_{0})$ denotes the projection to the range of $\mathcal{L}^{-}$ .

(0)

Let $\xi$ denote the unique solution of

$T_{\Phi_{N}}(\xi)=-\Pi_{0}(\mathfrak{e}_{N}),$ (2.18)

and set $\xi_{N+1}=\xi_{N}+\xi$ .
(1)

Let $\psi$ be the unique solution of

$\mathcal{L}^{-}(\psi)=-(1-\Pi_{0})\left(\mathfrak{e}_{N}+\text{d}\mathfrak{F}_{\Phi_{N}}(\xi,0)\right).$ (2.19)

Steps (2)–(4) then proceed as before, with the following minor adjustment. Due to the lack of uniform invertibility of $\mathcal{L}^{+}$ , the constant $C$ in the decay factor (written $C\delta$ in Step (3) in Section 2.2) may now depend on the gluing parameter $\varepsilon$ . Thus the condition for the sequence $\{\Phi_{N}\}$ for converge becomes that

|C(\varepsilon)\delta^{+}\delta^{-}|<1.

(2.20)

The decay constants $\delta^{\pm}=\delta^{\pm}(\varepsilon)$ may themselves depend on the gluing parameter.

The discussion of Section 2.3 carries over to the semi-Fredholm setting with analogous modifications. There are now three parametrices $P_{\xi},P^{+},P^{-}$ , with $P_{\xi}$ being a parametrix for $T_{\Phi_{N}}$ , and the cyclic iteration (2.17) may again be phrased as a non-linear analogue of the nested parametrix patching akin to how (2.14) is a non-linear version of (2.13). The linear version extending (2.13) in this case is constructed analogously to Proposition 2.4, replacing (2.9) by $\mathcal{H}\oplus\mathfrak{W}$ and (2.10) and (2.12) by a sequence

$\displaystyle P_{1}$	$\displaystyle:=$	$\displaystyle P_{\xi}$	(2.21)
$\displaystyle P_{2}$	$\displaystyle:=$	$\displaystyle P_{\xi}+P^{-}(\text{Id}-\text{d}\mathfrak{F}_{\Phi_{1}}P_{1})$	(2.22)
$\displaystyle P_{3}$	$\displaystyle:=$	$\displaystyle P_{\xi}+P^{-}(\text{Id}-\text{d}\mathfrak{F}_{\Phi_{1}}P_{1})+P^{+}(\text{Id}-\text{d}\mathfrak{F}_{\Phi_{1}}P_{2})$	(2.23)
$\displaystyle P_{4}$	$\displaystyle:=$	$\displaystyle P_{\xi}+P^{-}(\text{Id}-\text{d}\mathfrak{F}_{\Phi_{1}}P_{1})+P^{+}(\text{Id}-\text{d}\mathfrak{F}_{\Phi_{1}}P_{2})-P_{\xi}(\text{Id}-\text{d}\mathfrak{F}_{\Phi_{1}}P_{3})$

where $P_{\xi}=T_{\Phi_{1}}^{-1}\Pi_{0}\zeta^{-}$ , $P^{-}=\chi^{-}\mathcal{L}_{\Phi_{1}}^{-1}(1-\Pi_{0})\zeta^{-}$ and $P^{+}=\chi^{+}\mathcal{L}_{\Phi_{1}}^{-1}\zeta^{+}$ . Each successive approximation is defined by $P_{N+1}=P_{N}-P^{\circ}(\text{Id}-\text{d}\mathfrak{F}_{\Phi_{1}}P_{N})$ with $P^{\circ}$ cycling between $P_{\xi},P^{-}$ , and $P^{+}$ mod 3. Equivalently, $P_{3N}$ is characterized by

	$\displaystyle(\text{Id}-P_{3N}\text{d}\mathfrak{F}_{\Phi_{1}})$	$\displaystyle=$	$\displaystyle\left((\text{Id}-P^{+}\text{d}\mathfrak{F}_{\Phi_{1}})(\text{Id}-P^{-}\text{d}\mathfrak{F}_{\Phi_{1}})(\text{Id}-P_{\xi}\text{d}\mathfrak{F}_{\Phi_{1}})\right)^{N}$		(2.24)
		$\displaystyle=$	$\displaystyle(\text{Id}-P_{3}\text{d}\mathfrak{F}_{\Phi_{1}})^{N},$		(2.25)

which is $\text{d}T^{N}$ as in (2.13) for $g=0$ .

The main tasks in the proof of Theorem 1.6 are establishing the analogues of Lemma 2.3 and Proposition 2.4 in this setting. This is done in Sections 10–11 in the Seiberg–Witten setting, where $\xi$ in Item (IV’) parameterizes small deformations of the the singular set $\mathcal{Z}_{0}$ , and (2.15) is the operator in Theorem 1.2.

2.5. Key Technical Difficulties

Despite the fact that versions of the statements (I’)–(IV’) are established in [41, 43], there are still several key technical difficulties to overcome to carry out the alternating iteration. Addressing these in a systematic way to turn the scheme outlined in Section 2.4 into a proof of Theorem 1.6 occupies the bulk of Sections 4–11. In this subsection, we briefly sketch the most important technical problems that arise, and how they are solved.

2.5.1. Loss of Regularity

The main technicality that complicates the application of the cyclic iteration in Section 2.4 to prove Theorem 1.6 is the following:

(Difficulty I): The deformation operator (1.2) displays a loss of regularity of order $\tfrac{3}{2}$ .

Loss of regularity is an intriguing (though sometimes technical) pheomenon in certain non-linear PDEs [19, 1, 30]. While the linearized deformation operator (1.11) is a pseudo-elliptic operator of order $1/2$ , the non-linear terms and the $\text{Range}(\not{D}_{A_{0}})$ -components are of order $>1/2$ . Thus one cannot choose function spaces for which it is simultaneously the case that the linear operator is Fredholm and the non-linear terms are bounded. This loss of regularity necessitates the use of a version of the Nash-Moser Implicit Function Theorem in [43] (see Sections 7 and 8 therein).

Here, the loss of regularity manifests differently, and (perhaps surprisingly) the use of a Nash-Moser type iteration involving smoothing operators is not necessary. In Section 5, it is shown that $\text{Coker}(\not{D}_{\mathcal{Z}_{0}})\simeq\Gamma(\mathcal{Z}_{0};\mathcal{S})$ for a bundle $\mathcal{S}\to\mathcal{Z}_{0}$ and that the cokernel increasingly concentrates around $\mathcal{Z}_{0}$ as the Fourier index increases, so that the $\ell^{th}$ mode has characteristic length scale $1/|\ell|$ . Consequently, a restriction on the support of a spinor away from $\mathcal{Z}_{0}$ at distance $R_{0}$ leads to unexpectedly rapid decay on the Fourier modes of its cokernel component once the Fourier index exceeds $|\ell|\geq O(R_{0}^{-1})$ . The error terms in the alternating iteration are supported where the cut-off $d\chi^{\pm}\neq 0$ and therefore obey such a restriction. Consequently, it is possible to cancel the obstruction at each stage using deformations that lie in a large finite-dimensional space with Fourier modes only up to $O(\varepsilon^{-2/3})$ . In particular, each deformation is smooth, and the use of smoothing operators is not required.

This does not, however, mean that the loss of regularity is inconsequential. If $\xi_{N}$ is the deformation correction at the $N^{th}$ stage, given by (2.18), then the elliptic estimates for $T_{\Phi_{0}}$ gives a bound on $\|\xi_{N}\|_{1/2}$ in the Sobolev space of regularity $s=1/2$ . Due to the loss of regularity, however, bounding other terms requires control of $\|\xi\|_{s,2}$ in the space of regularity $s>2$ . Because of the restriction on Fourier modes of $\xi_{N}$ , this norm is larger by a power of $\varepsilon^{-1}$ . The loss of regularity therefore manifests as adverse powers of $\varepsilon^{-1}$ in certain error terms.

More specifically, the problematic term is $(1-\Pi_{0})\text{d}\mathfrak{F}_{\Phi_{N}}(\xi,0)$ in (2.19). In the proof of Theorem 1.6, this term takes the following form. Splitting the codomain of the linearization on $Y^{-}$ as $L^{2}(S_{2})=\text{Coker}(\mathcal{L}^{-})\oplus\text{Range}(\mathcal{L}^{-})$ and supplementing the domain to be $H^{1}(S_{1})\oplus\mathfrak{W}$ , the linearization can be written as the block matrix

\text{d}\mathfrak{F}_{\Phi_{1}}(\xi,\psi)=\begin{pmatrix}T_{\Phi_{1}}&0\\ \boxed{(1-\Pi_{0})\text{d}\mathfrak{F}_{\Phi_{1}}}&\mathcal{L}^{-}_{(\Phi_{1},A_{1})}\end{pmatrix}\begin{pmatrix}\xi\\ \psi\end{pmatrix}.

(2.26)

In this guise, there first two arrows of (2.17) (i.e. Step (0)–(1) in (2.18)) are precisely solving the first and second rows of (2.26) respectively. The problematic term is indicated by a box: in the Seiberg–Witten setting bounding this term requires a bound on $\|\xi\|_{5/2,2}$ . Without careful set-up of the entire iteration scheme, the extraneous powers of $\varepsilon^{-1}$ from the loss of regularity make this term too large for the iteration to converge.

As explained briefly in Section 1.6, the key idea that overcomes this difficulty is a more sophisticated method of deforming the singular set:

(Solution I): Use mode-dependent deformations of the singular set.

These are introduced in Section 6.3. Just as the $\ell^{th}$ Fourier mode of the cokernel has characteristic length scale $1/|\ell|$ as explained above, the mode-dependent deformations purposely manufacture a similar relationship for the diffeomorphisms used in (1.15). This link between the spatial domain on $Y$ and the Fourier domain of the deformations results in unexpectedly strong control of the boxed term (the crucial result for this being Lemma 6.12), without disrupting the solvability of $T_{\Phi_{0}}$ .

2.5.2. Non-local Deformation Operator

The alternating iteration scheme only converges because of the decay property (II) and (II’) in Sections 2.2 and 2.4, which causes each iteration to reduce the error by a factor $\delta$ of the decay across the neck region. In particular, in Section 2.2 the error on $Y^{-}$ is supported where $d\chi^{+}\neq 0$ and the solution in Step (1) decays across the region between the supports of $d\chi^{\pm}$ . The first stage in (2.17) disrupts this property:

(Difficulty II): The projection and deformation operator $(1-\Pi_{0})$ and $T_{\Phi_{1}}$ are non-local.

In particular, both terms of (2.19) are supported globally on $Y$ , regardless of the support of the error term in Step (0).

This at first makes it seem that there can be no decay factor $\delta^{-}$ when solving on $Y^{-}$ . Were the decay factor $\delta^{+}$ on $Y^{+}$ sufficiently strong, this would present no issue, but in our situation $C(\varepsilon)\delta^{+}=\varepsilon^{-1/24}$ in (2.20) grows as $\varepsilon\to 0$ due to the non-invertibility of the linearization on $Y^{+}$ and so $\delta^{-}$ must at least compensate for this factor. This is achieved by

(Solution II): Sobolev spaces of spinors with “effective” support.

The notion of effective support (Definition 4.6) generalizes the standard support, and imposes constraints on how the norms of a spinor grow as a weight increases. The results of Sections 6, and 9.2 show that the mode-dependent deformations from Subsection 2.5.1 above mean that the deformation operator preserves effective support, and this is enough for a decay result.

2.5.3. Non-Uniform Invertibility

This final difficulty is the problem that was solved in [41], and the solution may be treated as a black box here. It still bears mentioning, however, because it makes Difficulties I and II above more unforgiving. Since the model solutions $(\Phi_{\varepsilon}^{+},A_{\varepsilon}^{+})$ on $Y^{+}$ converge to $(\Phi_{0},A_{0})$ , the family of elliptic linearizations $\mathcal{L}^{+}$ converge to the degenerate elliptic linearization at $\varepsilon=0$ with infinite-dimensional obstruction.

(Difficulty III): The linearizations $\mathcal{L}^{+}_{\varepsilon}$ are not uniformly invertible in $\varepsilon$ .

This situation is effectively mandated by the fact that there must be an infinite-dimensional subspace that becomes the infinite-dimensional obstruction in the limit, and the accompanying elliptic estimates necessarily fail to be uniform on this subspace (in any norms with locally defined norm density).

It is not overly difficult to achieve estimates showing that the norm of $(\mathcal{L}^{+})^{-1}$ is $O(\varepsilon^{-\beta})$ for some $\beta>0$ . All naive approaches, however, frustratingly result in the constant $\beta$ that makes the powers of $\varepsilon$ in (2.20) precisely cancel, preventing convergence (even once Solutions I and II are optimized). In order to achieve stronger control, one must identify quite precisely the subspace from which the infinite-dimensional obstruction arises, and treat this subspace independently. This was achieved in [41] by introducing boundary conditions which capture a certain high-dimensional subspace that approaches the obstruction.

(Solution III): Use twisted APS boundary conditions.

These boundary conditions are packaged in Lemma 7.9. Details may be found in [41, Section 7].

Appendix of Gluing Parameters

This appendix collects the notation and gluing parameters used throughout the remainder of the article. References to the section where the items are defined in the body of the text are included.

(1) Notation for the Seiberg–Witten Equations

•

$S_{E}=S\otimes_{\mathbb{C}}E$ the two-spinor bundle (Section 3.1).
•

$S^{\text{Re}},S^{\text{Im}}$ the real and imaginary subbundles of $S_{E}$ , defined over $Y-\mathcal{Z}$ (Section 3.2).
•

$\Omega=\Omega^{0}(i\mathbb{R})\oplus\Omega^{1}(i\mathbb{R})$ the bundle of $\mathfrak{u}(1)=i\mathbb{R}$ -valued $0$ and $1$ -forms.
•

$h_{0}=(\Phi_{\tau},A_{\tau})$ the family of $\mathbb{Z}_{2}$ -eigenvectors from Theorem (1.4), assumed to satisfy (1.5).
•

$h_{1}=(\Phi_{1},A_{1})$ the initial global approximate solutions on $Y$ (Section 7.2).
•

$(\varphi,a)$ used to denote corrections on $Y^{+}$ (Section 7.1).
•

$(\psi,b)$ used to denote corrections on $Y^{-}$ (Sections 4.2 and 7.3).
•

$\mathcal{L}_{(\Phi,A)}$ the linearized SW equations at the re-normalized configuration $(\varepsilon^{-1}\Phi,A)$ (Section 4.1).
•

$\not{\mathbb{D}},\text{d}\not{\mathbb{D}}$ the universal Dirac operator and its linearization (Section 6.1).
•

$\mathbb{SW},\text{d}\mathbb{SW}$ the universal Seiberg–Witten equations and their linearization (Section 8.2).

(2) Gluing Parameters

•

$\varepsilon\in(0,\varepsilon_{0})$ the $L^{2}$ -norm parameter.
•

$\tau\in(-\tau_{0},\tau_{0})$ the coordinate along the parameter path $p_{\tau}=(g_{\tau},B_{\tau})$ . Assumed to satisfy (1.5).
•

$\delta=\varepsilon^{1/48}$ the convergence factor from a single cycle of (1.14).
•

$\gamma,\gamma^{\pm},\underline{\gamma}<<1$ small numbers, say $O(10^{-6})$ .
•

$\eta$ a linearized deformation of $\mathcal{Z}_{\tau}$ (Section 6.1).
•

$\xi=\varepsilon\eta$ a re-normalized linearized deformation (Section 9.1).
•

$\nu^{+}=\tfrac{1}{4}-10^{-6}$ the inside weight.
•

$\nu^{-}=\tfrac{1}{2}-10^{-6}$ the outside weight.

(3) Regions and Cut-offs

•

$Y^{+}=N_{\lambda}(\mathcal{Z}_{\tau})$ the inside region: a tubular neighborhood of radius $\lambda=\varepsilon^{1/2}$ (Section 7).
•

$Y^{-}\!=\!Y\!-\!N_{\lambda^{-}}(\mathcal{Z}_{\tau})$ the outside region: the complement of a neighborhood of radius $\lambda^{-}=\varepsilon^{2/3-\gamma^{-}}$ .
•

$\chi^{+}$ a logarithmic cut-off equal to $1$ for $r\leq\lambda/4$ and vanishing for $r\geq\lambda/2$ .
•

$\chi^{-}$ a logarithmic cut-off equal to $1$ for $r\geq\lambda^{-}/2$ and vanishing for $r\leq\lambda^{-}/4$ .
•

$\mathbb{1}^{+}$ the indicator function of the set $r\leq\varepsilon^{2/3-\gamma^{-}}$ .
•

$\mathbb{1}^{-}=1-\mathbb{1}^{+}$ the indicator function of the complementary region.

Important Remark.

The small numbers $\varepsilon_{0},\tau_{0}$ and $\gamma$ are allowed to decrease a finite number of times between successive appearances over the course of the proof of Theorem 1.6. The proof of Proposition 11.3 in Section 11 collects the accumulated factors of $\gamma$ . The numbers $\gamma^{\pm},\underline{\gamma}$ remain fixed throughout.

3. $\mathbb{Z}_{2}$ -Harmonic Spinors and Compactness

This section reviews the compactness properties of the two-spinor Seiberg–Witten equations from [32], and begins the set-up of the gluing analysis. More detailed expositions of the same material may be found in [41, 16, 61, 32, 52].

3.1. Compactness Theorem

Let $Y$ be a compact, oriented, 3-manifold. With $S,E\to Y$ as in Section 1.1 and $p=(g,B)$ as in (1.8), set $S_{E}:=S\otimes_{\mathbb{C}}E$ . Clifford multiplication on $S$ induces a Clifford multiplication $\gamma:\underline{\mathbb{R}}\oplus T^{*}Y\to\text{End}(S_{E})$ which acts trivially on $E$ . Define the moment map $\mu:S_{E}\to\Omega^{1}(i\mathbb{R})$ by

\frac{1}{2}\mu(\Psi,\Psi)=\sum_{j=1}^{3}\frac{i}{2}\langle i\gamma(e^{j})\Psi,\Psi\rangle e^{j}.

(3.1)

where $\{e^{j}\}$ is a local orthonormal frame of $T^{*}Y$ . Unlike for the single-spinor Seiberg–Witten equations, there are $0\neq\Psi\in\Gamma(S_{E})$ with $\mu(\Psi,\Psi)=0$ .

For the two-spinor Seiberg–Witten equations (1.1–1.2) to be an elliptic system on a 3-manifold, the an auxiliary $0$ -form is required. Revising the notation slightly, let $A=(a_{0},A_{1})\in\Omega^{0}(i\mathbb{R})\otimes\mathcal{A}_{U(1)}$ , and denote

\not{D}_{A}=\not{D}_{A_{1}}+\gamma(a_{0})\hskip 56.9055pt\star F_{A}=\star F_{A_{1}}-da_{0}.

(3.2)

where $\not{D}_{A_{1}}$ is the Dirac operator on $S_{E}$ formed using the spin connection of $g$ , the connection $A_{1}$ , and the background connection $B$ on $E$ , and $F_{A_{1}}$ is the curvature of $A_{1}$ .

Definition 3.1.

The (extended) two-spinor Seiberg-Witten Equations for the parameter $p$ are the following equations for configurations $(\Psi,A)\in\Gamma(S_{E})\times(\Omega^{0}(i\mathbb{R})\times\mathcal{A}_{U(1)})$ :

	$\displaystyle\not{D}_{A}\Psi$	$\displaystyle=$	$\displaystyle 0$		(3.3)
	$\displaystyle\star F_{A}+\tfrac{1}{2}\mu(\Psi,\Psi)$	$\displaystyle=$	$\displaystyle 0$		(3.4)

where $\star F_{A}$ and $\not{D}_{A}$ are as in (3.2). These equations are invariant under the action of the gauge group $\mathcal{G}=\text{Maps}(Y;U(1))$ .

Notice that definition (3.1) of $\mu$ differs by a sign from what is used by many other authors [37, 34, 15]. Consequently, the sign in the curvature equation (3.4) also reverses, leaving the equations unaltered. If $(\Psi,A)$ solves (3.3–3.4) and $\Psi\neq 0$ , then integration by parts shows that $a_{0}=0$ . For the purposes of Theorem 1.6, it therefore suffices to solve the extended equations. From here onward, we work exclusively with the extended equations.

Standard elliptic theory shows that a sequence of solutions to (3.3–3.4) with an a priori bound on the spinors’ $L^{2}$ -norm admits a convergent subsequence [37, 34, 32]. In the case of the standard (one-spinor) Seiberg–Witten equations, such an a priori bound follows from the Weitzenböck formula, the maximum principle, and a pointwise identity for $\mu$ . In fact, the bound in that case is $\|\Psi\|_{L^{2}}\leq\tfrac{s_{0}}{2}$ where $s_{0}$ is the $C^{0}$ -norm of the scalar curvature. In the case of two-spinors, the same pointwise identity for $\mu$ fails, and there may be sequences of solutions $(\Psi_{i},A_{i})$ such that $\|\Psi_{i}\|_{L^{2}}\to\infty$ which therefore admit no convergent subsequence.

The behavior of such sequences can be understood by renormalizing the spinor to have unit $L^{2}$ -norm. Thus, with $\varepsilon=\frac{1}{\|\Psi\|_{L^{2}}}$ we define renormalized spinors

\boxed{\Phi:=\varepsilon\Psi}

(3.5)

so that $\|\Phi\|_{L^{2}(Y)}=1$ . As in the introduction, the re-normalized or blown-up Seiberg-Witten equations for a triple $(\Phi,A,\varepsilon)$ become (1.3–1.5), where $A=(a_{0},A_{1})$ is now as in (3.2) . The following theorem of Haydys–Walpuski describes the convergence behavior of sequences of solutions to the blown-up equations. The theorem states that sequences of solutions along which $\varepsilon\to 0$ converge to solution of the $\varepsilon=0$ -version of (1.3–1.5) away from a singular set $\mathcal{Z}_{0}$ .

Theorem 3.2.

(Haydys–Walpuski [32, 42, 66, 51]) Suppose that $(\Phi_{i},A_{i},\varepsilon_{i})$ is a sequence of solutions to (1.3–1.5) with respect to a converging sequence of parameters $p_{i}\to p_{0}=(g_{0},B_{0})$ such that $\varepsilon_{i}\to 0$ . Then there exists a triple $(\mathcal{Z}_{0},A_{0},\Phi_{0})$ where

•

$\mathcal{Z}_{0}\subset Y$ is a closed, rectifiable subset of Hausdorff codimension 2,
•

$A_{0}$ is a flat $U(1)$ -connection on $Y-\mathcal{Z}_{0}$ with holonomy in $\mathbb{Z}_{2}$ ,
•

$\Phi_{0}$ is a spinor on $Y-\mathcal{Z}_{0}$ such that $|\Phi_{0}|$ extends continuously to $Y$ with $\mathcal{Z}_{0}=|\Phi_{0}|^{-1}(0)$ ,

satisfying

\not{D}_{A_{0}}\Phi_{0}=0\hskip 42.67912pt\mu(\Phi_{0},\Phi_{0})=0\hskip 42.67912pt\|\Phi_{0}\|_{L^{2}}=1

(3.6)

on $Y-\mathcal{Z}_{0}$ , and after passing to a subsequence, $\Phi_{i}\to\Phi_{0}$ , and $A_{i}\to A_{0}$ in $C^{\infty}_{loc}(Y-\mathcal{Z}_{0})$ modulo gauge transformations, and $|\Phi_{i}|\to|\Phi_{0}|$ in $C^{0,\alpha}(Y)$ for some $\alpha>0$ .

Remark 3.3.

The above statement combines the original result of Haydys–Walpuski with refinements proved by Taubes [51], Zhang [66] and the author [42]. In [51], Taubes showed that the singular set $\mathcal{Z}_{0}$ has finite 1-dimensional Hausdorff content; building on this Zhang showed in [66] that the singular set is rectifiable. In [42], the author improved the convergence to the limit from weak $L^{2,2}_{loc}$ for the spinor and weak $L^{1,2}_{loc}$ for the connection to $C^{\infty}_{loc}$ for both. Taubes also proved a four-dimensional version of Theorem 3.2 in [52], to which the same refinements apply.

Remark 3.4.

Theorem 3.2 is a particular instance of a family of similar compactness results for generalized Seiberg–Witten equations which originated with C. Taubes’s generalization of Uhlenbeck Compactness to $\text{PSL}(2,\mathbb{C})$ connections. Generalizations and similar results for other generalized Seiberg–Witten equations can be found in [54, 53, 52, 49, 65, 61, 45].

3.2. The Haydys Corresondence

The limiting configurations $(\Phi_{0},A_{0})$ in Theorem 3.2 are equivalent to $\mathbb{Z}_{2}$ -harmonic spinors as defined in the Section 1.2. This equivalence is a particular instance of the Haydys Correspondence [21, 15].

A limiting configuration as in Theorem 3.2 induces a decomposition of the restriction of the two-spinor bundle $S_{E}$ to $Y-\mathcal{Z}_{0}$ as follows. Since $A_{0}$ is flat with holonomy in $\mathbb{Z}_{2}$ , $\det(S)|_{Y-\mathcal{Z}_{0}}\simeq\underline{\mathbb{C}}$ is trivial, and $S|_{Y-\mathcal{Z}_{0}}$ admits a reduction of structure to $SU(2)$ . Thus, there is a “charge conjugation” map $J\in\text{End}(S|_{Y-\mathcal{Z}_{0}})$ such that $J^{2}=-\text{Id}$ ; since $E$ is an $SU(2)$ -bundle it admits a similar map, denoted $j$ . The product $\sigma=J\otimes_{\mathbb{C}}j$ satisfies $\sigma^{2}=\text{Id}$ , i.e. it is a real structure on $S_{E}|_{Y-\mathcal{Z}_{0}}$ . Consequently, there is a decomposition

S_{E}|_{Y-\mathcal{Z}_{0}}=S^{\text{Re}}\oplus S^{\text{Im}}

(3.7)

where

S^{\text{Re}}=\{\tfrac{1}{2}(\Psi+\sigma\Psi)\ |\ \Psi\in\Gamma(S_{E}|_{Y-\mathcal{Z}_{0}})\}\hskip 56.9055ptS^{\text{Im}}=\{\tfrac{1}{2}(\Psi-\sigma\Psi)\ |\ \Psi\in\Gamma(S_{E}|_{Y-\mathcal{Z}_{0}})\}

are the “real” and “imaginary” subbundles respectively.

These subbundles have the following useful characterization, which is proved in [41, Sec. 2], and [16, Sec. 3].

Lemma 3.5.

Let $(\mathcal{Z}_{0},A_{0},\Phi_{0})$ be a limiting configuration as in Theorem 3.2. The decomposition (3.7) satisfies the following:

(1)

The decomposition is parallel with respect to the connection $\nabla_{A_{0}}$ induced by $A_{0}$ .
(2)

Clifford multiplication by $\mathbb{R}$ -valued forms preserves the decomposition, i.e.

$\gamma:(\Omega^{0}\oplus\Omega^{1})(\mathbb{R})\times S^{\text{Re}}\to S^{\text{Re}}$

and likewise for $S^{\text{Im}}$ . Conversely, Clifford multiplication by $i\mathbb{R}$ -valued forms reverses it.
(3)

$\Phi_{0}\in\Gamma(S^{\text{Re}})$ , and there exists a spin structure on $Y$ with spinor bundle $S_{0}$ and a real Euclidean line bundle $\ell\to Y-\mathcal{Z}_{0}$ such that

$S^{\text{Re}}\simeq S_{0}\otimes_{\mathbb{R}}\ell$

on $Y-\mathcal{Z}_{0}$ . Moreover, under this isomorphism, $\nabla_{A_{0}}$ is taken to the connection formed from the spin connection on $S_{0}$ and the unique flat connection on $\ell$ , with an $\mathbb{R}$ -linear perturbation commuting with $\gamma$ arising from $B_{0}$ . ∎

As a consequence of items (1) and (2) above, the Dirac operator on $S_{E}$ restricts to a Dirac operator

\not{D}_{A_{0}}^{\text{Re}}:\Gamma(S^{\text{Re}})\to\Gamma(S^{\text{Re}}),

(3.8)

and likewise for the imaginary part. When the subbundle in question is evident, we will omit the superscript from the notation. The isomorphism in Item (3) intertwines (3.8) and the Dirac operator on $S_{0}\otimes_{\mathbb{R}}\ell$ formed using the connection in Item (3). This leads to the following equivalence, which is the manifestation of the Haydys Correspondence in this particular setting.

Corollary 3.6.

Suppose that $\mathcal{Z}_{0}\subset Y$ is a smooth, embedded link. Then the data of a limiting configuration satisfying (3.6) as in Theorem 3.2 is equivalent to a $\mathbb{Z}_{2}$ -Harmonic spinor $(\mathcal{Z}_{0},A_{0},\Phi_{0})$ on $S^{\text{Re}}\simeq S_{0}\otimes_{\mathbb{R}}\ell$ satisfying

\not{D}_{A_{0}}\Phi_{0}=0\hskip 56.9055pt\nabla_{A_{0}}\Phi_{0}\in L^{2}

with respect to $(g_{0},B_{0})$ .

Proof.

Except for the integrability condition, the corollary follows directly from isomorphism of item (3) in Lemma 3.5. The fact that $\nabla\Phi_{0}\in L^{2}$ will follow from Lemma 4.5 in Section 4, which shows requirement that $\nabla\Phi_{0}\in L^{2}$ is equivalent (for regular $\mathbb{Z}_{2}$ -harmonic spinors) to requiring that $|\Phi_{0}|$ extend continuously over $\mathcal{Z}_{0}$ with $\mathcal{Z}_{0}\subset|\Phi_{0}|^{-1}(0)$ . ∎

The purpose of the Haydys Correspondence is that it takes advantage of the gauge freedom to temper the singular nature of the limiting equations. To explain this further, limiting configurations are considered up to $U(1)$ gauge transformations and solve the globally degenerate system of equations (3.6); indeed, the symbol (after gauge-fixing) in the curvature equation of (1.4) vanishes everywhere as $\varepsilon\to 0$ , leading to a loss of ellipticity. On the other side of the Haydys correspondence, $\mathbb{Z}_{2}$ -harmonic spinors are considered only up to the action of $\mathbb{Z}_{2}$ by sign on $S_{0}\otimes_{\mathbb{R}}\ell$ , and solve the Dirac equation on $S_{0}$ , which is a singular elliptic equation whose symbol degenerates only locally along $\mathcal{Z}_{0}$ (see the local description in Section 4.1). While the first type of degeneracy appears to at first be rather intractable, the latter description places the problem in the well-studied class of elliptic edge problems [36, 39].

Remark 3.7.

Items (1) and (2) show that $S^{\text{Re}}$ is a 4-dimensional real Clifford module on $Y-\mathcal{Z}_{0}$ . The isomorphism in Item (3) of Lemma 3.5 endows it with a complex structure, but not canonically so. In particular, the induced Dirac operator (3.8) is only $\mathbb{R}$ -linear if the $SU(2)$ -connection $B$ is non-trivial (as it must be for condition (2) of Definition 1.1 to be met). In four-dimensions, the bundle $S^{\text{Re}}$ need not be topologically isomorphic to the spinor bundle twisted by a real line bundle.

3.3. Recovering $\text{Spin}^{c}$ Structures

By Corollary 3.6, a limiting configuration $(\mathcal{Z}_{0},A_{0},\Phi_{0})$ gives rise to a $\mathbb{Z}_{2}$ -harmonic spinor. The reverse is not immediately true because, in contrast to the Seiberg–Witten equations, the definition of a $\mathbb{Z}_{2}$ -harmonic spinor makes no references to a $\text{Spin}^{c}$ structure. The topological information of the $\text{Spin}^{c}$ structure is lost in the limiting process of Theorem 3.2 and must be reconstructed before the gluing analysis begins.

Specifically, we seek a $\text{Spin}^{c}$ structure with spinor bundle $S$ such that $S^{\text{Re}}$ as defined by (3.7) satisfies the isomorphism of Item (3) from Lemma 3.5 for the twisted spinor bundle $S_{0}\otimes_{\mathbb{R}}\ell$ that hosts $\Phi_{0}$ . Given such an $S$ , Corollary 3.6 implies that $(\mathcal{Z}_{0},A_{0},\Phi_{0})$ may be viewed as a (non-smooth) configuration on the subbundle $S^{\text{Re}}\subset S_{E}$ of two-spinor bundle formed from $S$ , and the gluing analysis begins from there.

The following lemma reconstructs the correct $\text{Spin}^{c}$ -structure for the gluing, given an orientation of $\mathcal{Z}_{0}$ . The proof may be found in Section 3 of [41]; see also [24] for more results in this direction.

Lemma 3.8.

Let $(\mathcal{Z}_{0},A_{0},\Phi_{0})$ be a regular $\mathbb{Z}_{2}$ -harmonic spinor on $S_{0}\otimes_{\mathbb{R}}\ell$ . An orientation of $\mathcal{Z}_{0}$ determines a unique $\text{Spin}^{c}$ -structure with spinor bundle $S\to Y$ satisfying the following criteria.

(1)

The first chern class satisfies

$c_{1}(S)=-\text{PD}[\mathcal{Z}_{0}]$

with the specified orientation of $\mathcal{Z}_{0}$ .
(2)

$S$ extends $S_{0}\otimes_{\mathbb{R}}\ell$ in the sense that $S|_{Y-\mathcal{Z}_{0}}\simeq S_{0}\otimes_{\mathbb{R}}\ell$ . Moreover, there is an isomorphism

$S_{0}\otimes_{\mathbb{R}}\ell\simeq S^{\text{Re}}\subset S_{E}$

where $S_{E}=S\otimes_{\mathbb{C}}E$ , which makes $\Phi_{0}$ a smooth section of $S^{\text{Re}}\to Y-\mathcal{Z}_{0}$ .

Notice that we do not assume $\mathcal{Z}_{0}$ is connected, thus there are $2^{k}$ possible choices of orientation when $\mathcal{Z}_{0}$ has $k$ components.

Given Lemma 3.8, the data of a regular $\mathbb{Z}_{2}$ -harmonic spinor (with an orientation of $\mathcal{Z}_{0}$ ) is equivalent to that of a limiting configuration $(\mathcal{Z}_{0},A_{0},\Phi_{0})$ using the induced $\text{Spin}^{c}$ -structure on $Y$ . From here on, we therefore cease to distinguish between a regular (oriented) $\mathbb{Z}_{2}$ -harmonic spinor $(\mathcal{Z}_{0},A_{0},\Phi_{0})$ and the corresponding limiting configuration denoted (purposefully) by the same triple.

3.4. Adapted Coordinates

In order to describe Seiberg–Witten configurations in a neighborhood of $\mathcal{Z}_{0}$ , we use adapted coordinate systems, constructed as follows.

Fix a component $\mathcal{Z}_{j}$ of $\mathcal{Z}_{0}$ with length $|\mathcal{Z}_{j}|$ and an arclength parameterization $p:S^{1}\to\mathcal{Z}_{j}$ . Choose an orthonormal frame $\{n_{1},n_{2}\}$ for the pullback $p^{*}N\mathcal{Z}_{0}$ of the normal bundle, ordered so that $\{\dot{p},n_{1},n_{2}\}$ is an oriented frame of $p^{*}TY$ along $\mathcal{Z}_{j}$ . Let $N_{r}(\mathcal{Z}_{0})$ denote the tubular neighborhood of radius $r$ around $\mathcal{Z}_{0}$ , measured in the geodesic distance of $g_{0}$ .

Definition 3.9.

A system of Fermi coordinates $(t,x,y)$ of radius $r_{0}<r_{\text{inj}}$ where $r_{\text{inj}}$ is the injectivity radius of $(Y,g_{0})$ is the diffeomorphism $\bigsqcup_{j}S^{1}\times D_{r_{0}}\simeq N_{r_{0}}(\mathcal{Z}_{0})$ defined by

(t,x,y)\mapsto\text{Exp}_{\mathfrak{p}(t)}(xn_{1}+yn_{2}).

on each component of $\mathcal{Z}_{0}$ , where $t\in[0,|\mathcal{Z}_{j}|)$ is the normalized coordinate in the $S^{1}$ direction. In these coordinates the metric $g_{0}$ has the form

g_{0}=dt^{2}+dx^{2}+dy^{2}\ +\ O(r).

(3.9)

We denote the corresponding cylindrical coordinates by $(t,r,\theta)$ . Given a smooth family $(g_{\tau},\mathcal{Z}_{\tau})$ as in Theorem 1.4 it may be arranged that the Fermi coordinate systems depend smoothly on $\tau$ .

A system of Fermi coordinates induces a trivialization of $(\Omega^{0}\oplus\Omega^{1})(i\mathbb{R})$ . The next lemma, which is proved in Section 3 of [41], provides a satisfactory trivialization of the two-spinor bundle $S_{E}$ .

Lemma 3.10.

In the neighborhood $N=N_{r_{0}}(\mathcal{Z}_{j})$ of each component of $\mathcal{Z}_{j}$ , there is a local trivialization

(S\otimes_{\mathbb{C}}E)|_{N}\simeq N\times(\mathbb{C}^{2}\otimes_{\mathbb{C}}\mathbb{H})

with the following properties.

(1)

The connection $A_{0}$ has the form

$A_{0}:=\text{d}\ +\ \frac{i}{2}d\theta\ +\ O(1).$

(2)

There is an $\epsilon_{j}\in\{0,1\}$ such that the restriction of $S^{\text{Re}}$ to $N$ is given by

S^{\text{Re}}\big{|}_{N_{r_{0}}(\mathcal{Z}_{j})}=\left\{\begin{pmatrix}\alpha\\ \beta\end{pmatrix}\otimes 1\ +\ e^{-i\theta}e^{-i\epsilon_{j}t}\begin{pmatrix}-\overline{\beta}\\ \overline{\alpha}\end{pmatrix}\otimes j\ \Big{|}\ \alpha,\beta:N\to\mathbb{C}\right\}.

Again, for the $\tau$ -parameterized family of Theorem 1.4, and it may be assumed that this family of trivializations depends smoothly on $\tau$ (using $\mathcal{Z}_{\tau}$ and $A_{\tau}$ respectively).

4. The Singular Linearization

The linearized Seiberg–Witten equations play an essential role in carrying out the alternating iteration outlined in Section 2.4. This section introduces the linearized equations, which are a singular elliptic system at a $\mathbb{Z}_{2}$ -harmonic spinor.

4.1. Singular Linearization

Differentiating, (3.3–3.4), the linearized (extended) Seiberg–Witten equations at a (renormalized) configuration $(\Phi,A)$ acting on a linearized deformation $(\varphi,a)$ are

	$\displaystyle\not{D}_{A}\varphi+\gamma(a)\tfrac{\Phi}{\varepsilon}$	$\displaystyle=$	$\displaystyle 0$		(4.1)
	$\displaystyle(\star d,-d)a+\tfrac{\mu(\varphi,\Phi)}{\varepsilon}$	$\displaystyle=$	$\displaystyle 0$		(4.2)

where $\varepsilon$ is as in (3.5).

To make (4.1–4.2) into an elliptic system, we impose the $\Omega^{0}(i\mathbb{R})$ -valued gauge-fixing condition

-d^{\star}a-\frac{i\langle i\varphi,\Phi\rangle}{\varepsilon}=0,

(4.3)

where $d^{\star}$ denotes the adjoint of the exterior derivative. Extend (the polarization of) $\mu$ to a bilinear map $\mu:S_{E}\otimes S_{E}\to(\Omega^{0}\oplus\Omega^{1})(i\mathbb{R})$ by

\displaystyle\mu(\varphi,\psi)

\displaystyle:=

\displaystyle(i\langle i\varphi,\psi\rangle,\mu_{1}(\varphi,\psi))

where $\mu_{1}$ is what were previously denoted by $\mu$ .

Lemma 4.1.

Suppose that $(\Phi,A)$ is a smooth configuration on $Y$ . Then the (extended, gauge-fixed) linearized Seiberg–Witten equations at $(\Phi,A)$ on a linearized deformation $(\varphi,a)\in\Gamma(S_{E})\oplus(\Omega^{0}\oplus\Omega^{1})(i\mathbb{R})$ take the form:

\mathcal{L}_{(\Phi,A)}(\varphi,a)=\begin{pmatrix}\not{D}_{A}&\gamma(\_)\tfrac{\Phi}{\varepsilon}\\ \tfrac{\mu(\_,\Phi)}{\varepsilon}&\mathbb{d}\end{pmatrix}\begin{pmatrix}\varphi\\ a\end{pmatrix}

(4.4)

where $\mathbb{d}a=\begin{pmatrix}0&-d^{\star}\\ -d&\star d\end{pmatrix}\begin{pmatrix}a_{0}\\ a_{1}\end{pmatrix}$ . Moreover, $\mathcal{L}_{(\Phi,A)}$ is a self-adjoint elliptic operator. ∎

Notice that the parameter $\varepsilon$ is kept implicit in the notation $\mathcal{L}_{(\Phi,A)}$

A $\mathbb{Z}_{2}$ -harmonic spinor $(\mathcal{Z}_{0},A_{0},\Phi_{0})$ (or eigenvector) is not a smooth configuration on $Y$ , and the ellipticity in Lemma 4.1 fails when linearizing at these. At a $\mathbb{Z}_{2}$ -harmonic spinor, however, the linearization admits a pleasing block form with respect to the decomposition of Lemma 3.5. Note in this case the operator acts on sections of bundles only defined over $Y-\mathcal{Z}_{0}$ .

Lemma 4.2.

The (extended, gauge-fixed) linearized Seiberg–Witten equations at $(\mathcal{Z}_{0},A_{0},\Phi_{0})$ take the following form on a linearized deformation $(\varphi_{1},\varphi_{2},a)\in\Gamma(S^{\text{Re}})\oplus\Gamma(S^{\text{Im}})\oplus(\Omega^{0}\oplus\Omega^{1})(i\mathbb{R})$ :

\mathcal{L}_{(\Phi_{0},A_{0})}(\varphi_{1},\varphi_{2},a)=\begin{pmatrix}\not{D}_{A_{0}}&0&0\\ 0&\not{D}_{A_{0}}&\gamma(\_)\frac{\Phi_{0}}{\varepsilon}\\ 0&\frac{\mu(\_,\Phi_{0})}{\varepsilon}.&\mathbb{d}\end{pmatrix}\begin{pmatrix}\varphi_{1}\\ \varphi_{2}\\ a\end{pmatrix}

(4.5)

The same applies at an eigenvector $(\mathcal{Z}_{\tau},A_{\tau},\Phi_{\tau})$ .

Proof.

The proof is an immediate consequence of Lemma 3.5. ∎

As explained in the introduction, the Dirac operator $\not{D}_{A_{0}}$ at the singular connection $A_{0}$ is a degenerate elliptic edge operator¹¹1An elliptic edge operator is an elliptic combination of the derivatives $r\partial_{r},\partial_{\theta},r\partial_{t}$ in Fermi coordinates; technically speaking, $r\not{D}_{A_{0}}$ is the edge operator in question, but the factor of $r$ only shifts the weight. [36]. In the local coordinates and trivializations of Lemma 3.10, the degenerate nature becomes manifest. Near $\mathcal{Z}_{0}$ , it has the form

\not{D}_{A_{0}}=\begin{pmatrix}i\partial_{t}&-2\partial\\ 2\overline{\partial}&-i\partial_{t}\end{pmatrix}\ +\ \frac{1}{4r}\begin{pmatrix}0&e^{-i\theta}\\ -e^{i\theta}&0\end{pmatrix}\ +\ \mathfrak{d}_{1}\ +\ \mathfrak{d}_{0}

(4.6)

where $\mathfrak{d}_{1}=O(r)\nabla$ is a first order operator vanishing along $\mathcal{Z}_{0}$ , and $\mathfrak{d}_{0}=O(1)$ is a bounded zeroth order operator. In particular, the second term, which arises from the non-trivial holonomy of $A_{0}$ , is unbounded on $L^{2}$ . Equivalently, $r\not{D}_{A_{0}}$ is a elliptic operator with $L^{2}$ -bounded terms whose symbol degenerates along $\mathcal{Z}_{0}$ . In fact, there are no function spaces with local norms for which the extension of this operator is Fredholm.

The remainder of Sections 4–5 develop the analysis of this operator. The lower $2\times 2$ block in (4.5) is analyzed in Section 7.3, using a trick that reduces it to standard elliptic theory.

4.2. The Singular Dirac Operator

This section summarizes results from the elliptic edge theory of the singular Dirac operator (4.6). More generally, Lemmas 3.5 and 4.2 (and the expression 4.6) apply for any $\tau\in(-\tau_{0},\tau_{0})$ , and we consider the family of operators

\not{D}_{A_{\tau}}:\Gamma(Y-\mathcal{Z}_{\tau};S^{\text{Re}})\longrightarrow\Gamma(Y-\mathcal{Z}_{\tau};S^{\text{Re}})

(4.7)

where the dependence of $S^{\text{Re}}$ on $\tau$ is suppressed in the notation. More detailed discussion and proofs of the statements in this subsection may be found in Sections 2–4 of [43], and a discussion from the perspective of the microlocal analysis of elliptic edge operators is contained in [28].

To begin, consider the following function spaces. Let $r$ be a smooth weight function such that

r(y)=\begin{cases}\text{dist}(y,\mathcal{Z}_{\tau})\hskip 28.45274pty\in N_{r_{0}/2}(\mathcal{Z}_{\tau})\\ \text{const.}\hskip 48.36958pty\in Y-N_{r_{0}}(\mathcal{Z}_{\tau})\end{cases}

(4.8)

where the distance is measured using the metric $g_{\tau}$ (though we omit this from the notation), and $r_{0}$ is as in Definition 3.9.

Definition 4.3.

For a constant $\nu\in\mathbb{R}$ , the weighted edge Sobolev spaces (of regularity $m=0,1$ ) are defined by

	$\displaystyle r^{1+\nu}H^{1}_{e}(Y-\mathcal{Z}_{\tau};S^{\text{Re}})$	$\displaystyle:=$	$\displaystyle\Big{\{}\ u\ \ \Big{\|}\ \ \int_{Y-\mathcal{Z}_{\tau}}\left(\|\nabla u\|^{2}+\frac{\|u\|^{2}}{r^{2}}\ \right)r^{-2\nu}dV\ <\ \infty\Big{\}}$		(4.9)
	$\displaystyle r^{\nu}L^{2}(Y-\mathcal{Z}_{\tau};S^{\text{Re}})$	$\displaystyle:=$	$\displaystyle\Big{\{}\ v\ \ \Big{\|}\ \ \int_{Y-\mathcal{Z}_{\tau}}{\|v\|^{2}}\ r^{-2\nu}dV\ <\ \infty\Big{\}}$		(4.10)

where $\nabla$ denotes the covariant derivative on $S^{\text{Re}}$ formed from $A_{\tau}$ and the background pair $(g_{\tau},B_{\tau})$ . These spaces are equipped with the norms given by the (positive) square root integrals required to be finite, and the Hilbert space structures arising from their polarizations.

$\not{D}_{A_{\tau}}$ extends to a bounded linear operator

\not{D}_{A_{\tau}}:r^{1+\nu}H^{1}_{e}(Y-\mathcal{Z}_{\tau};S^{\text{Re}})\longrightarrow r^{\nu}L^{2}(Y-\mathcal{Z}_{\tau};S^{\text{Re}}).

(4.11)

The results of [43, Prop 2.4], and [36, Thm 6.1] show the following.

Lemma 4.4.

For $-\tfrac{1}{2}<\nu<\tfrac{1}{2}$ ,

(A)

(4.11) is left semi-Fredholm (i.e. $\ker(\not{D}_{A_{\tau}})$ is finite-dimensional, and $\text{Range}(\not{D}_{A_{\tau}})$ is closed.)

(B)

There is a constant $C_{\nu}>0$ such that the following “semi-elliptic” estimate holds:

\|u\|_{r^{1+\nu}H^{1}_{e}}\leq C_{\nu}\Big{(}\|\not{D}_{A_{\tau}}u\|_{r^{\nu}L^{2}}\ +\ \|\pi_{\nu}(u)\|_{r^{\nu}L^{2}}\Big{)}\hskip 42.67912pt\text{ for all }\ \ u\in r^{1+\nu}H^{1}_{e}

(4.12)

where $\pi_{\nu}$ is the $r^{\nu}L^{2}$ -orthogonal projection onto the finite-dimensional kernel.

(C)

When the assumptions of Theorem 1.6 hold, (4.12) holds uniformly for $\tau\in(\tau_{0},\tau_{0})$ when $\pi_{\nu}$ is replaced by the projection to the 1-dimensional eigenspace spanned by $\Phi_{\tau}$ .

The finite-dimensional kernel for $\nu=0$ is, by definition, the set of $\mathbb{Z}_{2}$ -harmonic spinors. The upcoming Lemma 4.5 implies that this space is independent of $\nu$ in the range $-\tfrac{1}{2}<\nu<\tfrac{1}{2}$ .

A $\mathbb{Z}_{2}$ -harmonic spinor is not necessarily smooth. Notice that the estimate (4.12) assumes a priori that $u\in r^{1+\nu}H^{1}_{e}$ and does not imply that an $r^{\nu}L^{2}$ -solution can be bootstrapped to $u\in r^{1+\nu}H^{1}_{e}$ . Thus elliptic bootstrapping in the standard sense fails for $\not{D}_{A_{\tau}}$ . Consequently, even for $\nu=0$ , the kernel and cokernel of (4.11) need not coincide, despite the fact that $\not{D}_{A_{\tau}}$ is formally self-adjoint. This is a general phenomenon for elliptic edge operators; the appropriate substitute for smoothness in this setting is the existence of polyhomogeneous expansions along the singular set $\mathcal{Z}_{\tau}$ as we now explain.

Fix a choice of Fermi coordinates near $\mathcal{Z}_{\tau}$ as in Definition 3.9. The results of [36, Sec. 7] imply the following regularity result about $\mathbb{Z}_{2}$ -harmonic spinors (see also [26, App. A], [43, Section 3.3], and [18] for more general exposition).

Lemma 4.5.

Suppose that $\Phi\in r^{1+\nu}H^{1}_{e}(Y-\mathcal{Z}_{\tau};S^{\text{Re}})$ is a $\mathbb{Z}_{2}$ -harmonic spinor or eigenvector. Then $\Phi$ admits a local polyhomogenous expansion of the following form:

\displaystyle\Phi

\displaystyle\sim

\displaystyle\begin{pmatrix}c(t)\ \ \ \ \ \\ d(t)e^{-i\theta}\end{pmatrix}r^{1/2}\ +\ \sum_{n\geq 1}\sum_{k=-2n}^{2n+1}\sum_{p=0}^{n-1}\begin{pmatrix}\ \ c_{n,k,p}(t)e^{ik\theta}\ \ \ \ \ \\ \ \ d_{n,k,p}(t)e^{ik\theta}e^{-i\theta}\ \ \end{pmatrix}r^{n+1/2}(\log r)^{p}

(4.13)

where $c(t),d(t),c_{k,m,n}(t),d_{k,m,n}(t)\in C^{\infty}(S^{1};\mathbb{C})$ . Here, $\sim$ denotes convergence in the following sense: for every $N\in\mathbb{N}$ , the partial sums

\Phi_{N}=\sum_{n\leq N}\sum_{k=-2n}^{2n+1}\sum_{p=0}^{n-1}\begin{pmatrix}\ \ c_{n,k,p}(t)e^{ik\theta}\ \ \ \ \ \\ \ \ d_{n,k,p}(t)e^{ik\theta}e^{-i\theta}\ \ \end{pmatrix}r^{n+1/2}(\log r)^{p}

satisfy the pointwise bounds

\displaystyle|\Phi-\Phi_{N}|

\displaystyle\leq

\displaystyle C_{N}r^{N+1+\tfrac{1}{4}}\hskip 56.9055pt|\nabla_{t}^{\alpha}\nabla^{\beta}(\Phi-\Phi_{N})|\leq C_{N,\alpha,\beta}r^{N+1+\tfrac{1}{4}-|\beta|}

(4.14)

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 directions normal to $\mathcal{Z}_{\tau}$ . ∎

Notice that the non-degeneracy condition of Definition 1.1 is equivalent to the statement that the leading coefficients satisfy $|c(t)|^{2}+|d(t)|^{2}>0$ for all $t\in\mathcal{Z}_{\tau}$ . Note also that the existence of the expansion (4.13) implies that the second condition of (1.6) is equivalent to the requirement that $|\Phi_{0}|$ extends continuously to $Y$ with $\mathcal{Z}_{0}=|\Phi_{0}|^{-1}(0)$ as in Theorem 3.2.

4.3. Polynomial Decay

This section establishes that solution of the Dirac equation in $rH^{1}_{e}$ obey a decay property towards the singular set. This decay property is the precise manifestation of requirement (II) in the alternating method described in Section 2.2.

Because the projection to the obstruction bundle is a highly non-local operator on $Y$ , cancelling the obstruction using the deformations of the singular set (the horizontal arrow in 2.17) disrupts the property that the error is supported where $d\chi^{+}\neq 0$ . The following generalization of support is needed to address this.

Definition 4.6.

A spinor $g\in L^{2}(Y-\mathcal{Z}_{\tau};S^{\text{Re}})$ is said to have $\varepsilon^{\beta}$ -effective support if there is a constant $C$ such that for all $-\nu\in(-\tfrac{1}{2},0],$

\|g\|_{r^{\nu}L^{2}}\leq C\varepsilon^{-\gamma}\varepsilon^{-{\nu\beta}}\|g\|_{L^{2}}.

(4.15)

holds.

The definition (4.10) implies this property holds holds if $\text{supp}(g)$ lies in the region where $r\geq C\varepsilon^{\beta}$ .

The following lemma provides the relevant decay property for solutions of the Dirac equation in the outside region. In the statement, $\chi^{-},\nu^{-}$ are in Section 2.

Lemma 4.7.

Suppose that $g\in L^{2}\cap\text{Range}(\not{D}_{A_{\tau}})$ has $\varepsilon^{1/2}$ -effective support, and let $\psi\in rH^{1}_{e}(Y-\mathcal{Z}_{\tau};S^{\text{Re}})$ be the unique $\Phi_{\tau}$ -perpendicular solution to

\not{D}_{A_{\tau}}\psi=g.

Then

\|d\chi^{-}.\psi\|_{L^{2}}\ +\ \|(1-\chi^{-})g\|_{L^{2}}\leq C\varepsilon^{1/12-\gamma}\|g\|_{L^{2}}.

(4.16)

Proof.

For the duration of the proof set, $\nu=\nu^{-}=\tfrac{1}{2}-\gamma$ , and $\chi=\chi^{-}$ . The elliptic estimate (4.12) from Lemma 4.4 applies to show that

\|\psi\|_{r^{1+{\nu}}H^{1}_{e}}\ \leq\ C\|g\|_{r^{\nu}L^{2}}\ \leq\ C\varepsilon^{-\gamma}\varepsilon^{-\tfrac{\nu}{2}}\|g\|_{L^{2}}=C\varepsilon^{-\tfrac{1}{4}-\gamma^{\prime}}\|g\|_{L^{2}}.

Then, since $\chi$ is a logarithmic cut-off function with $|d\chi.\psi|\leq\frac{C}{r}|\psi|$ , and is supported where $r\sim\varepsilon^{2/3-\gamma}$ ,

\|d\chi.\psi\|_{L^{2}}\leq C\left(\int_{d\chi\neq 0}\frac{|\psi|^{2}}{r^{2}}\frac{r^{2\nu}}{r^{2\nu}}\ dV\right)^{\tfrac{1}{2}}\leq C\varepsilon^{\nu(\tfrac{2}{3}-\gamma)}\|\psi\|_{r^{1+\nu}H^{1}_{e}}\leq C\varepsilon^{\nu(\tfrac{2}{3}-\gamma)}\varepsilon^{-\tfrac{1}{4}-\gamma^{\prime}}\|g\|_{L^{2}}

Relabeling $\gamma\mapsto\gamma^{\prime}+\nu\gamma$ and combining power of $\varepsilon$ gives the bound on the first term of (4.16). Since $\|(1-\chi)g\|_{L^{2}}=\|(1-\chi)\not{D}_{A_{\tau}}\psi\|_{L^{2}}\leq\|\psi\|_{rH^{1}_{e}(\chi\neq 1)}$ , a similar argument covers the second term. ∎

Remark 4.8.

The decay of solutions of $\not{D}\psi=g$ away from the support of $g$ can alternatively be obtained using the decay of the operator’s Schwartz kernel as in [36]. For our purposes, the above approach using weights is easier, and also applies to the inside region, where the properties of the Schwartz kernel of the linearization become quite opaque.

5. The Obstruction Bundle

This section characterizes the infinite-dimensional cokernel of (4.11) for the weight $\nu=0$ . The cokernel may be canonically identified with the $L^{2}$ -solutions of the formal adjoint operator, which for $\nu=0$ is $\not{D}_{A_{\tau}}$ itself (now understood in a weak sense with domain $L^{2}$ , see Section 2.2 of [43]).

5.1. The Obstruction Basis

To understand the form of the cokernel it is instructive for consider the model case of $Y^{\circ}=S^{1}\times\mathbb{R}^{2}$ . Endow $Y^{\circ}$ with coordinates $(t,x,y)$ and the product metric where $S^{1}$ has length $2\pi$ , and set $\mathcal{Z}_{\circ}=S^{1}\times\{0\}$ . Let $E=\underline{\mathbb{H}}$ be the trivial bundle equipped with the product connection. As in Section 3.2, $S^{\text{Re}}\subset\underline{\mathbb{C}}^{2}\otimes\underline{H}$ is given elements of the form $\tfrac{1}{2}(\Psi+\sigma\Psi)$ . The Dirac operator takes the form (4.6) here with $\mathfrak{d}_{1},\mathfrak{d}_{0}=0$ . By Section 3 of [43] or by direct computation, the $L^{2}$ -kernel of $\not{D}_{A_{0}}$ on $S^{\text{Re}}$ is given by the $L^{2}$ span of

\Psi_{\ell}^{\circ}:=\frac{1}{2}\left(\text{Id}+\sigma\right)\left({\sqrt{|\ell|}}\frac{e^{i\ell t}e^{-|\ell|r}}{\sqrt{r}}\begin{pmatrix}e^{-i\theta}\\ \text{sgn}(\ell)\end{pmatrix}\right)

(5.1)

for $\ell\in\mathbb{Z}$ . Note that any $l^{2}$ -linear combination $\sum c_{\ell}\Psi_{\ell}^{\circ}$ lies in $L^{2}$ and is smooth away from $\mathcal{Z}_{\circ}$ , but fails to lie in $rH^{1}_{e}$ , because $\nabla r^{-1/2}\notin L^{2}$ . In this case, $\text{coker}(\not{D}_{A_{\tau}})$ may be identified with $L^{2}(\mathcal{Z}_{\circ};\underline{\mathbb{C}})$ via Fourier series with $\Psi_{\ell}^{\circ}\mapsto e^{i\ell t}$ . ²²2Here, we are glossing over the fact that the $\ell=0$ modes are not in $L^{2}$ on the non-compact $Y^{\circ}$ ; compactness of $Y$ ameliorates this, see Section 4.3 of [43].

For a general compact manifold $Y$ , the cokernel has a similar characterization given in the upcoming Propositions 5.2 and 5.3. In this case, there is the following caveat: because of the appearance of the $\mathbb{Z}_{2}$ -harmonic spinor $\Phi_{0}$ at $\tau=0$ , the family of cokernels has a discontinuity at $\tau=0$ (the cokernel would jump in dimension here, were it finite-dimensional). Instead, we work with the following “thickening” of the family of cokernels.

Definition 5.1.

The Obstruction Space associated to the data $(\mathcal{Z}_{\tau},g_{\tau},B_{\tau})$ is the (infinite-dimensional) subspace of $L^{2}(S^{\text{Re}})$ defined by

\displaystyle{\bf Ob}(\mathcal{Z}_{\tau})

\displaystyle:=

\displaystyle\text{Span}\left(\ker(\not{D}_{A_{\tau}}|_{L^{2}})\ ,\ \Phi_{\tau}\right)

(5.2)

This has an $L^{2}$ -orthogonal decomposition $\text{\bf Ob}(\mathcal{Z}_{\tau})=\text{\bf Ob}^{\perp}(\mathcal{Z}_{\tau})\oplus\mathbb{R}\Phi_{\tau}.$

The Obstruction Bundle is the family of obstruction spaces parameterized by $\tau\in(-\tau_{0},\tau_{0})$

\text{\bf Ob}:=\bigsqcup_{\tau}\text{\bf Ob}(\mathcal{Z}_{\tau})\to(-\tau_{0},\tau_{0}).

The below proposition shows that Ob is a smooth Banach vector bundle, with fiber modeled on a space of spinors on $\mathcal{Z}_{0}$ . For this purpose, let

\mathcal{S}_{\tau}\to\mathcal{Z}_{\tau}

(5.3)

be rank 1 complex Clifford module whose fiber is the $+i$ eigenspace of $\gamma(dt)$ where $dt$ is the normalized, oriented unit tangent vector to $\mathcal{Z}_{\tau}$ and $\gamma$ is Clifford multiplication both defined using the metric $g_{\tau}$ .

Proposition 5.2.

There is a family of continuous linear isomorphisms

\Xi:\text{\bf Ob}\to(-\tau_{0},\tau_{0})\times(L^{2}(\mathcal{Z}_{0};\mathcal{S}_{0})\oplus\mathbb{R})

that endow the obstruction bundle with the structure of a smooth Hilbert vector bundle over $(-\tau_{0},\tau_{0})$ . Moreover, $\Xi_{\tau}^{-1}$ restricted to the $\mathbb{R}$ factor is the inclusion of the span of $\Phi_{\tau}$ .

Proof.

See Propositions 4.2 and 8.5 in [43]. ∎

The next proposition gives an explicit basis for $\text{\bf Ob}^{\perp}(\mathcal{Z}_{\tau})$ indexed by $\mathbb{Z}\times\pi_{0}(\mathcal{Z}_{\tau})$ , and gives bounds showing that the support of these basis elements increasingly concentrates near $\mathcal{Z}_{\tau}$ as the first index increases, just as in the model case of $Y^{\circ}$ .

For the statement of the proposition, let $\mathcal{S}_{\tau}$ be as defined preceding Proposition 5.2). For simplicity, we state the proposition in the case that $\mathcal{Z}_{\tau}$ has a single component; the general case is the obvious extension with an additional index $\alpha$ ranging over $\pi_{0}(\mathcal{Z}_{\tau})$ . A choice of arclength coordinate $t$ on $\mathcal{Z}_{\tau}$ induces a trivialization of $\mathcal{Z}_{\tau}$ , and there is a Dirac operator on $\mathcal{S}_{\tau}$ given by $i\partial_{t}$ in this trivialization. Let $\phi_{\ell}$ denote the eigenfunctions of this Dirac operator, which are associated with $e^{i\ell t}$ for $\ell\in 2\pi\mathbb{Z}/|\mathcal{Z}_{\tau}|$ in the trivialization.

Proposition 5.3.

([43, Prop 4.3] For $\tau\in(-\tau_{0},\tau_{0})$ , there is a bounded linear isomorphism

\text{ob}_{\tau}\oplus\iota:L^{2}(\mathcal{Z}_{\tau};\mathcal{S}_{\tau})\oplus\mathbb{R}\longrightarrow\text{\bf Ob}(\mathcal{Z}_{\tau})

(5.4)

and a basis of $\text{\bf Ob}^{\perp}(\mathcal{Z}_{\tau})$ which satisfy the following properties.

(A)

When $\not{D}_{A_{\tau}}$ is complex linear, there is a complex basis $\Psi_{\ell}$ of $\text{\bf Ob}^{\perp}(\mathcal{Z}_{\tau})$ indexed by $\ell\in 2\pi\mathbb{Z}/|\mathcal{Z}_{\tau}|$ such that the $\text{\bf Ob}(\mathcal{Z}_{\tau})$ -component of a spinor $\psi\in L^{2}$ under (5.4) is given by

\text{ob}_{\tau}^{-1}(\Pi^{\text{Ob}}\psi)=\sum_{\ell}\langle\psi,\Psi_{\ell}\rangle_{{\mathbb{C}}}\phi_{\ell}.\hskip 85.35826pt\iota^{-1}(\Pi^{\text{Ob}}\psi)=\langle\psi,\Phi_{\tau}\rangle\Phi_{\tau}.

(5.5)

where $\langle-,-\rangle_{\mathbb{C}}$ is the hermitian inner product. Moreover,

\Psi_{\ell}=\chi\Psi^{\circ}_{\ell}+\zeta_{\ell}+\xi_{\ell}

where

•

$\Psi_{\ell}^{\circ}$ are the $L^{2}$ -orthonormalized Euclidean obstruction elements (5.1) and $\chi$ is a cutoff function supported on a tubular neighborhood of $\mathcal{Z}_{\tau}$ .
•

$\zeta_{\ell}$ is a perturbation with $L^{2}$ -norm $O(\tfrac{1}{|\ell|})$ which decays exponentially away from $\mathcal{Z}_{0}$ with exponent $e^{-c|\ell|}$ .
•

$\xi_{\ell}$ is a perturbation of $L^{2}$ -norm $O(\tfrac{1}{|\ell|^{N}})$ for any $N\geq 2$ .

(B)

In the case that $\not{D}_{A_{0}}$ is only $\mathbb{R}$ -linear, then there is a real basis

\Psi_{\ell}^{\text{re}}=\chi\Psi^{\circ}_{\ell}+\zeta^{\text{re}}_{\ell}+\xi^{\text{re}}_{\ell}\hskip 56.9055pt\Psi_{\ell}^{\text{im}}=i(\chi\Psi^{\circ}_{\ell})+\zeta^{\text{im}}_{\ell}+\xi^{\text{im}}_{\ell}

satisfying identical bounds where (5.5) uses the inner product $\langle\psi,\Psi_{\ell}\rangle_{\mathbb{C}}=\langle\psi,\Psi_{\ell}^{\text{re},{}}\rangle\ +\ i\langle\psi,\Psi_{\ell}^{\text{im},{}}\rangle.$

∎

A more precise meaning of the second and third bullet points is given in Proposition 4.3 of [43], but is not needed for our purposes here.

As a consequence of the concentration of the cokernel modes around $\mathcal{Z}_{\tau}$ as $|\ell|\to\infty$ , the obstruction space displays a rather different type of regularity than the Sobolev regularity of the spinors on $Y-\mathcal{Z}_{\tau}$ (see Section 6.1 of [43] for more discussion). Regularity of a spinor $\Pi_{\tau}(\Psi)\in\text{\bf Ob}(\mathcal{Z}_{\tau})$ , means that $\text{ob}_{\tau}^{-1}\circ\Pi_{\tau}(\Psi)\in L^{s,2}(\mathcal{Z}_{\tau};\mathcal{S}_{\tau})\subseteq L^{2}(\mathcal{Z}_{\tau};\mathcal{S}_{\tau})$ for some $s>0$ . By Proposition 5.3, this is a question of how fast the sequence of inner products (5.5) decays as $|\ell|\to\infty$ . Because of the concentration, latter depends both on the Sobolev regularity of $\Psi$ and its rate of decay toward $\mathcal{Z}_{\tau}$ ; in particular if a spinor $\Psi$ is compactly supported in $\mathcal{Z}_{\tau}$ ’s complement, its obstruction components enjoy high regularity, even if all its covariant derivatives fail to be integrable on $Y-\mathcal{Z}_{\tau}$ in the normal sense. This improved regularity is crucial in Sections 9–11, where it is applied to the error terms in the alternating iteration which are compactly supported away from $\mathcal{Z}_{\tau}$ in the neck region.

The next lemma makes this improvement of regularity precise. Given $L_{0}\in\mathbb{N}$ , let

{\pi}_{L_{0}}(\phi_{\ell})=\begin{cases}\phi_{\ell}\hskip 28.45274pt|\ell|\leq L_{0}\\ 0\ \ \hskip 27.03003pt|\ell|>L_{0}\end{cases}

denote the Fourier projector to modes less than $L_{0}$ in $L^{2}(\mathcal{Z}_{\tau};\mathcal{S}_{\tau})$ .

Lemma 5.4.

For any $N\in\mathbb{N}$ , there are $C_{N},R_{N}>0$ such that the following is satisfied. If $\Psi\in L^{2}(Y-\mathcal{Z}_{\tau};S^{\text{Re}})$ is a spinor and $R<R_{N}$ is such that $\text{dist}(\text{supp}(\Psi),\mathcal{Z}_{\tau})\geq R$ . Then for any $0<\gamma<<1$ ,

\|(1-\pi_{L_{0}})\circ\text{ob}^{-1}_{\tau}\circ\Pi_{\tau}(\Psi)\|_{L^{2}}\leq\frac{C_{N}}{(L_{0})^{N}}\|\Psi\|_{L^{2}}

holds for any $L_{0}>R^{-\tfrac{1}{1-\gamma}}$ . In particular, $\text{ob}_{\tau}^{-1}\circ\Pi_{\tau}(\Psi)\in C^{\infty}(\mathcal{Z}_{\tau};\mathcal{S}_{\tau})$ .

Proof.

By Proposition 5.3, the projection $(1-\pi^{\text{low}})\circ\text{ob}_{\tau}^{-1}\circ\Pi_{\tau}(\Psi)$ is given by

\sum_{|\ell|>L_{0}}\langle\Psi_{\ell},e\rangle_{\mathbb{C}}\phi_{\ell}.

(5.6)

The result then follows straightforwardly from using the decomposition $\Psi_{\ell}=\chi\Psi^{\circ}_{\ell}+\zeta_{\ell}+\xi_{\ell}$ . More specifically, using the first two bullet points of Item (A) in Proposition 5.3, the first two terms contribute $O(L_{0}\text{Exp}(-L_{0}R))\leq O(L_{0}\text{Exp}(-L_{0}^{\gamma}))$ since the assumption on $L_{0}$ implies $L_{0}R>L_{0}^{\gamma}$ . For $R_{N}$ sufficiently small, this dominates $L_{0}^{-N}$ . Using Cauchy-Schwartz on $\xi_{\ell}$ and the third bullet point of Item (A) with $N^{\prime}=N+2$ then gives the desired bound. Applying the estimate repeatedly for $L_{0}=2^{k}$ shows that $\text{ob}^{-1}_{\tau}\circ\Pi_{\tau}(\Psi)\in L^{N,2}(\mathcal{Z}_{\tau};\mathcal{S}_{\tau})$ for every $N$ . ∎

5.2. The Surjective Weights

By the general theory of elliptic edge operators ([36, Thm 6.1]), a first order elliptic edge operator $L$

L:r^{1+\nu}H^{1}_{e}\to r^{\nu}L^{2}

is semi-Fredholm provided that the weight $\nu$ lies outside the discrete set $I(L)$ of indicial roots. More specifically, there are critical weights $\underline{\nu}<\overline{\nu},$ such that (i) $L$ is left semi-Fredholm (finite-dimensional kernel and closed range) for $\overline{\nu}<\nu\notin I(L)$ , and (ii) is right semi-Fredholm (finite-dimensional cokernel) for $\underline{\nu}>\nu\notin I(L)$ . For the singular Dirac operator $\not{D}_{A_{\tau}}$ , the critical weights (by [28, Prop 3.9]) are:

I(r\not{D}_{A_{\tau}})=\mathbb{Z}+\tfrac{1}{2}\hskip 71.13188pt\overline{\nu}=\underline{\nu}=-\tfrac{1}{2}.

(5.7)

Thus when the weight $\nu$ decreases past the critical weight $-\tfrac{1}{2}$ , $\not{D}_{A_{\tau}}$ flips from being left semi-Fredholm to being right semi-Fredholm.

Lemma 4.4 shows (4.11) is left semi-Fredholm for $\nu\in(-\tfrac{1}{2},\tfrac{1}{2})$ . The next lemma shows it is right semi-Frehdolm in the specific case $\nu=-1$ .

Lemma 5.5.

For $\nu=-1$ ,

\not{D}_{A_{\tau}}:H^{1}_{e}(Y-\mathcal{Z}_{\tau};S^{\text{Re}})\longrightarrow r^{-1}L^{2}(Y-\mathcal{Z}_{\tau};S^{\text{Re}})

(5.8)

has a finite-dimensional cokernel. Moreover, for $u\perp_{L^{2}}\ker(\not{D}_{A_{\tau}}|_{H^{1}_{e}})$ , the elliptic estimate

\|u\|_{H^{1}_{e}}\leq C\|\not{D}_{A_{\tau}}u\|_{r^{-1}L^{2}}.

holds uniformly in $\tau$ .

Proof.

See [36, Thm 6.1] and [28, Thm 3.18]. ∎

Corollary 5.6.

There is a closed subspace $\mathcal{X}_{\tau}\subseteq H^{1}_{e}$ such that

\not{D}_{A_{\tau}}:\mathcal{X}_{\tau}\to\text{\bf Ob}^{\perp}(\mathcal{Z}_{\tau})

is an isomorphism. In particular, there is a $C>0$ such that if $\Psi\in\text{\bf Ob}^{\perp}(\mathcal{Z}_{\tau})$ , then there exists a unique $u_{\Psi}\in\mathcal{X}_{\tau}$ satisfying

\not{D}_{A_{\tau}}u_{\Psi}=\Psi,\hskip 56.9055pt\|u_{\Psi}\|_{H^{1}_{e}}\leq C\|\Psi\|_{L^{2}},

where $C$ is uniform in $\tau$ .

Proof.

Self-adjointness implies that the cokernel of (5.8) is spanned by the $\mathbb{Z}_{2}$ -harmonic spinors, which are orthogonal to $\text{\bf Ob}^{\perp}(\mathcal{Z}_{\tau})$ by definition. The conclusion then follows directly from Lemma 5.5. ∎

Corollary 5.6 shows that there is a subspace of $H^{1}_{e}$ consisting of singular spinors which hit the infinite-dimensional obstruction of (4.11). The next point is important, though. The spinors in this subspace of $H^{1}_{e}$ cannot be used to cancel the obstruction components in the alternating iteration. Since these spinors grow across the overlap region, rather than decay, an alternating iteration scheme employing them will not satisfy requirement (II) of Section 3.2, and will therefore not converge.

Remark 5.7.

In fact, [36, Thm. 7.14] describes the form of $u_{\Psi}$ more precisely. That result implies that the maximal domain $\mathcal{D}=\{u\in L^{2}\ |\ \not{D}_{A_{\tau}}u\in L^{2}\}$ consists of spinors of the form

\mathcal{D}=\left\{u=\begin{pmatrix}\tfrac{c(t)}{\sqrt{z}}\\ \tfrac{d(t)}{\sqrt{\overline{z}}}\end{pmatrix}e^{-i\theta/2}\ +\ v\ \ \Big{|}\ \ v\in rH^{1}_{e}\right\}.

and $\not{D}_{A_{\tau}}:\mathcal{D}\to L^{2}$ is right semi-Fredholm. Proposition 5.3 implies that $\text{\bf Ob}^{\perp}(\mathcal{Z}_{\tau})$ consists of $L^{2}$ -spinors whose leading coefficients (modulo a compact operator) lie in the subspace where $d(t)=iH(c(t))\in L^{-1/2,2}$ . The spinors $u_{\Psi}$ , modulo $rH^{1}_{e}$ and compact operator, fill the subspace where $d(t)=-iH(c(t))$ .

6. Deformations of Singular Sets

This section extends the theory of the singular Dirac operator to the case where the singular set varies. We begin by defining the universal Dirac operator $\not{\mathbb{D}}$ as in (1.10), which is the infinite-dimensional family of singular Dirac operators parameterized by embedded singular sets. We then calculate the derivative $\text{d}\not{\mathbb{D}}$ with respect to deformations of the singular set and give a more precise version of Theorem 1.2.

6.1. The Universal Dirac Operator

Let $\text{Emb}^{2,2}(\bigsqcup S^{1};Y)$ denote the Banach manifold of embedded singular sets with Sobolev regularity $(2,2)$ . The tangent space $T\text{Emb}^{2,2}(\bigsqcup S^{1};Y)$ at a singular set $\mathcal{Z}_{\tau}$ is naturally identified with the space $L^{2,2}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})$ of sections of the normal bundle $N\mathcal{Z}_{\tau}$ . To begin, we construct families of charts around $\mathcal{Z}_{\tau}\in\text{Emb}^{2,2}(\bigsqcup S^{1};Y)$ using families of diffeomorphisms deforming $\mathcal{Z}_{\tau}$ .

Choose a family of diffeomorphisms

	$\displaystyle\mathbb{F}_{\tau}:L^{2,2}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})$	$\displaystyle\to$	$\displaystyle\text{Diff}(Y)$		(6.1)
	$\displaystyle\eta$	$\displaystyle\mapsto$	$\displaystyle F_{\eta}:Y\to Y$		(6.2)

that associates to each $\eta\in L^{2,2}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})$ a diffeomorphism $F_{\eta}$ with $F_{0}=\text{Id}$ and such that $\frac{d}{ds}\big{|}_{s=0}F_{s\eta}$ is a vector field extending $\eta$ smoothly to $Y$ . We also assume that (6.1) depends smoothly on $\tau$ . Then set

	$\displaystyle\text{Exp}_{\tau}:L^{2,2}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})$	$\displaystyle\to$	$\displaystyle\text{Emb}^{2,2}(\mathcal{Z}_{\tau};Y)$		(6.3)
	$\displaystyle\eta$	$\displaystyle\mapsto$	$\displaystyle\mathcal{Z}_{\eta,\tau}:=F_{\eta}(\mathcal{Z}_{\tau}).$		(6.4)

$\text{Exp}_{\tau}$ is a local diffeomorphism on a neighborhood of $0$ by the Inverse Function Theorem (cf [43, Lem 5.3]). Fix $\rho_{0}$ small enough that $\text{Exp}_{\tau}$ is a diffeomorphism on the ball of radius $\rho_{0}$ for for every $\tau\in(-\tau_{0},\tau_{0})$ . In a slight abuse of notation, we use $\mathcal{E}_{\tau}$ to denote both this ball and its image under $\text{Exp}_{\tau}$ . We will use these charts with one choice of the family (6.1) in (6.12) and another more sophisticated choice in (6.21).

Each $\mathcal{Z}\in\mathcal{E}_{\tau}$ defines

(1)

A flat connection $A_{\mathcal{Z}}$ on $Y-\mathcal{Z}$ with holonomy in $\mathbb{Z}_{2}$ homotopic to that of $A_{\tau}$ .
(2)

A bundle $S^{\text{Re}}_{\mathcal{Z}}$ and Dirac operator $\not{D}_{\mathcal{Z}}:\Gamma(S^{\text{Re}}_{\mathcal{Z}})\to\Gamma(S^{\text{Re}}_{\mathcal{Z}})$ as in Lemma 3.5 and (3.8).
(3)

Hilbert spaces $rH^{1}_{e}(Y-\mathcal{Z};S^{\text{Re}}_{\mathcal{Z}})$ , and $L^{2}(Y-\mathcal{Z};S^{\text{Re}}_{\mathcal{Z}})$ as in Definition 4.3.

Note each of these depends implicitly on $\tau$ ; in particular the weights used in the norms are defined analogously to (4.8) using the geodesic distance of $g_{\tau}$ .

Using these, define families of Hilbert spaces $p_{1}:\mathbb{H}^{1}_{e}(\mathcal{E}_{\tau})\to\mathcal{E}_{\tau}$ and $p_{0}:\mathbb{L}^{2}(\mathcal{E}_{\tau})\to\mathcal{E}_{\tau}$ by

	$\displaystyle\mathbb{H}^{1}_{e}(\mathcal{E}_{\tau})$	$\displaystyle:=$	$\displaystyle\{(\mathcal{Z},u)\ \|\ \mathcal{Z}\in\mathcal{E}_{\tau}\ ,\ u\in rH^{1}_{e}(Y-\mathcal{Z};S^{\text{Re}}_{\mathcal{Z}})\}$
	$\displaystyle\mathbb{L}^{2}(\mathcal{E}_{\tau})$	$\displaystyle:=$	$\displaystyle\{(\mathcal{Z},v)\ \|\ \mathcal{Z}\in\mathcal{E}_{\tau}\ ,\ v\in L^{2}(Y-\mathcal{Z};S^{\text{Re}}_{\mathcal{Z}})\}$

with the $p_{0},p_{1}$ the obvious projections.

Lemma 6.1.

Each choice of a family $\mathbb{F}_{\tau}$ as in (6.1) determines trivializations

	$\displaystyle\Upsilon:\mathbb{H}^{1}_{e}(\mathcal{E}_{\tau})$	$\displaystyle\simeq$	$\displaystyle\mathcal{E}_{\tau}\times rH^{1}_{e}(Y-\mathcal{Z}_{\tau};S^{\text{Re}})$
	$\displaystyle\Upsilon:\mathbb{L}^{2}(\mathcal{E}_{\tau})$	$\displaystyle\simeq$	$\displaystyle\mathcal{E}_{\tau}\times\ L^{2}(Y-\mathcal{Z}_{\tau};S^{\text{Re}}).$

Together, these trivializations endow the spaces on the left with the structure of locally trivial Hilbert vector bundles over $\mathcal{E}_{\tau}$ .

Proof.

This is Lemma 5.1 in [43]. The proof is nearly identical to the proof of Lemma 8.2 in Section 8, which applies to the full Seiberg–Witten equations. ∎

The following definition gives a more precise meaning to the operator defined in (1.10).

Definition 6.2.

The Universal Dirac Operator is the section $\not{\mathbb{D}}$ defined by

The next proposition calculates the linearization of the universal Dirac operator

\text{d}\not{\mathbb{D}}_{(\mathcal{Z}_{\tau},\Phi_{\tau})}:L^{2,2}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})\oplus rH^{1}_{e}(Y-\mathcal{Z}_{\tau};S^{\text{Re}})\ \longrightarrow\ L^{2}(Y-\mathcal{Z}_{\tau};S^{\text{Re}})

(6.5)

at the pair $(\mathcal{Z}_{\tau},\Phi_{\tau})$ in the trivialization of Lemma 6.1. Since $\not{\mathbb{D}}$ is linear on fibers, differentiation in the fiber directions is trivial. As explained in Section 1.6, the naturality of the Dirac operator with respect to diffeomorphisms allows us to recast differentiation with respect to the embedding as differentiation with respect to the metrics obtained via pullback by the diffeomorphisms $F_{\eta}$ . For each fixed $\tau\in(-\tau_{0},\tau_{0})$ and $s\in[-1,1]$ , set

g_{s\eta}:=F_{s\eta}^{*}g_{\tau}

and denote the derivative by $\dot{g}_{\eta}=\frac{d}{ds}\Big{|}_{s=0}g_{s\eta}$ . There is again an implicit dependence on $\tau$ .

Proposition 6.3.

The linearization (6.5) is given by

\displaystyle\text{d}\not{\mathbb{D}}_{(\mathcal{Z}_{\tau},\Phi_{\tau})}(\eta,\psi)

\displaystyle=

\displaystyle\mathcal{B}_{\Phi_{\tau}}(\eta)\ +\ \not{D}_{A_{\tau}}\psi

(6.6)

where

\mathcal{B}_{\Phi_{\tau}}(\eta)=\left(-\frac{1}{2}\sum_{ij}\dot{g}_{\eta}(e_{i},e_{j})e^{i}.\nabla^{\tau}_{j}+\frac{1}{2}d\text{Tr}_{g_{\tau}}(\dot{g}_{\eta}).+\frac{1}{2}\text{div}_{g_{\tau}}(\dot{g}_{\eta}).+\mathcal{R}(B_{\tau},\eta).\right)\Phi_{\tau}\ \

(6.7)

in which $\mathcal{R}(B_{\tau},\eta)$ is a smooth term that involves first derivatives of $B_{\tau}$ and is zeroth order in $\eta$ , Clifford multiplication . is that of the metric $g_{\tau}$ , and $\nabla^{\tau}$ denotes the perturbation to the spin connection formed from $(g_{\tau},B_{\tau})$ .

Proof Sketch..

(See [43, Section 5.2] for complete details). The derivative with respect to $\psi$ is immediate, becuase $\not{\mathbb{D}}$ is linear in $\psi$ . Differentiating with respect to the family of metrics $g_{\eta}$ requires associating the spinor bundles for distinct metrics. This is done as follows. For a fixed $\eta$ , let

X=(Y\times[0,1],g_{s\eta}+ds^{2})

(6.8)

denote the metric cylinder on $Y$ . For $s=0$ , the positive spinor bundle $W^{+}\to X$ is isomorphic to the spinor bundle of $Y$ with the metric $g_{0}=g_{\tau}$ , while for $s=1$ it is isomorphic to that with the metric $g_{\eta}$ . Let $\mathfrak{T}_{g_{\tau}}^{g_{\eta}}$ denote the isomorphism between the two spinor bundles for $g_{\tau},g_{\eta}$ defined by parallel transport along rays $\{y\}\times[0,1]$ using the spin connection on $W^{+}$ (perturbed by $B_{s}=F_{s\eta}^{*}(B_{\tau})$ ).

The partial derivative with respect to $\eta$ is then given by

\text{d}\not{\mathbb{D}}_{(\mathcal{Z}_{\tau},\Phi_{\tau})}(\eta,0)=\left(\frac{d}{ds}\Big{|}_{s=0}\mathfrak{T}_{g_{\tau}}^{g_{s\eta}}\circ\not{D}_{g_{s\eta}}\circ(\mathfrak{T}_{g_{\tau}}^{g_{s\eta}})^{-1}\right)\Phi_{\tau}

(6.9)

where $\not{D}_{g_{s\eta}}$ denotes the singular Dirac operator using the metric $g_{s\eta}$ and twisted around the fixed singular set $\mathcal{Z}_{\tau}$ . Minus a few additional details regarding pulling back the perturbation $B_{\tau}$ , which gives rise to the term $\mathcal{R}(B_{\tau},\eta)$ , the proposition is now completed by the below theorem of Bourguignon-Gauduchon, which calculates (6.9). ∎

Generalizing the above situation slightly, let $g_{s}$ be a family of metrics on $Y$ , and denote by $\mathfrak{T}_{g_{0}}^{g_{s}}$ the association of spinor bundles in the proof of Proposition 6.3 by parallel transport on the cylinder (6.8).

Theorem 6.4.

(Bourguignon-Gauduchon [6]) The derivative of the Dirac operator with respect to the family of metrics $g_{s}$ at $s=0$ acting on a spinor $\Psi$ is given by

\left(\frac{d}{ds}\Big{|}_{s=0}\mathfrak{T}_{g_{0}}^{g_{s}}\circ\not{D}_{g_{s}}\circ(\mathfrak{T}_{g_{0}}^{g_{s}})^{-1}\right)\Psi=-\frac{1}{2}\sum_{ij}\dot{g}_{s}(e_{i},e_{j})e^{i}.\nabla^{g_{0}}_{j}\Psi+\frac{1}{2}d\text{Tr}_{g_{0}}(\dot{g}_{s}).\Psi+\frac{1}{2}\text{div}_{g_{0}}(\dot{g}_{s}).\Psi

(6.10)

where $\not{D}_{g_{s}}$ is the Dirac operator of the metric $g_{s}$ , and $e_{i},e^{i},\ .\ ,\text{div}_{g_{0}},\nabla^{g_{0}}$ are respectively an orthonormal frame and co-frame, Clifford multiplication, the divergence of a symmetric tensor, and the spin connection of the metric $g_{0}$ . ∎

In (6.10), the first term arises from differentiating the symbol/Clifford multiplication of the Dirac operator, while the last two terms arise from differentiating the spin connection.

6.2. The Deformation Operator

This subsection calculates the projection of the derivative (6.7) to the obstruction bundle.

Let $\pi_{\tau}=\langle\Phi_{\tau},-\rangle\Phi_{\tau}$ be the $L^{2}$ -orthogonal projection onto the span of the eigenvector $\Phi_{\tau}$ . Using the orthogonal splitting $L^{2}(Y-\mathcal{Z}_{\tau})\simeq\text{\bf Ob}(\mathcal{Z}_{\tau})\oplus\text{Range}^{\perp}_{\tau}$ where $\text{Range}^{\perp}_{\tau}=\{\psi\in\text{Range}(\not{D}_{A_{\tau}})\ |\ \pi_{\tau}(\psi)=0\}$ , the derivative (6.6) can be written as a block matrix:

\text{d}\not{\mathbb{D}}_{(\mathcal{Z}_{\tau},\Phi_{\tau})}=\begin{pmatrix}\Pi^{\text{Ob}}\mathcal{B}_{\Phi_{\tau}}&\Lambda(\tau)\pi_{\tau}\\ \\ (1-\Pi^{\text{Ob}})\mathcal{B}_{\Phi_{\tau}}&\not{D}_{A_{\tau}}\end{pmatrix}\ :\ \begin{matrix}L^{2,2}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})\\ \oplus\\ rH^{1}_{e}\end{matrix}\ \ \longrightarrow\ \ \begin{matrix}\text{\bf Ob}(\mathcal{Z}_{0})\\ \oplus\\ \text{Range}_{\tau}^{\perp}.\end{matrix}

(6.11)

Composing with the isomorphism $\text{ob}^{-1}_{\tau}\oplus\iota:\text{\bf Ob}(\mathcal{Z}_{\tau})\to L^{2}(\mathcal{Z}_{\tau};\mathcal{S}_{\tau})\oplus\mathbb{R}$ from Proposition 5.3 where $\mathcal{S}_{\tau}$ is as in (5.3), the top left block of (6.11) can be written as $(T_{\Phi_{\tau}},\pi_{\tau})$ where $T_{\Phi_{\tau}}$ is the composition:

In particular, $T_{\Phi_{\tau}}$ is an operator on sections of vector bundles over the fixed curve $\mathcal{Z}_{\tau}$ .

The form of $T_{\Phi_{\tau}}$ depends on the selection of the family of diffeomorphisms $\mathbb{F}_{\tau}$ . This is effectively a choice of gauge on the bundle $\mathbb{H}^{1}_{e}$ , as the trivializations of Lemma 6.1 also depend on $\mathbb{F}_{\tau}$ (cf Remark 6.11). One natural choice is as follows. Fix a system of Fermi coordinates as in Definition 3.9 with radius $r_{0}$ , and choose a cut-off function $\chi:[0,\infty)\to[0,1]$ equal to 1 for $r\leq r_{0}/2$ and vanishing for $r\geq r_{0}$ . Then define a family of diffeomorphisms (6.1) by setting

F_{\eta}\left(t,z\right):=(t,z+\chi(r)\eta(t)).

(6.12)

and extending $F_{\eta}$ by the identity outside of $r\geq r_{0}$ . This is a diffeomorphism provided $\|\eta\|_{2,2}$ is sufficiently small (see [43, Lem 5.3]).

The following theorem combines the statements of Theorem 6.1 and Lemma 6.7 from [43]. It is a refined version of Theorem 1.2 in the introduction.

Theorem 6.5.

([43]). For the family of diffeomorphisms (6.12), the operator $T_{\Phi_{\tau}}$ is an elliptic pseudo-differential operator of order $\tfrac{1}{2}$ , and its Fredholm extension

T_{\Phi_{\tau}}:L^{1/2,2}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})\longrightarrow L^{2}(\mathcal{Z}_{\tau};\mathcal{S}_{\tau})

(6.13)

has index 0. Specifically, on sections $\eta\in C^{\infty}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})$ , it is given by the formula

T_{\Phi_{\tau}}(\eta(t))=\left(-\tfrac{3|\mathcal{Z}_{\tau}|}{2}(\Delta+1)^{-\tfrac{3}{4}}\ \circ\ \mathcal{T}_{\Phi_{\tau}}\circ\frac{d}{dt^{2}}\right)\eta(t)+K\eta(t)

(6.14)

where $|\mathcal{Z}_{\tau}|$ is the length of $\mathcal{Z}_{\tau}$ , $\Delta$ is the Laplacian on $\mathcal{S}_{\tau}$ , $\frac{d}{dt^{2}}$ is the second covariant derivative on $\Gamma(N\mathcal{Z}_{\tau})$ induced by the Levi-Civita connection of $g_{\tau}$ , $K$ is a pseudo-differential operator of order $\tfrac{3}{8}$ (hence compact), and $\mathcal{T}_{\Phi_{\tau}}$ is the zeroth order operator given by

\mathcal{T}_{\Phi_{\tau}}(s(t))=H(c_{\tau}(t)s(t))-\overline{s}(t)d_{\tau}(t)

where $-iH$ is the Hilbert transform in the trivialization of $\mathcal{S}_{\tau}$ induced by the arclength parameterization, and $c_{\tau},d_{\tau}$ are the leading coefficients of $\Phi_{\tau}$ from the expansion (4.13). In particular, $\mathcal{T}_{\Phi_{\tau}}$ depends smoothly on $\tau$ . ∎

The unobstructed condition in Definition 1.3 can be restated in terms of the operator $T_{\Phi_{\tau}}$ .

Corollary 6.6.

If $\mathbb{Z}_{2}$ -harmonic spinor $(\mathcal{Z}_{0},A_{0},\Phi_{0})$ has unobstructed deformations, then $T_{\Phi_{\tau}}$ is invertible for $\tau$ sufficiently small.

Proof.

Definition 1.3 means that $(\mathcal{Z}_{0},A_{0},\Phi_{0})$ is unobstructed if $T_{\Phi_{0}}$ is injective. Since it is index 0, (6.13) is invertible. It follows (by Lemma 8.17 of [43]) that $T_{\Phi_{\tau}}$ is invertible for $\tau$ sufficiently small. ∎

As a consequence of Theorem 6.5, the following version of standard elliptic estimates hold. They are proved by repeated differentiation (or integration by parts for $m<2$ ).

Corollary 6.7.

For any $m\geq 0$ , the extension

T_{\Phi_{\tau}}:L^{m+1/2,2}(\mathcal{Z}_{\tau};\mathcal{S}_{\tau})\to L^{m,2}(\mathcal{Z}_{\tau};\mathcal{S}_{\tau})

is Fredholm of index 0 and there are constants $C_{m}$ such that it satisfies

\|\eta\|_{m+1/2,2}\leq C_{m}\ (\|T_{\Phi_{\tau}}(\eta)\|_{{m,2}}+\|\eta\|_{m+1/4,2}).

(6.15)

∎

A more quantitative version of these elliptic estimates will also be needed, which is given in the next proposition. It says, roughly, that the constants $C_{m}$ in the elliptic estimates (6.15) grow only as fast as the derivatives of the metric $g_{\tau}$ and $\Phi_{\tau}$ in the directions tangential to $\mathcal{Z}_{\tau}$ . In the statement of the proposition, $g_{0}$ is used to denote the product metric in Fermi coordinates on $N_{r_{0}}(\mathcal{Z}_{\tau})$ defined using $g_{\tau}$ . As in Definition 3.9, $g_{0}$ differs from $g_{\tau}$ by a symmetric tensor of size $O(r)$ .

Corollary 6.8.

Suppose that there is an $M>1$ such that for each $m\in\mathbb{N}$ the bounds

|\partial^{m}_{t}(g_{\tau}-g_{0})|\leq M^{m}\|g_{\tau}-g_{0}\|_{C^{3}(Y)}\hskip 42.67912pt|\partial^{m}_{t}\Phi_{\tau}|\leq M^{m}\|\Phi_{\tau}\|_{C^{1}(Y)}

hold on $N_{r_{0}}(\mathcal{Z}_{\tau})$ . Then there is a constant $C_{m}$ independent of $M$ such that if $T_{\Phi_{\tau}}(\eta)=f$ , then the following estimate holds for every $m\geq 0$ :

\|\eta\|_{m+1/2,2}\leq C_{m}\ \left(\|f\|_{m,2}\ +\ \|\eta\|_{m+1/4,2}\right)\ +\ C_{m}M^{m}\left(\|f\|_{2}\ +\ \|\eta\|_{1/4,2}\right).

(6.16)

∎

Remark 6.9.

Note that there are several distinct notions of derivative being used. In the assumption of Corollary 6.8, $\partial_{t}$ refers to the derivative using the product connection d in Fermi coordinates. Moreover, the derivatives in e.g. 6.10 are 3-dimensional covariant derivatives on $Y$ , while the derivatives in (6.14) and the norms $L^{m,2}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})$ are one-dimensional covariant derivatives denoted $\frac{d}{dt}$ on $\mathcal{Z}_{\tau}$ arising from the restriction of the three-dimensional covariant derivatives to $\mathcal{S}_{\tau},N\mathcal{Z}_{\tau}$ . It is not true that the isomorphism $\text{ob}_{\tau}$ intertwines the derivatives, i.e. $\nabla_{t}\text{ob}_{\tau}(\eta)\neq\text{ob}_{\tau}(\frac{d}{dt}\eta)$ , though Lemma 4.17 in [43] shows that the regularity of the two sides match, i.e.

\text{ob}^{-1}_{\tau}\Psi\in L^{m,2}(\mathcal{Z}_{\tau})\ \ \Leftrightarrow\ \ \Psi\in\text{\bf Ob}(\mathcal{Z}_{\tau})\cap H^{m}(Y-\mathcal{Z}_{\tau}).

6.3. Mode-Dependent Deformations

This section introduces a more sophisticated choice of a family of diffeomorphisms (6.1) to deform singular sets. These new diffeomorphisms are constructed by taking a dyadic decomposition of the disks normal to $\mathcal{Z}_{\tau}$ by annuli with radii $O((n+1)^{-1})\leq r\leq O(n^{-1})$ for $n\in\mathbb{N}$ . At $r=0$ , the action of the diffeomorphisms is identical to (6.12), but for $r\geq 0$ the action is tempered so that on each dyadic annulus it depends on the Fourier modes $\eta_{\ell}$ of $\eta$ only up to $|\ell|=O(n)$ . As explained in Section 1.4 these mode-dependent deformations are the crucial ingredient needed to make alternating iteration converge. More specifically, via the upcoming Proposition 6.12, they are tool that gives control over the loss of regularity described in Subsection 2.5.

To begin, we introduce the following notation. If $f_{\ell}:[0,r_{0})\to\mathbb{R}$ is a family of smooth functions indexed by $\ell\in\mathbb{Z}$ , let $\underline{f}$ denote the operator

	$\displaystyle\underline{f}:L^{2}(\mathcal{Z}_{\tau};\mathbb{C})$	$\displaystyle\longrightarrow$	$\displaystyle L^{2}(N_{r_{0}}(\mathcal{Z}_{\tau});\mathbb{C})$		(6.17)
	$\displaystyle\underline{f}[\eta]$	$\displaystyle:=$	$\displaystyle\sum_{p\in\mathbb{Z}}f_{p}(r)\eta_{p}e^{ipt}.$		(6.18)

where a fixed choice of Fermi coordinates is used to associate $N\mathcal{Z}_{\tau}\simeq\underline{\mathbb{C}}$ , and $\eta_{p}$ are the Fourier coefficients of $\eta(t)$ . $\underline{f}$ is a $[0,r_{0})$ -parameterized family of pseudo-differential operators on $L^{2}(\mathcal{Z}_{\tau};\mathbb{C})$ whose Fourier multiplier is given by $\{f_{\ell}(r)\}_{\ell\in\mathbb{Z}}$ for each fixed $r$ .

Next, let $R_{0}>0$ be a large positive number to be specified shortly, and denote by $\chi:[0,\infty)\to\mathbb{R}$ a smooth cut-off function equal to 1 for $r\leq R_{0}$ and supported in the region where $r\leq 2R_{0}$ . Assume it is chosen so that

|d\chi|\leq\frac{C}{R_{0}}.

(6.19)

Additionally, let $\chi_{r_{0}}$ denote a second smooth cut-ff function equal to 1 for $r\leq r_{0}/2$ and supported in $N_{r_{0}}(\mathcal{Z}_{\tau})$ . Then, for each $\ell\in\mathbb{Z}$ , set

\chi_{\ell}(r):=\chi(|\ell|r)\chi_{r_{0}}(r).

(6.20)

The family $\chi_{\ell}$ gives rise an operator $\underline{\chi}$ as in (6.17).

Using this, define a new family of diffeomorphisms by

\underline{F}_{\eta}(t,z):=\left(t,z+\underline{\chi}[\eta]\right)

(6.21)

on $N_{r_{0}}(\mathcal{Z}_{\tau})$ and extended to $Y$ by the identity. For $\|\eta\|_{2,2}$ and $r_{0}$ sufficiently small, the Inverse Function Theorem shows that $\underline{F}_{\eta}$ is indeed a diffeomorphism on $N_{r_{0}}(\mathcal{Z}_{\tau})$ . As in Sections 5.1–5.2, $\underline{F}_{\eta}$ gives rise to (i) an exponential map $\underline{\text{Exp}}$ , (ii) trivializations $\underline{\Upsilon}$ , a (iii) a family of pullback metrics $\underline{g}_{\eta}$ , (iv) a partial derivative $\underline{\mathcal{B}}_{\Phi_{\tau}}(\eta)$ , and (v) a deformation operator $\underline{T}_{\Phi_{\tau}}$ .

The following corollary states that the results of Section 6.2 carry over to the underlined versions. It also dictates a lower bound for the value of the constant $R_{0}$ used to define (6.17).

Corollary 6.10.

The operator $\underline{T}_{\Phi_{\tau}}$ is given by

\underline{T}_{\Phi_{\tau}}=T_{\Phi_{\tau}}+T_{R_{0}}

where $T_{R_{0}}$ is a pseudo-differential operator of order $1/2$ and for some $K\in\mathbb{N}$ and $M>10$ , it satisfies

\|T_{R_{0}}\|\leq C(R_{0})^{K}\text{Exp}(-R_{0}/c)\ +\ CR_{0}^{-M}

In particular, if $(\mathcal{Z}_{0},A_{0},\Phi_{0})$ has unobstructed deformations then for $R_{0}$ sufficiently large, $\underline{T}_{\Phi_{\tau}}$ is invertible for $\tau\leq\tau_{0}$ , and the results of Corollaries 6.7 and 6.8 continue to hold.

Proof Sketch.

The idea of the proof is straightforward, given the following observation: the cokernel elements $\Psi_{\ell}$ are nearly isolated in the $\ell^{th}$ Fourier mode and decay exponentially with with exponent $1/|\ell|$ . Therefore, altering the deformation of the $\ell^{th}$ mode outside $R_{0}/|\ell|$ for $R_{0}$ large enough is a small perturbation. ∎

Remark 6.11.

(Cf. Remark 5.6 of [43] and Section 4.1 of [12]) There is an infinite-dimensional space of possible choices of a family of diffeomorphisms $\mathbb{F}$ of (6.1). Any two choices differ by pre-composing with a family of diffeomorphisms $G_{\eta}\in\text{Diff}(Y;\mathcal{Z}_{\tau})$ that fix $\mathcal{Z}_{\tau}$ . By pullbacks, this latter group acts by infinite-dimensional gauge transformations on the bundles of Lemma 6.1. Thus the choice of a family of diffeomorphisms is effectively a choice of gauge, and the expressions for the operator (6.13) depend on this choice. The choice of mode-dependent deformations may be understood as a gauge in which certain stronger estimates are available, akin to the elliptic estimates provided by the Coulomb gauge in standard situations.

The next lemma provides key bounds that are used to control the loss of regularity in Sections 9–11. The collection of terms it will be necessary to bound have two general types.

(A)

Let $\Phi$ be a spinor such that there is an $\alpha>0$ , so that for all $k\in\mathbb{Z}^{\geq 0}$ , $|\nabla^{k}_{t}\Phi|\leq C_{k_{1}}r^{\alpha}$ and $|\nabla^{k}_{t}\nabla_{v}\Phi|\leq C_{k}r^{\alpha-1}$ , where $v=\partial_{x}$ , or $\partial_{y}$ in Fermi coordinates. Consider terms of the form

$M_{\Phi}(\eta):=\underline{\partial^{m}\chi}[d_{t}^{n}\eta]\sigma_{j}\nabla^{k}\Phi.$ (6.22)
(B)

Let $\psi\in rH^{1}_{e}$ and set $k=0,1$ . Consider terms of the form

$M_{\psi}(\eta)=\underline{\partial^{m}\chi}[d_{t}^{n}\eta]\sigma_{j}\nabla^{k}\psi.$ (6.23)

In these expressions, $\underline{\partial^{m}\chi}$ denotes the operator (6.17) formed analogously to (6.20) but using $\partial^{m}\chi$ for a a multi-index of order $m$ , $d_{t}^{n}\eta$ is the $n^{th}$ derivative of $\eta\in C^{\infty}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})$ , and $\sigma_{j}$ is chosen from a collection of fixed pointwise linear endomorphisms (e.g. $\gamma(e^{j})$ ). We say terms of the form (6.22) and (6.23) have weights

{w_{A}=m+n+k.}\hskip 56.9055pt{w_{B}=m+n}

respectively.

Lemma 6.12.

Let $\eta\in C^{\infty}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})$ be a deformation, then the following bounds hold.

(A)

Suppose that $M_{\Phi}$ is a term of Type A as in (6.22) having weight $w_{A}$ , and that $\beta\in\mathbb{R}$ . Then

$\|r^{\beta}M_{\Phi}(\eta)\|_{L^{2}(N_{r_{0}})}\leq C\|\eta\|_{w_{A}-(1+\alpha+\beta)}$
where $\|\eta\|_{s}$ denotes the $L^{s,2}$ -norm on $\mathcal{Z}_{\tau}$ . In particular, the terms of weight $w_{A}=2$ satisfy
- •
  
  $\Big{\|}\underline{\chi}[\eta^{\prime\prime}]\sigma_{j}r^{\beta}\Phi\Big{\|}_{L^{2}(N_{r_{0}})}+\Big{\|}\underline{\partial_{a}\chi}[\eta^{\prime}]\sigma_{j}r^{\beta}\Phi\Big{\|}_{L^{2}(N_{r_{0}})}+\Big{\|}\underline{\partial_{ab}\chi}[\eta]\sigma_{j}r^{\beta}\Phi\Big{\|}_{L^{2}(N_{r_{0}})}\leq C\|\eta\|_{1-\alpha-\beta}$
- •
  
  $\Big{\|}\underline{\chi}[\eta^{\prime}]\sigma_{j}r^{\beta}\nabla_{b}\Phi\Big{\|}_{L^{2}(N_{r_{0}})}+\Big{\|}\underline{\partial_{a}\chi}[\eta]\sigma_{j}r^{\beta}\nabla_{b}\Phi\Big{\|}_{L^{2}(N_{r_{0}})}\leq C\|\eta\|_{1-\alpha-\beta}$
where $a,b=t,x,y$ .

(B)

Suppose that $M_{\psi}(\eta)$ is a term of Type B as in (6.23) having weight $w_{B}$ . Then, for some $\underline{\gamma}<<1$

\|r^{\beta}M_{\psi}(\eta)\|_{L^{2}(N_{r_{0}})}\leq C\|\eta\|_{w_{B}+\underline{\gamma}-(1/2+\beta)}\|\psi\|_{rH^{1}_{e}}.

In particular, for $w_{B}=2$ ,

•

$\Big{\|}\underline{\chi}[\eta^{\prime\prime}]\sigma_{j}\psi r^{\beta}\Big{\|}_{L^{2}(N_{r_{0}})}+\Big{\|}\underline{\partial_{a}\chi}[\eta^{\prime}]\sigma_{j}\psi r^{\beta}\Big{\|}_{L^{2}(N_{r_{0}})}+\Big{\|}\underline{\partial_{ab}\chi}[\eta]\sigma_{j}\psi r^{\beta}\Big{\|}_{L^{2}(N_{r_{0}})}\leq C\|\eta\|_{3/2+\underline{\gamma}-\beta}\|\tfrac{\psi}{r}\|_{L^{2}}$
•

$\Big{\|}\underline{\chi}[\eta^{\prime}]\sigma_{j}r^{\beta}\nabla_{b}\psi\Big{\|}_{L^{2}(N_{r_{0}})}+\Big{\|}\underline{\partial_{a}\chi}[\eta]\sigma_{j}r^{\beta}\nabla_{b}\psi\Big{\|}_{L^{2}(N_{r_{0}})}\leq C\|\eta\|_{3/2+\underline{\gamma}-\beta}\|\nabla{\psi}\|_{L^{2}}.$

Proof.

The lemma is proved working locally in Fermi coordinates and an accompanying trivialization as in Section 3.4. Thus $\eta,\Phi$ become $\mathbb{C}$ and $\mathbb{C}^{2}\otimes\mathbb{H}$ -valued functions respectively. The lemma is proved in the cases of $w_{A}=w_{B}=2$ ; the general cases are similar.

(A) To begin, consider the term $\underline{\chi}[\eta^{\prime\prime}]$ and the case that $\beta=0$ . On each circle $S^{1}\times\{(x,y)\}\subseteq N_{r_{0}}(\mathcal{Z}_{\tau})$ the multiplication map $L^{1,2}(S^{1})\times L^{2}(S^{1})\to L^{2}(S^{1})$ is uniformly boundedin $(x,y)$ . Applying this to the product $\Phi(t,x,y)\underline{\chi}[\eta^{\prime\prime}]$ for each pair $(x,y)$ ,

$\displaystyle\Big{\\|}\underline{\chi}[\eta^{\prime\prime}]\sigma_{j}\Phi\Big{\\|}^{2}_{L^{2}(N_{r_{0}})}$	$\displaystyle=$	$\displaystyle\int_{D_{\mathbb{r}_{0}}}\int_{S^{1}}\|\underline{\chi}[\eta^{\prime\prime}]\sigma_{j}\Phi\|^{2}dt\ rd\theta dr$
	$\displaystyle\leq$	$\displaystyle\int_{D_{r_{0}}}\|C_{0}r^{\alpha}\|^{2}\cdot\int_{S^{1}}\|\underline{\chi}[\eta^{\prime\prime}]\|^{2}dt\ rd\theta dr$
	$\displaystyle\leq$	$\displaystyle C_{1}^{2}\int_{D_{r_{0}}}\|\sum_{p\in\mathbb{Z}}\eta_{p}(ip)^{2}e^{ipt}\chi_{p}(r)\|^{2}r^{2\alpha+1}d\theta drdt$

where we have replaced the integral over $S^{1}$ with a $D_{r_{0}}$ -parameterized sum over Fourier modes via Plancherel’s Theorem.

Since the integrand is uniformly bounded, and $\eta_{p}$ is independent of $(x,y)$ , the sum may be pulled outside, giving

$\displaystyle\leq$	$\displaystyle C\sum_{p\in\mathbb{Z}}\|\eta_{p}\|^{2}\|p\|^{4}\int_{D_{r_{0}}}\|\chi_{p}(r)\|^{2}r^{2\alpha+1}d\theta drdt$	(6.24)
$\displaystyle\leq$	$\displaystyle C\sum_{p\in\mathbb{Z}}\|\eta_{p}\|^{2}\|p\|^{4}[r^{2\alpha+2}]_{r=0}^{R_{0}/p}$	(6.25)
$\displaystyle\leq$	$\displaystyle C\sum_{p\in\mathbb{Z}}\|\eta_{p}\|^{2}\|p\|^{2-2\alpha}=C\\|\eta\\|_{1-\alpha,2}$	(6.26)

as desired.

For the $\underline{\partial_{a}\chi}[\eta^{\prime}]$ term, the proof is the same, except we use the bound $|\partial_{a}\chi_{p}|\leq C|p|$ ; this adds a factor of $|p|^{2}$ to the integrand, but the same factor is lost by replacing $\eta^{\prime\prime}$ by $\eta^{\prime}$ . For the term $\underline{\partial_{ab}\chi}[\eta]$ the same applies, now with $|p|^{4}$ . Likewise, for the terms involving $\nabla_{a}\Phi$ , one power of $|p|$ is gained from using $r^{\alpha-1}$ instead of $\alpha$ , but the same factor is again lost since $\underline{[}\eta^{\prime}],\underline{\partial_{a}\chi}[\eta]$ both have one less derivative. The case with $\beta\neq 0$ is again similar.

(B) The proof in this case is essentially the same, except we use that multiplication $C^{0}(S^{1})\times L^{2}(S^{1})\to L^{2}(S^{1})$ is bounded, and apply it to the term $\eta^{\prime\prime}\psi(t,x_{0},y_{0})$ since $\psi$ does not necessarily have pointwise bounds. First, divide $D_{r_{0}}$ into the sequence of annuli

A_{n}:=\{\tfrac{2R_{0}}{n+1}\leq r\leq\tfrac{2R_{0}}{n}\}

for $n\geq 1$ . Then

$\displaystyle\Big{\\|}\underline{\chi}[\eta^{\prime\prime}]\sigma_{j}\psi r^{\beta}\Big{\\|}^{2}_{L^{2}(N_{r_{0}})}$	$\displaystyle=$	$\displaystyle\sum_{n\geq 1}\int_{A_{n}}\int_{S^{1}}\|\underline{\chi}[\eta^{\prime\prime}]\sigma_{j}\psi\|^{2}r^{2\beta}dt\ rd\theta dr$
	$\displaystyle\leq$	$\displaystyle\sum_{n\geq 1}\sup_{A_{n}}\\|\underline{\chi}(\eta^{\prime\prime})\\|_{C^{0}(S^{1})}\int_{A_{n}}\int_{S^{1}}\|\psi\|^{2}r^{2\beta}dt\ rd\theta dr$
	$\displaystyle\leq$	$\displaystyle\sum_{n\geq 1}\sup_{A_{n}}\\|\underline{\chi}(\eta^{\prime\prime})\\|^{2}_{C^{0}(S^{1})}\sup_{A_{n}}r^{2+2\beta}\\|\tfrac{\psi}{r}\\|_{L^{2}(A_{n})}.$

Now, since $\chi_{p}=0$ on $A_{n}$ for $p\geq 2(n+1)$ , we see that

\sup_{A_{n}}\|\underline{\chi}(\eta^{\prime\prime})\|_{C^{0}}\leq C\|\pi^{2(n+1)}\eta^{\prime\prime}\|_{C^{0}}\leq C\|\pi^{2(n+1)}\eta\|_{{5/2+\underline{\gamma}}}

where $\pi^{2(n+1)}$ denotes the projection to Fourier modes $|p|\leq 2(n+1)$ . Since $\sup_{A_{n}}r^{2+2\beta}\leq\tfrac{C}{n^{2+2\beta}}$ , the restriction on Fourier modes implies $C\|\eta\|_{5/2+\underline{\gamma}}n^{-2-2\beta}\leq C\|\eta\|_{3/2+\underline{\gamma}-\beta}$ . Using this, the above is bounded by

		$\displaystyle\leq$	$\displaystyle C\sum_{n\geq 1}\\|\pi^{2n+1}\eta\\|_{{3/2+\underline{\gamma}-\beta}}\\|\tfrac{\psi}{r}\\|_{L^{2}(A_{n})}$
		$\displaystyle\leq$	$\displaystyle C\\|\eta\\|_{{3/2+\underline{\gamma}-\beta}}\\|\tfrac{\psi}{r}\\|_{L^{2}(N_{r_{0}})}$

as desired. Just as in (A), the $\underline{\partial_{a}\chi}[\eta^{\prime}],\underline{\partial_{ab}\chi}[\eta]$ terms lose a power of $n$ from the derivative but gain one from the derivatives of the cut-off. The terms with $\nabla\psi$ are the same without the $r^{2}$ factor. ∎

6.4. Non-Linear Terms

The universal Dirac operator $\not{\mathbb{D}}$ is a fully non-linear function of the singular set $\mathcal{Z}$ . This section describes the non-linear terms. When solving the linearized equations in every step of the gluing iteration, these non-linear terms contribute a portion of the error for the subsequent step.

Given a configuration $h_{0}=(\mathcal{Z},\Phi)\in\mathbb{H}^{1}_{e}$ ,

\not{\mathbb{D}}(h_{0}+(\eta,\psi))=\not{\mathbb{D}}h_{0}+\text{d}\not{\mathbb{D}}_{h_{0}}(\eta,\psi)+Q_{h_{0}}(\eta,\psi)

where $\text{d}_{h_{0}}\not{\mathbb{D}}$ is the linearization (6.6) and $Q_{h_{0}}$ is the non-linear term. Here, $\mathcal{Z}+\eta$ means $\mathcal{Z}_{\eta}$ as in (6.4) using the chart centered at $\mathcal{Z}$ . The following lemma characterizes the non-linear term $Q_{h_{0}}$ .

Lemma 6.13.

The non-linear term $Q$ has the following form:

\displaystyle Q_{h_{0}}(\eta,\psi)

\displaystyle=

\displaystyle\underline{\mathcal{B}}_{\psi}(\eta)\ \ +\ \ M_{\Phi}(\eta,\eta)\ \ +\ \ M_{\psi}(\eta,\eta)\ \ +\ \ F_{\Phi+\psi}(\eta)

where

•

$\underline{\mathcal{B}}_{\psi}(\eta)$ is as in (6.7) with $\psi$ in place of $\Phi_{\tau}$ .
•

$M_{\Phi}(\eta,\eta)$ is a term quadratic in $\eta$ and linear in $\Phi$ which is a finite sum of terms of the form

$m(y)\cdot a_{1}(\underline{\chi}[\eta])\cdot M_{\Phi}(\eta)$

where $m(y)\in C^{\infty}(Y;\text{End}(S^{\text{Re}}))$ , $M_{\Phi}(\eta)$ is a linear term of type A in the sense of Lemma 6.12 with weight $w_{A}=2$ , and $a_{1}(\underline{\chi}[\eta])$ is a linear combination of $\underline{\chi}[\eta^{\prime}],\underline{\partial_{a}\chi}[\eta]$ , and $\underline{\chi}[\eta]$ .
•

$M_{\psi}(\eta,\eta)$ is the same with the final term replace by a term $M_{\psi}(\eta)$ of Type B in the sense of Lemma 6.12 with weight $w_{B}=2$ .
•

$F_{\Phi+\psi}(\eta)$ has the same form but with higher-order dependence on $\eta$ , so that it satisfies a bound

$|F_{\Phi+\psi}(\eta)|\leq C\|\eta\|_{C^{1}}\Big{(}M_{\Phi}^{\prime}(\eta,\eta)+M_{\psi}^{\prime}(\eta,\eta)\Big{)}$

where $M_{\Phi}^{\prime},M_{\psi}^{\prime}$ are of the same form as $M_{\Phi},M_{\psi}$ respectively.

Proof.

This formula is derived by substituting the formula for the pullback metric $g_{\eta}=F_{\underline{\chi}}(\eta)^{*}g$ into the non-linear version of Bourguignon-Gauduchon’s formula in Theorem 6.4. See Section 8.3 of [43]. ∎

To bound the non-linear terms later, we have the following analogue of Lemma 6.12. In the statement of the lemma, we tacitly use $\underline{\chi^{\prime}}[\eta]$ to denote a term having one derivative, i.e. a linear combination of $\underline{\chi}[\eta^{\prime}]$ and $\underline{\partial_{a}\chi}[\eta]$ . $\underline{\chi}^{\prime\prime}[\eta]$ denotes the same but with up to second derivatives.

Lemma 6.14.

Retaining the assumptions and notation of Lemma 6.12, the following bounds hold for $\eta,\xi\in C^{\infty}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})$ .

(A’)

For a fixed small number number $\underline{\gamma}<<1$ ,

	$\displaystyle\Big{\\|}\underline{\chi^{\prime}}[\eta]\underline{\chi^{\prime}}[\xi]\nabla\Phi\Big{\\|}_{L^{2}(N_{r_{0}})}$	$\displaystyle\leq$	$\displaystyle C\\|\eta\\|_{3/2+\underline{\gamma}}\cdot\\|\xi\\|_{1-\alpha}$
	$\displaystyle\Big{\\|}\underline{\chi^{\prime\prime}}[\eta]\underline{\chi^{\prime}}[\xi]\Phi\Big{\\|}_{L^{2}(N_{r_{0}})}+\Big{\\|}\underline{\chi^{\prime}}[\eta]\underline{\chi^{\prime\prime}}[\xi]\Phi\Big{\\|}_{L^{2}(N_{r_{0}})}$	$\displaystyle\leq$	$\displaystyle C(\\|\eta\\|_{3/2+\underline{\gamma}}\cdot\\|\xi\\|_{1-\alpha}+\\|\eta\\|_{1-\alpha}\cdot\\|\xi\\|_{3/2+\underline{\gamma}})$

(B’)

Likewise,

	$\displaystyle\Big{\\|}\underline{\chi^{\prime}}[\eta]\underline{\chi^{\prime}}[\xi]r^{\beta}\nabla\psi\Big{\\|}_{L^{2}(N_{r_{0}})}$	$\displaystyle\leq$	$\displaystyle C\\|\eta\\|_{3/2+\underline{\gamma}-\beta}\\|\xi\\|_{3/2+\underline{\gamma}}\\|\tfrac{\psi}{r}\\|_{L^{2}}$
	$\displaystyle\Big{\\|}\underline{\chi^{\prime\prime}}[\eta]\underline{\chi^{\prime}}[\xi]r^{\beta}\psi\Big{\\|}_{L^{2}(N_{r_{0}})}+\Big{\\|}\underline{\chi^{\prime}}[\eta]\underline{\chi^{\prime\prime}}[\xi]r^{\beta}\psi\Big{\\|}_{L^{2}(N_{r_{0}})}$	$\displaystyle\leq$	$\displaystyle C(\\|\xi\\|_{3/2+\underline{\gamma}}\\|\eta\\|_{3/2+\underline{\gamma}-\beta}+\\|\eta\\|_{3/2+\underline{\gamma}}\\|\xi\\|_{3/2+\underline{\gamma}-\beta})\\|\tfrac{\psi}{r}\\|_{L^{2}}$

Proof.

Notice that the bound $d\chi_{p}\leq C|p|$ for each $p\in\mathbb{Z}$ implies that $\|\underline{\partial_{a}\chi}[\eta]\|_{L^{2}}\lesssim\|\eta\|_{1,2}$ . In each case above, a factor with one derivative can therefore be pulled out using the Sobolev embedding $\|\chi^{\prime}[\eta]\|_{C^{0}}\leq C\|\eta\|_{L^{3/2+\underline{\gamma},2}}$ or equivalently for $\xi$ , after which the proofs proceed as in Lemma 6.12. ∎

7. Concentrating Local Solutions

This section reviews the model solutions constructed in [41]. This $\varepsilon$ -parameterized family of model solutions solve the Seiberg–Witten equations on a tubular neighborhood of the singular set, and locally converge to $(\mathcal{Z}_{0},A_{0},\Phi_{0})$ , with curvature concentrating along $\mathcal{Z}_{0}$ as $\varepsilon\to 0$ . Results are stated here without proof, with references to the relevant sections of [41]. These result here also include the trivial extension of [41] from a single $\mathbb{Z}_{2}$ -harmonic spinor to the $\tau$ -parameterized family of eigenvectors (1.12).

7.1. Hilbert Spaces and Boundary Conditions

This subsection defines Sobolev spaces with weights and boundary conditions on a tubular neighborhoods of each singular set $\mathcal{Z}_{\tau}$ . These $(\varepsilon,\tau)$ -parameterized tubular neighborhoods — which eventually host the model solutions — shrink in diameter as $\varepsilon\to 0$ for each fixed $\tau$ . More specifically, let

\boxed{\lambda(\varepsilon)=\varepsilon^{1/2}}

(7.1)

and, immediately suppressing the dependence on $\varepsilon\in(0,\varepsilon_{0})$ , denote by $Y^{+}=N_{\lambda}(\mathcal{Z}_{\tau})$ the tubular neighborhood of the singular set for $\tau\in(-\tau_{0},\tau_{0})$ . It is equipped with Fermi coordinates as in Definition 3.9 using the metric $g_{\tau}$ .

The Sobolev norms depend on two weight functions, which we now describe. The latter of these is defined in terms of a “de-singularized spinor” $\Phi_{\tau}^{h_{\varepsilon}}$ , which is a precursor to the model solutions. Denote by $r_{0}$ the radius of the Fermi coordinate chart.

(1)

With $r=\text{dist}(-,\mathcal{Z}_{\tau})$ , let $R_{\varepsilon}$ be a smooth function defined as follows:

$R_{\varepsilon}=\begin{cases}\sqrt{\kappa^{2}\varepsilon^{4/3}+r^{2}}\hskip 22.76228ptr\leq r_{0}/2\\ \text{const}\hskip 56.9055ptr\geq r_{0},\end{cases}$

where $\kappa=\kappa(\tau)$ is a uniformly bounded constant depending depending smoothly on $\tau\in(-\tau_{0},\tau_{0})$ defined in terms of the leading coefficients of $\Phi_{\tau}$ .
(2)

For each $\tau\in(-\tau_{0},\tau_{0})$ , let $\Phi_{\tau}^{h_{\varepsilon}}$ be the de-singularized spinor as defined in Definition 4.5 of [41]. The norm $|\Phi^{h_{\varepsilon}}_{\tau}|$ is a smooth function on $Y$ with the following properties for constants $c_{1},c_{2}$ (see [41, Sec. 4]):
- (2a)
  
  For $r\geq c_{1}\varepsilon^{2/3}$ , it satisfies $\big{|}|\Phi^{h_{\varepsilon}}_{\tau}|-|\Phi_{\tau}|\big{|}\leq C\text{Exp}(-\tfrac{r^{3/2}}{\varepsilon})$ .
- (2b)
  
  For $r\leq\lambda$ , it is a monotonically increasing function of $r$ with a uniform lower bound $|\Phi_{\tau}^{h_{\varepsilon}}|\geq c_{2}\varepsilon^{1/3}$ .

The weight function $R_{\varepsilon}$ is approximately equal to $r$ on a tubular neighborhood of $\mathcal{Z}_{\tau}$ , being constant outside this neighborhood, and leveling off at $r\sim\varepsilon^{2/3}$ so that there is a uniform lower bound $R_{\varepsilon}\geq c_{1}\varepsilon^{2/3}$ . The weight function $|\Phi_{\tau}^{h_{\varepsilon}}|$ is exponentially close to $|\Phi_{\tau}|\sim r^{1/2}$ for $c_{1}\varepsilon^{2/3}\leq r$ . This latter weight function is effectively commensurate with $|\Phi_{\tau}^{h_{\varepsilon}}|\sim\sqrt{R_{\varepsilon}}$ , but is used instead because it naturally appears in the Weitzenböck formula for the linearized operator.

Definition 7.1.

Let $\nu\in\mathbb{R}$ be a weight. The “inside” Sobolev norms are defined by

	$\displaystyle\\|(\varphi,a)\\|_{H^{1,+}_{\varepsilon,\nu}}$	$\displaystyle:=$	$\displaystyle\left(\int_{Y^{+}}\left(\|\nabla\varphi\|^{2}+\|\nabla a\|^{2}+\frac{\|\varphi\|^{2}}{R_{\varepsilon}^{2}}+\frac{\|\mu(\varphi,\Phi^{h_{\varepsilon}}_{\tau})\|^{2}}{\varepsilon^{2}}+\frac{\|a\|^{2}\|\Phi^{h_{\varepsilon}}_{\tau}\|^{2}}{\varepsilon^{2}}\right)R_{\varepsilon}^{2\nu}\ dV\right)^{1/2}$		(7.2)
	$\displaystyle\\|(\varphi,a)\\|_{L^{2,+}_{\varepsilon,\nu}}$	$\displaystyle:=$	$\displaystyle\left(\int_{Y^{+}}\left(\|\varphi\|^{2}+\|a\|^{2}\right)R_{\varepsilon}^{2\nu}\ dV\right)^{1/2}$		(7.3)

where the dependence of $\Phi^{h_{\varepsilon}},dV$ , and $R_{\varepsilon}$ on $\tau$ is suppressed in the notation. Both norms give rise to inner products via their polarizations. Because $N_{\lambda}(\mathcal{Z}_{\tau})$ is compact, these norms are equivalent to the standard $L^{1,2}$ and $L^{2}$ norms respectively (though not uniformly in $\varepsilon,\nu$ ).

Next, Sobolev spaces are defined using these norms along with a set of boundary conditions on $\partial Y^{+}$ and orthogonality constraints in the interior. The boundary conditions are a twisted variation of APS boundary conditions; the details of these conditions are crucial for the proofs of the theorems in [41], but not needed here (see Remark 7.4 below). The reader is referred to Sections 7.1–7.3 of [41] for details, and in particular to Figure 2 of [41] which illustrates these conditions.

The following lemma is a restatement of Proposition 7.15 in [41] with unnecessary details omitted.

Lemma 7.2.

There exists a Hilbert subspace

H^{+}_{\Phi_{\tau}}\subseteq L^{1/2,2}(\partial Y^{+})\oplus V

where $V\subseteq L^{2}(Y^{+})$ is a subspace of complex dimension $1+2L_{0}\varepsilon^{-1/2}$ with $L_{0}\in\mathbb{R}$ , and a projection operator $\Pi^{+}:L^{1,2}(Y^{+})\to H^{+}_{\Phi_{\tau}}$ given by the direct sum of the boundary restriction and the $L^{2}$ -orthogonal projection to $V$ , such that

(\mathcal{L}_{(\Phi,A)}\ ,\ \Pi^{+}):L^{1,2}(Y^{+})\longrightarrow L^{2}(Y^{+})\oplus H^{+}_{\Phi_{\tau}}

(7.4)

is Fredholm of Index 0 for any configuration $(\Phi,A)\in C^{\infty}(Y^{+})$ .

Definition 7.3.

Define the “inside” Hilbert spaces by

	$\displaystyle H^{1,+}_{\varepsilon,\nu}$	$\displaystyle:=$	$\displaystyle\Big{\{}(\varphi,a)\ \big{\|}\ \\|(\varphi,a)\\|_{H^{1,+}_{\varepsilon,\nu}}<\infty\ \ ,\ \ \Pi^{\mathcal{L}}(\varphi,a)=0\Big{\}}$
	$\displaystyle L^{2,+}_{\varepsilon,\nu}$	$\displaystyle:=$	$\displaystyle\Big{\{}(\varphi,a)\ \big{\|}\ \\|(\varphi,a)\\|_{L^{2,+}_{\varepsilon,\nu}}<\infty\Big{\}}.$

They are equipped with the norms from Definition 7.1, and the inner products arising from their polarizations. They are defined over the domain $Y^{+}=N_{\lambda}(\mathcal{Z}_{\tau})$ , and depend implicitly on $\tau$ .

Remark 7.4.

Although the details of the boundary conditions defined by Lemma 7.2 are peripheral to our purposes here, their importance for the overall gluing construction cannot be overstated, nor should the reader be lulled into dismissing the result of the lemma as an application of standard APS theory. Although the existence of some space $H$ such that the result of Lemma 7.2 about the index holds is a consequence of standard APS theory [2, 34, 4], the linearized equations at the model solutions (as in the upcoming Theorem 7.7) will necessarily fail to be invertible in a suitable sense for any but a very precisely crafted boundary condition. The reasons for this are explained in detail in Section 7.1.3 of [41], and the correct definition of $\Pi^{+}$ is the main challenge in [41].

7.2. Concentrating Local Solutions

This subsection states the main results of [41] about the model solutions and the linearization at these.

Theorem 1.2 of [41] first constructs model solutions $(\Phi^{+}_{\varepsilon,\tau},A^{+}_{\varepsilon,\tau})$ on $N_{\lambda}(\mathcal{Z}_{\tau})$ that satisfy the Seiberg–Witten equations and the boundary conditions of Lemma 7.9. Using the cut-off function $\chi^{+}$ (recall this is equal to $1$ for $r\leq\lambda/2$ and vanishes for $r\geq 3\lambda/4$ ), these are are extended by the limiting eigenvector and connection to form global approximate solutions:

(\Phi_{1},A_{1}):=\chi^{+}\cdot(\Phi^{+}_{\varepsilon,\tau},A^{+}_{\varepsilon,\tau})+(1-\chi^{+})\cdot(\Phi_{\tau},A_{\tau}).

(7.5)

On the left, the dependence on $(\varepsilon,\tau)$ is suppressed. The subscript “1” indicates that these are the first in the sequence of approximate solutions (2.2) arising in the alternating iteration.

Theorem 7.5.

([41], Theorem 1.2 ) Suppose that $(\mathcal{Z}_{\tau},A_{\tau},\Phi_{\tau})$ for $\tau\in(-\tau_{0},\tau_{0})$ are a family of $\mathbb{Z}_{2}$ -harmonic eigenvectors satisfying the hypotheses of Theorem 1.6. Then, there exist approximate solutions $(\Phi_{1},A_{1})$ smoothly parameterized by $(\varepsilon,\tau)$ and constructed as in (7.5) with the following properties.

(I)

They satisfy

$\text{SW}\left(\frac{\Phi_{1}}{\varepsilon},A_{1}\right)=\frac{\chi^{-}\Lambda(\tau)\Phi_{\tau}}{\varepsilon}+e_{1}+f_{1}$

where SW denotes the extended, gauge-fixed Seiberg–Witten equations with respect to $(g_{\tau},B_{\tau})$ , and $e_{1},f_{1}$ are error terms obeying
- (1)
  
  $e_{1}\in\Gamma(S^{\text{Re}})$ , and $\text{supp}(e_{1})\subseteq\text{supp}(d\chi^{+})$
- (2)
  
  $\|e_{1}\|_{L^{2}(Y)}\leq C\varepsilon^{-1/24+\gamma}.$
- (3)
  
  $\|f_{1}\|_{L^{2}(Y)}\leq C\varepsilon^{M}$ for $M>10$ , and $\text{supp}(f_{1})\subseteq\text{supp}(d\chi^{+})$ .
for $\gamma<<1$ . Moreover, the derivatives $\partial_{\tau}e_{1},\partial_{\tau}f_{1}$ also satisfy (1)–(3).

(II)

There is a configuration $(\varphi,a)\in H^{1,+}_{\varepsilon,0}$ such that

(\Phi_{1},A_{1})=(\Phi^{h_{\varepsilon}}_{\tau},A^{h_{\varepsilon}}_{\tau})\ +\ \chi^{+}\cdot(\varphi,a)

where

\|(\varphi,a)\|_{H^{1,+}_{\varepsilon,0}}\ +\ \|\partial_{\tau}(\varphi,a)\|_{H^{1,+}_{\varepsilon,0}}\leq C\varepsilon^{-1/12-\gamma}.

(7.6)

(III)

The $L^{2}$ -norm satisfies $\|\tfrac{\Phi_{1}}{\varepsilon}\|_{L^{2}(Y)}=\tfrac{1}{\varepsilon}\ +\ o(1)$ .

∎

Remark 7.6.

The de-singularization process used to obtain $(\Phi^{h_{\varepsilon}}_{\tau},A^{h_{\varepsilon}}_{\tau})$ smoothes the $\mathbb{Z}_{2}$ -harmonic spinor and the accompanying singular connection $A_{\tau}$ , thereby re-introducing a highly concentrated “bubble” of curvature. Briefly, these de-singularized configurations are exponentially close to $(\Phi_{\tau},A_{\tau})$ outside a neighborhood of radius $r=c\varepsilon^{2/3}$ , and for $r<c\varepsilon^{2/3}$ may be characterized as follows. $\Phi^{h_{\varepsilon}}_{\tau}$ is smooth and non-vanishing with $|\Phi^{h_{\varepsilon}}_{\tau}|>c\varepsilon^{1/3}$ , and $A^{h_{\varepsilon}}_{\tau}$ is smooth and its connection form in the trivialization (3.10) vanishes at $r=0$ . The curvature $F_{A^{h_{\varepsilon}}}$ is smooth with $C^{0}$ -norm of size $O(\varepsilon^{-4/3})$ , and $L^{2}$ -norm of size $O(\varepsilon^{-2/3})$ . See [41, Fig.1, pg.30]. The model solutions $(\Phi_{1},A_{1})$ are a small perturbations a as in Item (II) and do not disrupt this qualitative behavior.

The next theorem gives a precise statement of the invertibility of the linearization at the approximate solutions from Theorem 7.5. Let $\mathcal{L}_{(\Phi_{1},A_{1})}$ denote the (extended, gauge-fixed) linearization at the model solutions (recall from Section 4.1 that this means the linearization at the un-renormalized spinor with norm $O(\varepsilon^{-1})$ ), and let $\mathcal{L}^{\pm}$ denote its restrictions to $Y^{\pm}$ .

Theorem 7.7.

([41], Theorems 1.4 & 7.1) For any $\nu\in\left[0,\tfrac{1}{4}\right)$ ,

\mathcal{L}^{+}_{(\Phi_{1},A_{1})}:H^{1,+}_{\varepsilon,\nu}(Y^{+})\longrightarrow L^{2,+}_{\varepsilon,\nu}(Y^{+})

(7.7)

is Fredholm of Index 0. Moreover, there exists an $\varepsilon_{0}$ such that for $\varepsilon\in(0,\varepsilon_{0})$ , it is invertible, and there is a constant $C_{\nu}$ such that the bound

\|(\varphi,a)\|_{H^{1,+}_{\varepsilon,\nu}}\leq\frac{C_{\nu}}{\varepsilon^{1/12+\gamma^{\prime}}}\|\mathcal{L}^{+}_{(\Phi_{1},A_{1})}(\varphi,a)\|_{L^{2,+}_{\varepsilon,\nu}}

(7.8)

holds uniformly for $\tau\in(-\tau_{0},\tau_{0})$ where $\gamma^{\prime}=\tfrac{2}{3}\left(\tfrac{1}{4}-\nu\right)+\gamma\nu$ . Moreover, $\partial_{\tau}\mathcal{L}^{+}_{(\Phi_{1},A_{1})}$ is uniformly bounded on the spaces (7.7). ∎

Note in particular for $\nu^{+}=\tfrac{1}{4}-10^{-6}$ is as in Section 2, then $\gamma^{\prime}<<1$ .

7.3. Outside Boundary Conditions

This subsection departs from [41] to discuss the linear theory for the Seiberg–Witten equations in $Y^{-}$ . Recall from Lemma 4.2 that the linearized Seiberg–Witten equations at the limiting $\mathbb{Z}_{2}$ -harmonic spinor take the form

\mathcal{L}_{(\Phi_{0},A_{0})}(\varphi_{1},\varphi_{2},a)=\begin{pmatrix}\not{D}_{A_{0}}&0&0\\ 0&\not{D}_{A_{0}}&\gamma(\_)\frac{\Phi_{0}}{\varepsilon}\\ 0&\frac{\mu(\_,\Phi_{0})}{\varepsilon}.&\mathbb{d}\end{pmatrix}\begin{pmatrix}\varphi_{1}\\ \varphi_{2}\\ a\end{pmatrix}.

(7.9)

where $\varphi=(\varphi_{1},\varphi_{2})\in S^{\text{Re}}\oplus S^{\text{Im}}$ . In particular, the real component decouples from the imaginary and form components. The top left block is the operator that was studied in Section 4; the bottom block is a copy of the standard (i.e. single spinor) SW equations, can be reduced to standard elliptic theory by viewing it as a boundary-value problem using an adaptation of the boundary conditions of Lemma 7.9.

First, define the outside Sobolev norms with boundary conditions complementary to those of Definition 7.1. Let $\lambda^{-}=\varepsilon^{2/3-\gamma}$ , and set $Y^{-}=Y-N_{\lambda^{-}}(\mathcal{Z}_{\tau})$ .

Definition 7.8.

Let $\nu\in\mathbb{R}$ be a weight. The “outside” Sobolev norms are defined by

	$\displaystyle\\|(\varphi_{1},\varphi_{2},a)\\|_{H^{1,-}_{\varepsilon,\nu}}$	$\displaystyle:=$	$\displaystyle\left(\\|\varphi_{1}\\|_{r^{1-\nu}H^{1}_{e}(Y)}^{2}\ +\ \int_{Y^{-}}\left(\|\nabla\varphi_{2}\|^{2}+\|\nabla a\|^{2}+\frac{\|\varphi_{2}\|^{2}\|\Phi\|^{2}}{\varepsilon^{2}}+\frac{\|a\|^{2}\|\Phi\|^{2}}{\varepsilon^{2}}\right)R_{\varepsilon}^{2\nu}\ dV\right)^{1/2}$
	$\displaystyle\\|(\varphi_{1},\varphi_{2},a)\\|_{L^{2,-}_{\varepsilon,\nu}}$	$\displaystyle:=$	$\displaystyle\left(\\|\varphi_{1}\\|^{2}_{r^{-\nu}L^{2}(Y)}\ +\ \int_{Y^{-}}\left(\|\varphi\|^{2}+\|a\|^{2}\right)R_{\varepsilon}^{2\nu}\ dV\right)^{1/2}$

where the dependence of $\Phi_{\tau},dV$ , and $R_{\varepsilon}$ on $\tau$ is suppressed in the notation. Here, $\nabla=\nabla_{A_{\tau}}$ .

Notice this norm is not, technically speaking, a norm on sections of a vector bundle over the manifold $Y$ ; in particular the section $\varphi_{1}$ in the real component is integrated over all $Y-\mathcal{Z}_{\tau}$ whereas the other two components only over $Y^{-}$ . Notice also the difference in the sign convention for the weight between these spaces and those of Section 4 (which adopts the conventions standard for edge operators); in particular $r^{\nu}L^{2}=L^{2}_{\varepsilon,-\nu}$ . Finally, observe that $|\varphi_{2}||\Phi|^{2}=|\mu(\varphi,\Phi)|^{2}$ since $\Phi\in S^{\text{Re}}$ and that $R_{\varepsilon}\sim r$ on $Y^{-}$ ; thus this norm is equivalent to the $H^{1,+}_{\varepsilon,\nu}$ -norm on $Y^{+}\cap Y^{-}$ (since $\Phi-\Phi^{h_{\varepsilon}}$ is exponentially small there).

The following lemma describes the boundary conditions on $(\varphi_{2},a)$ ; no boundary condition is imposed on the $\varphi_{1}$ -component. In it, $\mathcal{L}^{\text{Im}}_{(\Phi,A)}$ denotes the linearized Seiberg-Witten equations at a smooth configuration $(\Phi,A)$ acting on tuples $(0,\varphi_{2},a)$ .

Lemma 7.9.

There exists a Hilbert subspace

H^{-}_{\Phi_{\tau}}\subseteq L^{1/2,2}(\partial Y^{-};S^{\text{Im}}\oplus\Omega)

and a projection operator $\Pi^{-}:L^{1,2}(Y^{-};S^{\text{Im}}\oplus\Omega)\to H^{-}_{\Phi_{\tau}}$ given by the composition of restriction to the boundary and the $L^{2}$ -orthogonal projection such that

(\mathcal{L}^{\text{Im}}_{(\Phi,A)}\ ,\ \Pi^{-}):L^{1,2}(Y^{-},S^{\text{Im}}\oplus\Omega)\longrightarrow L^{2}(Y^{-};S^{\text{Im}}\oplus\Omega)\oplus H^{-}_{\Phi_{\tau}}

(7.10)

is Fredholm of Index 0.∎

Definition 7.10.

Define the “outside” Hilbert spaces by

	$\displaystyle H^{1,-}_{\varepsilon,\nu}$	$\displaystyle:=$	$\displaystyle\Big{\{}(\varphi_{1},\varphi_{2},a)\ \big{\|}\ \\|(\varphi,a)\\|_{H^{1,-}_{\varepsilon,\nu}}<\infty\ \ ,\ \ \Pi^{-}(\varphi,a)=0\Big{\}}$
	$\displaystyle L^{2,-}_{\varepsilon,\nu}$	$\displaystyle:=$	$\displaystyle\Big{\{}(\varphi_{1},\varphi_{2},a)\ \big{\|}\ \\|(\varphi,a)\\|_{L^{2,-}_{\varepsilon,\nu}}<\infty\Big{\}}.$

They are equipped with the norms from Definition 7.1, and the inner products arising from their polarizations. They depend implicitly on $\tau$ .

The following lemma establishes the invertibility of the bottom block matrix in (7.9), which we continue denote by $\mathcal{L}^{\text{Im}}_{(\Phi_{\tau},A_{\tau})}$ .

Lemma 7.11.

For $-\tfrac{1}{2}<\nu<\tfrac{1}{2}$ , the boundary-value problem

\mathcal{L}^{\text{Im}}_{(\Phi_{\tau},A_{\tau})}:H^{1,-}_{\varepsilon,\nu}(Y^{-};S^{\text{Im}}\oplus\Omega)\longrightarrow L^{2,-}_{\varepsilon,\nu}(Y^{-};S^{\text{Im}}\oplus\Omega)

(7.11)

is invertible for $\varepsilon_{0}$ sufficiently small. Moreover, the estimate

\|(\varphi_{2},a)\|_{H^{1,-}_{\varepsilon,\nu}}\leq C\|\mathcal{L}_{(\Phi_{\tau},A_{\tau})}^{\text{Im}}(\varphi_{2},a)\|_{L^{2,-}_{\varepsilon,\nu}}

holds uniformly in $\varepsilon,\tau$ , and $\partial_{\tau}\mathcal{L}^{\text{Im}}_{(\Phi_{\tau},A_{\tau})}$ is uniformly bounded on the spaces (7.11). ∎

Proof Sketch.

The Weitzenböck formula ([41, Prop 2.13]) shows that

\mathcal{L}^{\text{Im}}\mathcal{L}^{\text{Im}}\begin{pmatrix}\varphi_{2}\\ a\end{pmatrix}=\begin{pmatrix}\not{D}_{A_{\tau}}\not{D}_{A_{\tau}}\varphi_{2}\\ \mathbb{d}\mathbb{d}a\end{pmatrix}+\frac{1}{\varepsilon^{2}}\begin{pmatrix}\gamma(\mu(\varphi_{2},\Phi_{\tau}))\Phi_{\tau}\\ \mu(\gamma(a)\Phi_{\tau},\Phi_{\tau}))\end{pmatrix}+\frac{1}{\varepsilon}\mathfrak{B}(\varphi_{2},a)

(7.12)

where $\mathfrak{B}(\varphi_{2},a)$ schematically has terms of the form $a\cdot\nabla_{A_{\tau}}\Phi_{\tau}$ and $\varphi\cdot\nabla_{A_{\tau}}\Phi_{\tau}$ . Taking the inner product of (7.12) with $(\varphi_{2},a)$ shows that

\|\mathcal{L}^{\text{Im}}(\varphi_{2},a)\|^{2}_{L^{2}}=\|(\varphi_{2},a)\|^{2}_{H^{1,-}_{\varepsilon}}\ +\ \frac{1}{\varepsilon}\langle(\varphi_{2},a),\mathfrak{B}(\varphi_{2},a)\rangle\ +\ \text{b.d. terms}.

Using that $\Phi_{\tau}\sim r^{1/2}$ while $\nabla\Phi_{\tau},A_{\tau}\sim r^{-1/2}$ , the terms involving $\mathfrak{B}$ and $A_{\tau}$ are dominated by the $\varepsilon^{-2}|\Phi_{\tau}|^{2}$ weight in (7.8) on $Y^{-}$ where $r\geq\varepsilon^{2/3-\gamma}$ , and may be absorbed. The remainder of the proof is showing that the subspace of Lemma 7.9 may be chosen so that the boundary terms vanish (this is non-trivial because of factor of $e^{-i\theta}$ in the local expression (4.13), but is similar to the arguments in [41, Section 7]). ∎

7.4. Exponential and Polynomial Decay

Solutions of

\displaystyle\mathcal{L}_{(\Phi_{1},A_{1})}^{+}(\varphi,a)\

\displaystyle=

\displaystyle g^{+}

(7.13)

decay away from the support of $g^{+}$ . This section gives precise decay results that provide the operative version of requirement (II) from Section 2.2. On $Y^{+}$ , real and imaginary/form components manifest two different types decay, respectively: (1) polynomial decay, akin to that of Lemma 4.7 and (2) exponential decay with exponent $\varepsilon^{-1}$ .

This next lemma address the polynomial decay. An analogous result holds for solutions of the equation in Lemma (7.11), but is not needed.

Lemma 7.12.

Suppose $(\varphi,a)$ is the unique solutions of (7.13). If $\text{supp}(g^{+})\subseteq\{r\leq c\varepsilon^{2/3-\gamma}\}$ , then

\|d\chi^{+}.(\varphi,a)\|_{L^{2}}\leq C\varepsilon^{-1/24-\gamma}\|g\|_{L^{2}}.

Proof.

The proof follows an identical strategy to that of Lemma 4.7, with the following minor adjustments. First, Theorem 7.7 only applies for weights $\nu\in[0,\tfrac{1}{4})$ , the factor gained in the decay is reduced by half to $\varepsilon^{1/24-\gamma}$ . Second, the lack of uniform invertibility in (7.8) leads to an adverse factor of $\varepsilon^{-1/12-\gamma}$ . Multiplying these and redefining $\gamma$ yields the assertion of the lemma. ∎

One could (perhaps justifiably) gripe that Lemma 7.12 does not, at first glance, constitute a “decay” result because the decay factor diverges as $\varepsilon\to 0$ . The point, however, is that this result still tempers that growth compared to the factor expected from (7.8), and does so sufficiently much that when combined with Lemma 4.7 the power of $\varepsilon$ for a full cycle of (2.17) is positive.

The imaginary/form components of a solution to (7.13) obey a far stronger decay property. That this is so is a consequence of the intrinsic structure of generalized Seiberg–Witten equations, which allows them to be interpreted as a non-linear concentrating Dirac operator with degeneracy. This interpretation is developed extensively in [42], which establishes decay results in a quite general setting.

The following lemma specializes these results to the present case; it is proved in Appendix A of [42]. For the statement of the lemma, let $K_{\varepsilon}\Subset Y^{+}-\mathcal{Z}_{\tau}$ denote a family of compact subsets of the complement of the singular set. Set $r_{K}:=\text{dist}(K_{\varepsilon},\mathcal{Z}_{\tau})$ . Let $K^{\prime}_{\varepsilon}$ be a slightly larger family of compact sets so $Y^{+}-N_{r_{K}/2}(\mathcal{Z}_{\tau})\subseteq K_{\varepsilon}^{\prime}$ .

Lemma 7.13.

Let $(\varphi,a)$ be the unique solution to (7.13), and write $(\varphi,a)=(\varphi_{1},\varphi_{2},a)$ on $Y-\mathcal{Z}_{\tau}$ . There exist constants $C,c$ such that if $\text{supp}(g)\subseteq(K_{\varepsilon}^{\prime})^{c}$ , then $S^{\text{Im}}\oplus\Omega$ -components satisfy

\|(\varphi_{2},a)\|_{C^{m}(K_{\varepsilon})}\leq\frac{C}{\varepsilon^{2m+1}r_{K}^{3/2}}\text{Exp}\left(-\frac{cr_{K}^{3/2}}{\varepsilon}\right)\|(\varphi,a)\|_{H^{1,+}_{\varepsilon,0}}

(7.14)

uniformly for $\tau\in(-\tau_{0},\tau_{0})$ . Moreover, the configuration denoted by $(\varphi,a)$ in Item (II) of Theorem 7.5 obeys the above.

Proof.

Both statements follow directly from Corollary A.2 of [42], which includes a similar conclusion for solutions of non-linear equations of the form in Theorem 7.5. ∎

8. Universal Seiberg–Witten Equations

This section uses the concentrating local solutions defined in Section 7 to construct an infinite-dimensional family of model solutions parameterized by deformations of the singular sets $\mathcal{Z}_{\tau}$ . This family is used to define a universal version of the Seiberg–Witten equations akin to the universal Dirac operator (1.10).

8.1. Hilbert Bundles

This subsection defines Hilbert bundles of Seiberg–Witten configurations over the space of embedded singular sets, analogously to Section 6.1.

Let $\mathcal{E}_{\tau}\subseteq L^{2,2}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})$ denote the ball on which $\underline{\text{Exp}}$ is a diffeomorphism as in Section 6.3. For each $\xi\in\mathcal{E}_{\tau}$ , let $\underline{F}_{\xi}$ , and $\mathcal{Z}_{\xi,\tau}$ be as in (6.3)–(6.4), and define

Y^{+}_{\varepsilon,\tau,\xi}:=\underline{F}_{\xi}[N_{\lambda}(\mathcal{Z}_{\tau})]\hskip 56.9055ptY^{-}_{\varepsilon,\tau,\xi}:=Y-\underline{F}_{\xi}[N_{\lambda^{-}}(\mathcal{Z}_{\tau})]

where $\lambda,\lambda^{-}$ are as in (7.1) and Definition 7.8 respectively. For each triple $(\varepsilon,\tau,\xi)$ there are weight functions defined analogously to those in Section 7.1:

	$\displaystyle R_{\varepsilon,\tau,\xi}$	$\displaystyle:=$	$\displaystyle R_{\varepsilon}\circ\underline{F}^{-1}_{\xi}$		(8.1)
	$\displaystyle\Phi_{\tau,\xi}^{h_{\varepsilon}}$	$\displaystyle:=$	$\displaystyle\underline{\Upsilon}^{-1}(\xi,\Phi^{h_{\varepsilon}}_{\tau})$		(8.2)

where $R_{\varepsilon}$ is the weight from Item (1) in Section 7.1 (which depends implicitly on $\tau$ ), and $\underline{\Upsilon}$ is the analogue of 6.1 defined in the proof of Lemma 8.2 below.

Definition 8.1.

Let $\nu\in\mathbb{R}$ be a weight. For each $(\varepsilon,\tau)\in(0,\varepsilon_{0})\times(-\tau_{0},\tau_{0})$ and each $\xi\in\mathcal{E}_{\tau}$ , define Hilbert spaces

	$\displaystyle H^{1,\pm}_{\varepsilon,\nu}(Y^{\pm}_{\varepsilon,\tau,\xi})$	$\displaystyle:=$	$\displaystyle\Big{\{}\ \ \ \ \ (\varphi,a)\ \ \ \ \ \big{\|}\ \ \ \ \ \ \ \\|(\varphi,a)\\|_{H^{1,\pm}_{\varepsilon,\nu}}<\infty\ \ \ \Big{\}}$
	$\displaystyle L^{2,\pm}_{\varepsilon,\nu}(Y^{\pm}_{\varepsilon,\tau,\xi})$	$\displaystyle:=$	$\displaystyle\Big{\{}\ \ \ \ \ (\varphi,a)\ \ \ \ \ \big{\|}\ \ \ \ \ \ \ \\|(\varphi,a)\\|_{L^{2,\pm}_{\varepsilon,\nu}}<\infty\ \ \ \ \Big{\}}$

where the norms are defined analogously to Definitions 7.3 and 7.10 using the domains indicated and the weights (8.1–8.2). For each $(\varepsilon,\tau)$ , denote the $\mathcal{E}_{\tau}$ -parameterized families by

	$\displaystyle\mathbb{H}^{1,\pm}_{\varepsilon,\nu}(\mathcal{E}_{\tau})$	$\displaystyle:=$	$\displaystyle\Big{\{}(\mathcal{Z}_{\xi,\tau},\varphi,a)\ \ \ \hskip 11.38092pt\|\ \ \xi\in\mathcal{E}_{\tau}\ ,\ (\varphi,a)\in H^{1,\pm}_{\varepsilon,\nu}(Y^{\pm}_{\varepsilon,\tau,\xi})\ \ \Big{\}}$		(8.3)
	$\displaystyle\mathbb{L}^{2,\pm}_{\varepsilon,\nu}(\mathcal{E}_{\tau})$	$\displaystyle:=$	$\displaystyle\Big{\{}(\mathcal{Z}_{\xi,\tau},\varphi,a)\ \ \ \hskip 11.38092pt\|\ \ \xi\in\mathcal{E}_{\tau}\ ,\ (\varphi,a)\in L^{2,\pm}_{\varepsilon,\nu}(Y^{\pm}_{\varepsilon,\tau,\xi})\ \ \ \Big{\}}.$		(8.4)

Finally, define global spaces by

\mathbb{H}^{1}_{\varepsilon,\nu}:=\mathbb{H}^{1,+}_{\varepsilon,\nu}(\mathcal{E}_{\tau})\oplus\mathbb{H}^{1,-}_{\varepsilon,\nu}(\mathcal{E}_{\tau})\hskip 56.9055pt\mathbb{L}^{2}_{\varepsilon,\nu}:=\Big{\{}(\mathcal{Z}_{\xi,\tau},\varphi,a)\ |\ \xi\in\mathcal{E}_{\tau}\ ,\ \|(\varphi,a)\|_{L^{2}_{\varepsilon,\nu}(Y)}<\infty\Big{\}}

(8.5)

where the $L^{2}_{\varepsilon,\nu}(Y)$ -norm is defined identically to the $L^{2,+}_{\varepsilon,\nu}(Y^{+})$ -norm in Definition 7.1, but integrated over all $Y$ .

The following lemma shows that pulling back by the family of diffeomorphisms $\underline{F}_{\xi}$ trivializes these vector bundles.

Lemma 8.2.

The family of diffeomorphisms $\underline{F}_{\xi}$ determines trivializations

	$\displaystyle\underline{\Upsilon}:\mathbb{H}^{1,\pm}_{\varepsilon,\nu}(\mathcal{E}_{\tau})$	$\displaystyle\longrightarrow$	$\displaystyle\mathcal{E}_{\tau}\ \times\ H^{1,\pm}_{\varepsilon,\nu}(Y^{\pm}_{\varepsilon,\tau,0})$		(8.6)
	$\displaystyle\underline{\Upsilon}:\mathbb{L}^{2,\pm}_{\varepsilon,\nu}(\mathcal{E}_{\tau})$	$\displaystyle\longrightarrow$	$\displaystyle\mathcal{E}_{\tau}\ \times\ L^{2,\pm}_{\varepsilon,\nu}(Y^{\pm}_{\varepsilon,\tau,0})$		(8.7)

which endow the spaces on the left with the structure of smooth Hilbert vector bundles. The same applies to $\mathbb{L}^{2}_{\varepsilon,\nu},\mathbb{H}^{1}_{\varepsilon,\nu}$ .

Proof Sketch.

(See [43, Sec. 5.1] for further details). Fix $\varepsilon,\tau$ . To begin, we define $\underline{\Upsilon}$ on the spinor components. For each $\xi\in\mathcal{E}_{\tau}$ , the $\underline{\Upsilon}_{\xi}$ is the map induced on sections by a fiberwise isomorphism $\underline{\upsilon}_{\xi}:(S_{E})_{y_{1}}\to(S_{E})_{y_{2}}$ where $y_{1}=\underline{F}_{\xi}(y_{2})$ .

For clarity, fix a spin structure with spinor bundle $S_{g_{\tau}}$ and a complex line bundle $L$ so that the spinor bundle of the $\text{Spin}^{c}$ -structure is $S=S_{g_{\tau}}\otimes_{\mathbb{C}}L$ . $\underline{\upsilon}_{\xi}$ is defined as the composition of the following three maps:

where

•

$\underline{F}_{\xi}^{*}$ is the pullback by the diffeomorphism $\underline{F}_{\xi}$ .
•

$\mathfrak{S}_{\xi}$ is the canonical isomorphism $\underline{F}_{\xi}^{*}(S_{g_{\tau}})\simeq S_{g_{\xi}}$ on the first factor, and the identity on $L\otimes E$ .
•

$\mathfrak{T}_{g_{\xi}}^{g_{\tau}}$ is the parallel transport map from the proof of Proposition (6.3).

In the third bullet, parallel transport over $Z=Y\times[0,1]$ on the bundles $S_{g_{s\xi}}$ , $\underline{F}_{s\xi}^{*}(L\otimes E)$ using the connections $\underline{F}_{s\xi}^{*}B_{\tau}$ and $\underline{F}_{s\xi}^{*}A_{\circ}$ on the second and third factors where $A_{\circ}$ is a fixed smooth background connection on $L$ .

The definition of $\underline{\Upsilon}$ on the forms component is identical with the simplification that the bundle of forms is canonically identified with its pullback so the second map $\mathfrak{S}_{\xi}$ is not necessary. ∎

8.2. Concentrating Local Families

This subsection defines the universal Seiberg–Witten equations as sections of the vector bundles from Section 8.1. The equations are viewed as deformation equations around a universal family of concentrating approximate solutions.

Definition 8.3.

For $\tau\in(-\tau_{0},\tau_{0})$ and $\xi\in\mathcal{E}_{\tau}$ , define

(A)

The universal family of concentrating local solutions by

\left(\frac{\Phi_{1}({\varepsilon,\tau,\xi})}{\varepsilon},A_{1}({\varepsilon,\tau,\xi})\right):=\underline{\Upsilon}^{-1}\left(\xi,\left(\frac{\Phi_{1}(\varepsilon,\tau)}{\varepsilon},A_{1}(\varepsilon,\tau)\right)\right)

(8.8)

i.e. as the pullback by $\underline{\Upsilon}$ of the constant section at the approximate solutions $(\Phi_{1}(\varepsilon,\tau),A_{1}(\varepsilon,\tau))$ of Theorem 7.5.

(B)

The universal families of de-singularized configurations and $\mathbb{Z}_{2}$ -eigenvectors by

\left(\Phi^{h_{\varepsilon}}_{\tau,\xi},A^{h_{\varepsilon}}_{\tau,\xi}\right):=\underline{\Upsilon}^{-1}\left(\xi,(\Phi^{h_{\varepsilon}}_{\tau},A^{h_{\varepsilon}}_{\tau})\right)\hskip 42.67912pt\left(\Phi_{\tau,\xi},A_{\tau,\xi}\right):=\underline{\Upsilon}^{-1}\left(\xi,(\Phi_{\tau},A_{\tau})\right)

(8.9)

using de-singularized configurations $(\Phi^{h_{\varepsilon}}_{\tau},A^{h_{\varepsilon}}_{\tau})$ and the $\mathbb{Z}_{2}$ -eigenvectors $(\Phi_{\tau},A_{\tau})$ respectively.

The family of de-singularized configurations was already used as the weight in definition (8.2). Strictly speaking, $\underline{\Upsilon}$ is defined to act on forms $a\in(\Omega^{0}\oplus\Omega^{1})(i\mathbb{R})$ rather than connections. It is easy to verify, via the definition of $\underline{\Upsilon}$ in the proof of Lemma 8.2 that a connection can be pulled back equally well. By construction, (8.8) have curvature highly concentrated around the deformed curves $\mathcal{Z}_{\xi,\tau}$ .

For each triple $(\varepsilon,\tau,\xi)$ , there are accompanying cut-off functions $\chi^{\pm}_{\varepsilon,\tau,\xi}:=\chi^{\pm}\circ\underline{F}_{\xi}$ . To simplify notation, the latter two subscripts are omitted when they are clear in context. Consider the deformation equation of the Seiberg–Witten equations at the approximate solution (8.8), given by

\text{SW}_{\varepsilon,\tau,\xi}(\varphi,a,\psi,b):=\text{SW}_{\tau}\left(\left(\frac{\Phi_{1}({\varepsilon,\tau,\xi})}{\varepsilon},A_{1}({\varepsilon,\tau,\xi})\right)+\chi^{+}_{\varepsilon}\cdot(\varphi,a)+\chi^{-}_{\varepsilon}\cdot(\psi,b)\right)

(8.10)

for $(\varphi,a)\in(H^{1,+}_{\varepsilon})_{\xi}$ and $(\psi,b)\in(H^{1,-}_{\varepsilon})_{\xi}$ in the fibers over $\xi\in\mathcal{E}_{\tau}$ , where $\text{SW}_{\tau}$ denotes the Seiberg–Witten equations with parameter $p_{\tau}$ . Additionally, denote the projection by $p:\mathbb{H}^{1}_{\varepsilon,\nu}\to\mathcal{E}_{\tau}$ .

Definition 8.4.

For each $(\varepsilon,\tau)\in(0,\varepsilon_{0})\times(-\tau_{0},\tau_{0})$ , the universal Seiberg–Witten equations (resp. eigenvector equation) are the sections

defined by

	$\displaystyle\mathbb{SW}(\xi,\varphi,a,\psi,b)\ \$	$\displaystyle:=$	$\displaystyle\text{SW}_{\varepsilon,\tau,\xi}(\varphi,a,\psi,b)$		(8.11)
	$\displaystyle\overline{\mathbb{SW}}(\xi,\varphi,a,\psi,b,\mu)$	$\displaystyle:=$	$\displaystyle\text{SW}_{\varepsilon,\tau,\xi}(\varphi,a,\psi,b)-\mu\chi^{-}_{\varepsilon}\frac{\Phi_{\tau,\xi}}{\varepsilon}$		(8.12)

where the domain of the latter also includes the trivial summand $\mu\in\underline{\mathbb{R}}$ .

To be more precise, (8.10) means the extended, gauge-fixed Seiberg–Witten equations subject to the gauge-fixing condition

-d^{\star}c-i\frac{\langle i\zeta,{\Phi_{\xi,\tau}^{h_{\varepsilon}}}\rangle}{\varepsilon}=0,

(8.13)

where $(\zeta,c)=(\phi_{1},a_{1})+\chi^{+}(\varphi,a)+\chi^{-}(\psi,b)$ where $(\varepsilon\phi_{1},a_{1})=(\Phi_{1}-\Phi^{h_{\varepsilon}},A_{1}-A^{h_{\varepsilon}})$ for each $\xi,\tau$ . For $\xi=0$ , this is the gauge-fixing condition coincides with the one used in the proof of Theorem 7.5.

The purpose of introducing the family is that one only expects the deformation needed to correct (8.8) to a true solution to be small only in the weighted spaces for the correct deformation $\xi$ . Ultimately, each equation in the universal family (8.11) is simply the Seiberg–Witten equations on $Y$ using the parameters $(g_{\tau},B_{\tau})$ acting on spaces with varying weights. In particular, to solve the SW equation on $Y$ , it suffices to solve it for any parameter $\xi$ .

Corollary 8.5.

Suppose that $\xi\in\mathcal{E}_{\tau}\cap C^{\infty}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})$ and that $g\in C^{\infty}(Y)$ , then

{\mathbb{SW}_{\tau}}(\xi,\varphi,a,\psi,b)=g\ \ \Leftrightarrow\ \ \text{SW}_{\tau}(\Phi,A)=g

where $(\Phi,A)$ is the configuration on the right side of (8.10), and $\text{SW}_{\tau}$ is the Seiberg–Witten equation using the parameters $(g_{\tau},B_{\tau})$ . In particular, $(\Phi,A)$ is smooth.

The equivalent statement holds for the eigenvector equations $\overline{\mathbb{SW}}$ .

Proof.

The first statement is immediate from Definition 8.4. By Definition 8.1, $(\varphi,a)\in H^{1,+}_{\varepsilon,\nu}$ and $(\psi,b)\in H^{1,-}_{\varepsilon,\nu}$ imply that $(\Phi,A)\in L^{1,2}(Y)$ . Since the background parameter $(g_{\tau},B_{\tau})$ is smooth, elliptic bootstrapping shows that $(\Phi,A)$ is also smooth.

In the case of the eigenvector equation, $\chi^{-}\Phi_{\tau}\in C^{\infty}$ because $\Phi_{\tau}$ is smooth away from the singular set $\mathcal{Z}_{\tau}$ where $\chi^{-}=0$ . Since $\xi$ is smooth, $\chi^{-}\Phi_{\tau,\xi}$ is as well and elliptic bootstrapping applies again. ∎

The universal Seiberg–Witten equations can also be extended to a map

\overline{\mathbb{SW}}:\mathbb{H}^{1}_{\varepsilon,\nu}(\mathcal{E}_{\tau}\times\mathcal{X}_{\tau})\to p_{1}^{*}\mathbb{L}^{2}_{\varepsilon,\nu}

(8.14)

where $\mathcal{X}_{\tau}$ is the subspace from Lemma 5.6, and the bundle is the pullback of $\mathbb{H}^{1}_{\varepsilon,\nu}$ by the projection $\mathcal{E}_{\tau}\times\mathcal{X}_{\tau}\to\mathcal{E}_{\tau}$ . The extended map is defined by replacing $\psi$ in (8.10) by $\psi+u$ for $u\in\mathcal{X}_{\tau}$ . This map factors through (8.12) since the map $\mathcal{X}_{\tau}\hookrightarrow H^{1,-}_{\varepsilon,\nu}$ by $u\mapsto\chi^{-}u$ is an inclusion.

8.3. Universal Linearization

This section calculates of the derivative of the universal Seiberg–Witten equations with respect to the deformation parameter.

Let $h_{1}=(\mathcal{Z}_{\tau},0,0)\in\mathbb{H}^{1}_{\varepsilon,\nu}$ . The (fiberwise component of) the linearization is a map

\text{d}{\mathbb{SW}}_{h_{1}}:L^{2,2}(\mathcal{Z}_{\tau})\ \oplus\ H^{1,+}_{\varepsilon,\nu}(Y^{+})\ \oplus\ H^{1,-}_{\varepsilon,\nu}(Y^{-})\ \longrightarrow\ L^{2}(Y),

(8.15)

where we have used the canonical splitting of $T_{h_{1}}\mathbb{H}^{1}_{\varepsilon,\nu}$ along the zero-section. The next two propositions calculate this derivative in two steps, giving the analogues of Proposition 6.3 and Theorem 6.4 for the Seiberg–Witten equations.

In the following proposition, recall that $(g_{\xi},B_{\xi}):=\underline{F}_{\xi}^{*}(g_{\tau},B_{\tau})$ . Remember also that the model solution $(\Phi_{1},A_{1})$ depends implicitly on $(\varepsilon,\tau)$ .

Proposition 8.6.

In the local trivializations of Lemma 8.2, the derivative (8.15) is given by

\text{d}{\mathbb{SW}}_{h_{1}}(\xi,\varphi,a,\psi,b)=\mathfrak{B}_{h_{1}}(\xi)\ +\ \mathcal{L}_{h_{1}}\left(\chi_{\varepsilon}^{+}(\varphi,a)\ +\ \chi_{\varepsilon}^{-}(\psi,b)\right)

(8.16)

where

•

$\mathfrak{B}_{h_{1}}$ is defined by

\mathfrak{B}_{h_{1}}(\xi)=\frac{d}{ds}\Big{|}_{s=0}\left((\mathfrak{T}_{g_{\tau}}^{g_{\xi}(s)})^{-1}\circ\text{SW}_{p_{\xi}(s)}\circ\mathfrak{T}_{g_{\tau}}^{g_{\xi}(s)}\right)\left(\frac{\Phi_{1}}{\varepsilon},A_{1}\right)

(8.17)

where $\mathfrak{T}^{g_{\xi}(s)}_{g_{\tau}}$ is the parallel transport map from Lemma 8.2, and $\text{SW}_{p_{\xi(s)}}$ denotes the Seiberg-Witten equations with parameter $p_{\xi}(s)=(g_{s\xi},B_{s\xi})$ .

•

$\mathcal{L}_{h_{1}}$ is the linearized Seiberg-Witten equations at $(\Phi_{1},A_{1})$ , given by (4.4)

Proof.

(see also [43, Prop. 5.5]) For the duration of the proof, we suppress the dependence on $\tau,\varepsilon,\nu$ from the notation. Choose a path

	$\displaystyle\gamma:(-s_{0},s_{0})$	$\displaystyle\to$	$\displaystyle\mathbb{H}^{1}(\mathcal{E})$
	$\displaystyle s$	$\displaystyle\mapsto$	$\displaystyle(\mathcal{Z}_{\zeta(s)},\mathfrak{p}(s))$

such that $\gamma(0)=h_{1}$ , where $\zeta(s)=s\xi+O(s^{2})$ . We may write $\mathfrak{p}(s)=\underline{\Upsilon}_{\zeta(s)}^{-1}\mathfrak{q}(s)$ , where $\mathfrak{q}(s)=(\varphi_{s},a_{s},\psi_{s},b_{s})\in H^{1,+}\oplus H^{1,-}$ . Using Definition 8.4 and (8.10), and then substituting Definition 8.8, the derivative (8.16) is then given by

	$\displaystyle\frac{d}{ds}\Big{\|}_{s=0}\underline{\Upsilon}_{\zeta(s)}\circ{\mathbb{SW}}(\mathcal{Z}_{\zeta(s)},\mathfrak{p}(s))$	$\displaystyle=$	$\displaystyle\frac{d}{ds}\Big{\|}_{s=0}\underline{\Upsilon}_{\zeta(s)}\circ\text{SW}\left(\left(\frac{\Phi_{1}({\zeta(s)})}{\varepsilon},A_{1}({\zeta(s)})\right)+\chi^{\pm}\mathfrak{p}(s)\right)$
		$\displaystyle=$	$\displaystyle\frac{d}{ds}\Big{\|}_{s=0}\underline{\Upsilon}_{\zeta(s)}\circ\text{SW}\circ\underline{\Upsilon}_{\zeta(s)}^{-1}\left(\left(\frac{\Phi_{1}}{\varepsilon},A_{1}\right)+\chi^{+}(\varphi_{s},a_{s})+\chi^{-}(\psi_{s},b_{s})\right)$

where $\underline{\Upsilon}$ is used to denote the trivialization of both $\mathbb{H}^{1}$ and $\mathbb{L}^{2}$ .

(LABEL:tocalculate6.6) appears as the rightmost vertical arrow in the commuting diagram below. The diagram decomposes $\underline{\Upsilon}=\mathfrak{T}_{g_{\zeta}}^{g}\circ\mathfrak{S}\circ\underline{F}_{\zeta}^{*}$ as in the proof of Lemma 8.2. It also abbreviates $H^{1}=H^{1,+}\oplus H^{1,-}$ and $\Omega=(\Omega^{0}\oplus\Omega^{1})(i\mathbb{R})$ , and $p_{\zeta}=(g_{\zeta}(s),B_{\zeta(s)})$ , and also uses $S_{E}^{h}$ to denote the spinor bundle formed using the metric $h$ .

Expressing (LABEL:tocalculate6.6) as a composition using the vertical middle arrow in the diagram, and writing the Seiberg-Witten equations near a configuration $(\Phi,A)$ as

\text{SW}\left((\Phi,A)+(\varphi,a)\right)=\text{SW}(\Phi,A)+\mathcal{L}_{(\Phi,A)}(\varphi,a)+Q(\varphi,a),

(8.19)

the derivative is given by

$\displaystyle=$	$\displaystyle\frac{d}{ds}\Big{\|}_{s=0}\Upsilon_{\zeta(s)}\circ\text{SW}\circ\Upsilon_{\zeta(s)}^{-1}\left(\left(\frac{\Phi_{1}}{\varepsilon},A_{1}\right)+\chi^{+}(\varphi_{s},a_{s})+\chi^{-}(\psi_{s},b_{s})\right)$	(8.21)
$\displaystyle=$	$\displaystyle\frac{d}{ds}\Big{\|}_{s=0}(\mathfrak{T}_{g_{\tau}}^{g_{\zeta(s)}})^{-1}\circ\text{SW}_{p_{\zeta}(s)}\circ(\mathfrak{T}_{g_{\tau}}^{g_{\zeta(s)}})\left(\left(\frac{\Phi_{1}}{\varepsilon},A_{1}\right)+\chi^{+}(\varphi_{s},a_{s})+\chi^{-}(\psi_{s},b_{s})\right)$
$\displaystyle=$	$\displaystyle\frac{d}{ds}\Big{\|}_{s=0}(\mathfrak{T}_{g}^{g_{\zeta(s)}})^{-1}\circ\text{SW}_{p_{\zeta}(s)}\circ(\mathfrak{T}_{g}^{g_{\zeta(s)}})\left(\frac{\Phi_{1}}{\varepsilon},A_{1}\right)$
	$\displaystyle+\ \frac{d}{ds}\Big{\|}_{s=0}(\mathfrak{T}_{g}^{g_{\zeta(s)}})^{-1}\circ(\mathcal{L}^{p_{\zeta}(s)}_{h_{1}(s)}+Q^{p_{\zeta(s)}})\circ(\mathfrak{T}_{g}^{g_{\zeta(s)}})\left(\chi^{+}(\varphi_{s},a_{s})+\chi^{-}(\psi_{s},b_{s})\right),$

where the last equality is an instance of (8.19). Here, $\mathcal{L}_{h_{1}(s)}^{p_{\zeta(s)}}$ is the linearization of the Seiberg–Witten equations at $h_{1}(s)=\mathfrak{T}_{g}^{g_{\zeta(s)}}(h_{1})$ using the parameter $p_{\zeta}(s)$ .

Since $\zeta(s)=s\xi+O(s^{2})$ , (8.21) is by definition $\mathfrak{B}_{h_{1}}(\xi)$ after dropping $O(s^{2})$ terms. Differentiating (8.21) using the product rule, all terms vanish except those differentiating $(\varphi_{s},a_{s},\psi_{s},b_{s})$ since $\mathfrak{p}(0)=0$ . Because $\mathfrak{T}_{g}^{g_{\zeta(0)}}=\text{Id}$ and $Q$ is quadratic, what remains is simply $\mathcal{L}_{h_{1}}^{p_{0}}(\dot{\varphi},\dot{a},\dot{\psi},\dot{b})$ , giving the second bullet point. ∎

The next proposition gives a concrete formula for the term $\mathfrak{B}_{h_{1}}(\xi)$ using Theorem 6.4. Set

\dot{g}_{\xi}=\frac{d}{ds}\Big{|}_{s=0}\underline{F}_{s\xi}^{*}g_{\tau}\hskip 56.9055pt\dot{B}_{\xi}=\frac{d}{ds}\Big{|}_{s=0}\underline{F}_{s\xi}^{*}B_{\tau}.

Proposition 8.7.

The term $\mathfrak{B}_{h_{1}}(\xi)$ in (8.16) is given by

\mathfrak{B}_{h_{1}}(\xi)=\begin{pmatrix}\frac{1}{\varepsilon}\underline{\mathcal{B}}_{\Phi_{1}}(\xi)\\ \underline{\mathcal{D}}_{A_{1}}(\xi)+\underline{\mu}_{\Phi_{1}}(\xi)\end{pmatrix}

(8.22)

where

(1)

$\underline{\mathcal{B}}_{\Phi_{1}}(\xi)$ is the metric variation of the Dirac operator from Corollary 6.3,

\underline{\mathcal{B}}_{\Phi_{1}}(\xi)=\left(-\frac{1}{2}\sum_{ij}\dot{g}_{\xi}(e_{i},e_{j})e^{i}.\nabla_{j}+\frac{1}{2}d\text{Tr}_{g_{\tau}}(\dot{g}_{\xi}).+\frac{1}{2}\text{div}_{g_{\tau}}(\dot{g}_{\xi}).+\mathcal{R}(B_{\tau},\xi).\right)\Phi_{1}

(8.23)

where $\Phi_{1}=\Phi_{1}({\varepsilon,\tau})$ is as in (8.8), . denotes the Clifford multiplication of $g_{\tau}$ , and $\nabla_{j}$ is the covariant derivative on $S_{E}$ formed using the spin connection of $g_{\tau}$ , $A_{1}$ , and $B_{\tau}$ .

(2)

$\underline{\mathfrak{D}}_{A_{1}}(\xi)$ is the metric variation of the de-Rham operator $\mathbb{d}$ given by

{\bf\underline{\mathfrak{D}}}_{A_{1}}(\xi)=\left(-\frac{1}{2}\sum_{ij}\dot{g}_{\xi}(e_{i},e_{j})\mathbb{c}(e^{i})\nabla^{\text{LC}}_{j}+\frac{1}{2}\mathbb{c}(d\text{Tr}_{g_{\tau}}(\dot{g}_{\xi}))+\frac{1}{2}\mathbb{c}(\text{div}_{g_{\tau}}(\dot{g}_{\xi}))\right)A_{1}

(8.24)

where $A_{1}=A_{1}(\varepsilon,\tau)$ is the connection form of (8.8) in the trivialization of Lemma 3.10, $\mathbb{c}$ is the symbol of $\mathbb{d}$ , and $\nabla^{\text{LC}}$ is the Levi-Civitas connection of $g_{\tau}$ .

(3)

$\underline{\mu}_{\Phi_{1}}(\xi)$ is the metric variation of the moment map given by

\frac{1}{2}\underline{\mu}_{\Phi_{1}}(\xi)=\left(0,-\frac{1}{2}\sum_{jk}\ \frac{\langle i\dot{g}_{\xi}(e_{j},e_{k})e^{j}.\Phi_{1},\Phi_{1}\rangle}{\varepsilon^{2}}ie^{k}\right)

(8.25)

where . and $\Phi_{1}$ are as in Item (1). Here $j,k=1,2,3$ are indices and $i=\sqrt{-1}$ .

Proof.

(1) The metric variation formula of Bourguignon-Gauduchon (Theorem 6.4) applies equally well to twisted Dirac operators, provided the connection on the twisting bundle remains fixed. The $U(1)$ -connection in (8.17) (i.e. in the middle arrow of the diagram in the proof of Proposition 8.6) is the fixed connection $A_{1}$ by Definition 8.8. The connection on $E$ differs from the fixed connection $B_{\tau}$ by a zeroth order (in both $\xi$ and $\Phi_{1}$ ) term

B_{\tau}-\dot{B}_{\xi}:=\mathcal{R}(B_{\tau},\xi).

In fact, a quick calculation shows $\mathcal{R}(B_{\tau},\xi)=\iota_{\xi}F_{B_{\tau}}$ is the contraction of the curvature with $\frac{d}{ds}\Big{|}_{s=0}\underline{F}_{s\xi}$ .

(2) Let $L$ be as in the proof of Lemma 8.2. Let $A_{\circ}$ denote a smooth connection on $L$ that extends the product connection in the trivialization of Lemma 3.10. Then

\star_{\xi}F_{A_{1}}=\star_{\xi}F_{A_{\circ}}+\star_{\xi}d(A_{\circ}-A_{1})

where $\star_{\xi}$ is the Hodge star of $g_{\xi}$ . Since $g_{\xi}=g_{\tau}$ outside the neighborhood $N_{r_{0}}(\mathcal{Z}_{\tau})$ where $\underline{F}_{\xi}=\text{Id}$ , while $F_{A_{\circ}}$ vanishes inside $N_{r_{0}}(\mathcal{Z}_{\tau})$ , the first term is zero. Consequently, the variation of the curvature (when supplemented with the $\Omega^{0}(i\mathbb{R})$ component and gauge-fixing) reduces to the metric variation of $\mathbb{d}$ . The variation of this Dirac-type operator follows equally well from (Theorem 6.4), with the additional simplification that the form bundle does not depend on the metric.

(3) Let $\mathfrak{a}_{\xi}$ be such that $g_{\xi}=g_{\tau}(\mathfrak{a}_{\xi}X,Y)$ , then $\widetilde{e}_{j}=\mathfrak{a}_{\xi}^{-1/2}e_{j}$ is an orthonormal frame for $g_{\xi}$ where $\{e_{j}\}$ is one of $g_{\tau}$ . Expanding in Taylor series in the frame of $g_{\tau}$ and differentiating yields $\dot{a}_{\xi}(s)=-\tfrac{1}{2}\dot{g}_{\xi}$ , just as in the symbol term of (6.4). The variation formula then follows from the definition of $\mu$ (3.1). ∎

8.4. Non-Linear Terms

This section characterizes the non-linear terms in the universal Seiberg-Witten equations. The equations have quadratic non-linearities in fiber directions of $H^{1}_{\varepsilon,\nu}$ —these simply being the non-linearities of the original Seiberg-Witten equations, but are fully non-linear in the deformation parameter $\xi$ . There are also mixed terms fully non-linear in $\xi$ and linear or quadratic in the fiber directions.

The universal Seiberg-Witten equations at $h=\mathbb{H}^{1}_{\varepsilon}\oplus\underline{\mathbb{R}}$ may be written

\displaystyle\mathbb{SW}\left(h\right)

\displaystyle=

\displaystyle\text{SW}\left(\frac{\Phi_{1}}{\varepsilon},A_{1}\right)\ +\ \text{d}{\mathbb{SW}}_{h_{1}}(h)\ +\ \mathbb{Q}_{h_{1}}(h)

(8.26)

where $\mathbb{Q}_{h_{1}}$ consists of the non-linear terms. Write $h=(\xi,\varphi,a)$ where $(\varphi,a)=\chi^{+}(\varphi^{+},a^{+})+\chi^{-}(\psi,b)$ for $(\varphi^{+},a^{+},\psi,b)\in H^{1,+}_{\varepsilon}\oplus H^{1,-}_{\varepsilon}$ to simplify notation.

Proposition 8.8.

The non-linear term has the form

\mathbb{Q}_{h_{1}}(h)=Q_{\text{SW}}(\varphi,a)\ +\ \varepsilon Q_{\Phi_{1}}(\xi,\varphi)\ +\ \varepsilon Q_{A_{1}}(\xi,a)\ +\ Q_{a.\varphi}(\xi,\varphi,a)\ +\ Q_{\mu}(\xi,\varphi,\varphi)

(8.27)

where

(1)

$Q_{\text{SW}}$ is the standard non-linearity of the Seiberg–Witten equations given by $(a.\varphi,\mu(\varphi,\varphi))$ using Clifford multiplication with respect to $g_{\tau}$ .
(2)

$Q_{\Phi_{1}}$ is the non-linear portion of the metric variation of the Dirac operator $\not{D}_{A_{1}}$ on $S_{E}$ as in Lemma 6.13, taking $\Phi=\varepsilon^{-1}\Phi_{1}$ and $\psi=\varphi$ in the notation of that lemma.
(3)

$Q_{A_{1}}$ is the non-linear portion of the metric variation of the de-Rham operator $\mathbb{d}$ as in Lemma 6.13 taking $\Phi=A_{1}$ and $\psi=a$ and appropriately substituting ${\bf{cl}}$ for Clifford multiplication.
(4)

$Q_{a.\varphi}$ is the non-linearity arising from the metric variation on the first component of $Q_{\text{SW}}$ , which has the form

$Q_{a.\varphi}=F_{1}(\underline{\chi}^{\prime}[\xi])\cdot M_{a.\varphi}(\xi)$

where $M_{a.\varphi}(\xi)$ is a term of Type B in the sense of (6.23) taking $\psi=a.\varphi$ with weight $w_{B}=1$ , and $F_{1}$ is a $C^{\infty}(Y)$ -linear combination of $1,\underline{\chi}[\xi^{\prime}],\underline{\partial_{a}\chi}[\xi],$ and higher order functions of these.

(5)

$Q_{\mu}$ is the non-linearity arising from the metric variation of the moment map. Schematically, it has the form

-\frac{1}{2}\frac{\langle\dot{g}_{\xi}(e^{j},e^{k})ie^{j}(\Phi_{1}+\varepsilon\varphi),\varepsilon\varphi\rangle}{\varepsilon^{2}}ie^{k}+\frac{\langle q_{jk}(\xi)ie^{j}(\Phi_{1}+\varepsilon\varphi),\Phi_{1}+\varepsilon\varphi\rangle}{\varepsilon^{2}}ie^{k}

where $q_{jk}$ consists of quadratic and higher combinations of $\underline{\chi}[\xi^{\prime}],\underline{\partial_{a}\chi}[\xi]$ .

Proof.

(1) is immediate, (2)–(3) follow directly from Lemma 6.13. (4)–(5) are derived by expanding the formula for $\dot{g}_{\xi}$ as in Proposition 8.7 in Taylor series (see [43, Section 5.3]). ∎

9. Relating Deformation Operators

This section relates the deformation operator for the universal Seiberg–Witten equations from Section 8.3 (Proposition 8.6) to the deformation operator for the universal Dirac operator from Section 6.1 (Proposition 6.3). In the outside region $Y^{-}$ , the former is a small perturbation of the latter.

9.1. Two Deformation Operators

Because of the renormalization of the spinor in the model solutions (8.8), it is convenient to introduce the following inverse normalization for the deformation. For $\xi\in L^{2,2}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})$ , define $\eta$ by

\boxed{\xi(t)=\varepsilon\eta(t).}

(9.1)

Next, with $h_{0}=(\mathcal{Z}_{\tau},\Phi_{\tau},A_{\tau})$ and $h_{1}=(\mathcal{Z}_{\tau},\Phi_{1},A_{1})$ , define $\Xi:L^{2,2}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})\to L^{2}_{\varepsilon,\nu}(Y)$ by

	$\displaystyle\text{d}\mathbb{SW}_{h_{1}}(\xi,0,0)$	$\displaystyle=$	$\displaystyle\text{d}\not{\mathbb{D}}_{h_{0}}(\eta,0)\mathbb{\ }+\ \Xi(\eta)$		(9.2)
		$\displaystyle=$	$\displaystyle\text{d}\not{\mathbb{D}}_{h_{0}}(\eta,0)\mathbb{\ }+\ \Xi^{+}(\eta)\ +\ \Xi^{-}(\eta)$		(9.3)

where $\Xi^{\pm}(\eta)=\Xi(\eta)\mathbb{1}^{\pm}$ for $\mathbb{1}^{\pm}$ as in Section 2.

Because of the exponential decay in Lemma 7.13, the (re-normalized) approximate solution $(\Phi_{1},A_{1})$ is a small perturbation of the original $\mathbb{Z}_{2}$ -eigenvector $(\Phi_{\tau},A_{\tau})$ outside the invariant scale of $r=\varepsilon^{2/3}$ , thus one expects $\text{d}\mathbb{SW}_{h_{1}}(\varepsilon\eta)$ to also be a small perturbation of $\text{d}\not{\mathbb{D}}_{h_{0}}(\eta)$ in this region. The following lemma gives a precise bound.

Lemma 9.1.

Let $\eta(t)=\varepsilon^{-1}\xi(t)\in C^{\infty}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})$ be a linearized deformation. There exists a constant $C$ such that

\|\Xi^{-}(\eta)\|_{L^{2}_{\varepsilon,\nu}}\leq C\varepsilon^{11/12-\gamma}\|\eta\|_{3/2+\underline{\gamma}-\nu}

(9.4)

where $\gamma,\underline{\gamma}<<1$ , with $\underline{\gamma}$ as in Lemma 6.12.

Proof.

By Proposition 6.3, $\text{d}\not{\mathbb{D}}_{h_{0}}(\eta,0)=\underline{\mathcal{B}}_{\Phi_{\tau}}(\eta)$ and $\tfrac{1}{\varepsilon}\underline{\mathcal{B}}_{\Phi_{1}}(\xi)=\underline{\mathcal{B}}_{\Phi_{1}}(\eta)$ . By Proposition 8.7, we can therefore write

\Xi(\eta)=\begin{pmatrix}\underline{\mathcal{B}}_{\Phi_{1}}(\eta)-\underline{\mathcal{B}}_{\Phi_{\tau}}(\eta)\\ \underline{\mathcal{D}}_{A_{1}}(\eta)+\underline{\mu}_{\Phi_{1}}(\eta)\end{pmatrix}.

We first compare the spinor components. Let $\varphi_{1}$ denote the difference

\frac{\Phi_{1}}{\varepsilon}=\frac{\Phi_{\tau}}{\varepsilon}+\varphi_{1},

(9.5)

which satisfies $\|\varphi_{1}\|_{H^{1,+}_{\varepsilon}}\leq\varepsilon^{-1/12-\gamma}$ on the support of $\mathbb{1}^{-}$ by (7.6) in Item (II) of Theorem 7.5 and the exponential bound on $\Phi^{h_{\varepsilon}}-\Phi_{\tau}$ from Item(2b) in Section 7.1.

Then

	$\displaystyle\mathcal{B}_{\Phi_{1}}(\xi)-\mathcal{B}_{\Phi_{\tau}}(\xi)$	$\displaystyle=$	$\displaystyle\varepsilon\mathcal{B}_{\varphi_{1}}(\eta)$		(9.6)
		$\displaystyle=$	$\displaystyle\varepsilon\left(-\frac{1}{2}\sum_{ij}\dot{g}_{\eta}(e_{i},e_{j})e^{i}.\nabla_{j}+\frac{1}{2}d\text{Tr}_{g_{\tau}}(\dot{g}_{\eta}).+\frac{1}{2}\text{div}_{g_{\tau}}(\dot{g}_{\eta}).+\mathcal{R}(B_{\tau},\eta).\right)\varphi_{1}$		(9.6)

Each term in (9.6) is of Type B in the sense of Lemma 6.12, each with weight $w_{B}\leq 2$ . Applying Item (B) of that lemma with $\beta=\nu$ shows that

\|\varepsilon\mathcal{B}_{\varphi}(\eta)\mathbb{1}^{-}\|_{L^{2,-}_{\varepsilon,\nu}}\leq C\varepsilon\|\eta\|_{3/2+\underline{\gamma}-\nu}\Big{(}\|\varphi_{1}\|_{rH^{1}_{e}(Y^{-})}\Big{)}\leq C\varepsilon^{11/12-\gamma}\|\eta\|_{3/2+\underline{\gamma}-\nu}.

(9.7)

as desired. In the final inequality, we have used that the $rH^{1}_{e}$ and $H^{1,+}_{\varepsilon}$ norms are comparable for $r\geq\varepsilon^{2/3}$ . This establishes the desired bound for the spinor components.

For the $S^{\text{Im}}\oplus(\Omega^{0}\oplus\Omega^{1})$ , components, note that that the difference from the (flat) limiting connection $A_{\tau}$ and the imaginary components of $\Phi_{1}$ are exponentially small on $\text{supp}(\mathbb{1}^{-})$ by Lemma 7.13; exponentially small here meaning $O(\varepsilon^{-3}\text{Exp}(-\tfrac{1}{\varepsilon^{\gamma}}))$ for $\gamma>0$ . Since $\underline{\mathcal{D}}_{A_{\tau}}$ depends only on the $\Omega^{0}\oplus\Omega^{1}$ -component and $\underline{\mu}_{\Phi_{1}}$ only depend on the $S^{\text{Im}}$ -component by Item (2) of Lemma 3.5 (up to an exponentially small factor coming from the difference in (2a) in Section 7.1), the same argument as the spinor components shows these components satisfy (9.4) with an exponential factor in place of $\varepsilon^{11/12}$ , which may be absorbed once $\varepsilon$ is sufficiently small. ∎

The situation for radii less than the invariant scale, i.e. for $\Xi^{+}$ , stands in contrast to that of Lemma 9.1: there, the two deformation operators bear no meaningful relation. Since the invariant scale shrinks rapidly as $\varepsilon\to 0$ , however, the norm of either deformation operator in this region decreases like a fixed positive power of $\varepsilon$ so that $\Xi^{+}$ is effectively negligible, as the upcoming lemma shows. Since the weight function $R_{\varepsilon}$ is almost constant (up to a factor of $\varepsilon^{-\gamma}$ ) on $\text{supp}(\mathbb{1}^{+})$ , the lemma considers only the unweighted norms.

The proof utilizes the following re-scaling, which plays an essential role in the proof of Theorem 7.5 (see [41, Sec. 5.3]). There is a re-scaled coordinate $\rho$ satisfying $c_{1}\varepsilon^{2/3}\leq\rho\leq c_{2}\varepsilon^{2/3}$ for constants $c_{1},c_{2}$ such that de-singularized solutions $(\Phi^{h_{\varepsilon}},A^{h_{\varepsilon}})$ are given by

(\Phi^{h_{\varepsilon}},A^{h_{\varepsilon}})=(\varepsilon^{1/3}\Phi^{H}(\rho),A^{H}(\rho))

(9.8)

where $\Phi^{H},A^{H}$ are fixed, smooth, $\varepsilon$ -independent functions on $\mathcal{Z}_{\tau}\times\mathbb{R}^{2}$ in Fermi coordinates (3.9). Moreover, $\Phi^{H}\sim\rho^{1/2}$ for $\rho>>1$ , and $A^{H}=f(\rho)\left(\tfrac{dz}{z}-\tfrac{d\overline{z}}{\overline{z}}\right)$ , where $f(\rho)$ vanishes to second order at $\rho=0$ . See Section 4 of [41] for detailed proofs.

Lemma 9.2.

Let $\eta(t)=\varepsilon^{-1}\xi(t)\in C^{\infty}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})$ be a linearized deformation. There exists a constant $C$ such that

	$\displaystyle\\|\Xi^{+}(\eta)\\|_{L^{2}}$	$\displaystyle\leq$	$\displaystyle\\|\text{d}\mathbb{SW}_{h_{1}}(\xi,0)\mathbb{1}^{+}\\|_{L^{2}}+\\|\text{d}\not{\mathbb{D}}_{h_{0}}(\eta,0)\mathbb{1}^{+}\\|_{L^{2}}$		(9.9)
		$\displaystyle\leq$	$\displaystyle C\varepsilon^{-\gamma}\Big{(}\varepsilon^{1/3}\\|\eta\\|_{1,2}\ +\ \varepsilon^{11/12}\\|\eta\\|_{3/2+\underline{\gamma},2}\ +\ \ \varepsilon\\|\eta\\|_{2,2}\ +\ \varepsilon^{19/12}\\|\eta\\|_{5/2+\underline{\gamma},2}\Big{)}.$		(9.10)

where $\gamma,\underline{\gamma}<<1$ , with $\underline{\gamma}$ as in Lemma 6.12.

Proof.

(9.9) is immediate from (9.2) and the triangle inequality, so it suffices to show both terms on the right side of (9.9) satisfy the desired bound.

To begin with the deformations of the singular Dirac operator, the formula (6.7) shows

\|\text{d}\not{\mathbb{D}}_{h_{0}}(\eta,0)\mathbb{1}^{+}\|_{L^{2}}\leq\|(\dot{g}_{\eta})_{ij}e^{i}.\nabla\Phi_{\tau}\|_{L^{2}}+\|d\text{Tr}_{g_{\tau}}(\dot{g}_{\eta}).\Phi_{\tau}\|_{L^{2}}+\|\text{div}_{g_{\tau}}(\dot{g}_{\eta}).\Phi_{\tau}\|_{L^{2}}+\|\mathcal{R}(B_{\tau},\eta).\Phi_{\tau}\|_{L^{2}}.

where summation is implicit in the first term. Since $\Phi_{\tau}$ is polyhomogeneous with leading order $r^{1/2}$ by Lemma 4.5, each of these terms is of Type A in the sense of Lemma 6.12, with weight $w_{A}\leq 2$ . Copying the proof of Item (A) of Lemma 6.12, but now only integrating over the support of $\mathbb{1}^{+}$ shows that

\|(\dot{g}_{\eta})_{ij}e^{i}.\nabla\Phi_{\tau}\|_{L^{2}}\leq C\varepsilon^{1/3-\gamma}\|\eta\|_{1,2}\hskip 42.67912pt\|d\text{Tr}_{g_{\tau}}(\dot{g}_{\eta}).\Phi_{\tau}\|_{L^{2}}+\|\text{div}_{g_{\tau}}(\dot{g}_{\eta}).\Phi_{\tau}\|_{L^{2}}\leq C\varepsilon^{1-\gamma}\|\eta\|_{2,2}.

as desired. The term $\mathcal{R}(\mathcal{B}_{\tau},\eta)$ is subsumed by these because it has weight $w_{A}=1$ . This shows the second term of (9.9) satisfies the desired bound.

By (8.22) and the triangle inequality, the second term of (9.9) is bounded by

\displaystyle\|\text{d}\mathbb{SW}_{h_{1}}(\xi)\mathbb{1}^{+}\|_{L^{2}}\leq\|\underbrace{\underline{\mathcal{B}}_{\Phi_{1}}(\eta)}_{\text{(I)}}\|_{L^{2}}\ +\ \varepsilon\|\underbrace{\underline{\mathcal{D}}_{A_{1}}(\eta)}_{\text{(II)}}\|_{L^{2}}\ +\ \varepsilon\|\underbrace{\underline{\mu}_{\Phi_{1}}(\eta)}_{\text{(III)}}\|_{L^{2}}.

We bound each term (I)–(III) separately. Beginning with (I), write $(\Phi_{1},A_{1})=(\Phi^{h_{\varepsilon}}_{\tau},A^{h_{\varepsilon}}_{\tau})+(\varepsilon\varphi_{1},a_{1})$ where $(\varphi_{1},a_{1})$ are as in Item (2) of Theorem 7.5. The term (I) is comprised of four subterms (Ia)–(Id) as in (8.23); for each of these there is a leading order part coming from $(\Phi^{h_{\varepsilon}},A^{h_{\varepsilon}})$ and a perturbation coming from $(\varphi_{1},a_{1})$ . We begin with the leading order part of (Ia). Omitting indices and subscripts and writing $N^{+}=\text{supp}(\mathbb{1}^{+})$ for clarity,

$\displaystyle\\|\dot{g}_{\eta}.\nabla\Phi^{h_{\varepsilon}}\\|^{2}_{L^{2}}$	$\displaystyle\leq$	$\displaystyle C\int_{N^{+}}\left(\|\underline{\chi}(\eta^{\prime})\|^{2}+\|\underline{d\chi}(\eta)\|^{2}\right)\|\nabla\Phi^{h_{\varepsilon}}\|^{2}rdrd\theta dt$
	$\displaystyle\leq$	$\displaystyle C\int_{N^{+}}\left(\|\underline{\chi}(\eta^{\prime})\|^{2}+\|\underline{d\chi}(\eta)\|^{2}\right)\|\nabla_{\rho}\Phi^{h_{\varepsilon}}\|^{2}\rho d\rho d\theta dt$
	$\displaystyle\leq$	$\displaystyle C\varepsilon^{2/3}\int_{\mathcal{Z}_{\tau}}\left(\|\underline{\chi}(\eta^{\prime})\|^{2}+\|\underline{d\chi}(\eta)\|^{2}\right)\int_{\rho\leq\varepsilon^{-\gamma}}\|\nabla_{\rho}\Phi^{H}\|^{2}\rho d\rho d\theta dt$
	$\displaystyle\leq$	$\displaystyle C\varepsilon^{2/3-\gamma}\\|\eta\\|_{1,2}^{2}$

where we have changed variables to the rescaled coordinate $\rho$ (in both the volume and $\nabla$ ) and then substituted (9.8). The last inequality follows from the fact that $\|\underline{d\chi}(\eta)\|_{L^{2}(S^{1})}\leq C\|\eta^{\prime}\|_{L^{2}(S^{1})}$ (as in the proof of Lemma 6.14), and the fact that $\Phi^{H}\sim\rho^{1/2}$ for $\rho\geq\text{const}$ . The same argument applies to the terms (Ib)–(Id), except there is an additional factor of $\varepsilon^{2/3}$ because there is no derivative to rescale, but the norm $\|\eta\|_{L^{2,2}}$ is needed.

Turning now to the leading order of term (II), which is again comprised of three sub-terms (IIa)–(IIc) as in (8.24), a similar rescaling argument applies to show

	$\displaystyle\varepsilon^{2}\\|\dot{g}_{\eta}\nabla A^{h_{\varepsilon}}\\|_{L^{2}}^{2}$	$\displaystyle\leq$	$\displaystyle\varepsilon^{2}\int_{N^{+}}\left(\|\underline{\chi}(\eta^{\prime})\|^{2}+\|\underline{d\chi}(\eta)\|^{2}\right)\|\nabla_{\rho}f(\rho)\left(\tfrac{dz}{z}-\tfrac{d\overline{z}}{\overline{z}}\right)\|^{2}\rho d\rho d\theta dt$
		$\displaystyle=$	$\displaystyle\varepsilon^{2}\varepsilon^{-4/3}\int_{N^{+}}\left(\|\underline{\chi}(\eta^{\prime})\|^{2}+\|\underline{d\chi}(\eta)\|^{2}\right)\|G(\rho)\|^{2}\rho d\rho d\theta dt\leq\varepsilon^{2/3}\\|\eta\\|^{2}_{1,2}$

where we have substituted $\tfrac{1}{z}=\tfrac{\varepsilon^{2/3}}{\rho e^{i\theta}}$ (and likewise for $\overline{z}$ ), then used the fact that $f(\rho)$ vanishes to second order at $\rho=0$ to combine these into a smooth bounded function $G(\rho)$ . The last inequality follows just as in term (I). Terms (IIb)–(IIc) proceed with the same alterations as (Ib)–(Id).

In both terms (I)–(II) the perturbation terms arising from $(\varphi_{1},a_{1})$ , the same integration as for the leading order terms is used. First, for terms (Ia) and (IIa), a factor of $\|\eta\|_{3/2+\underline{\gamma},2}$ can be pulled out, reducing the integral to the $H^{1,+}_{\varepsilon}$ -norm; for the other terms, the same applies to (Ib–Id) and (IIb)–(IIc) with $\|\eta\|_{5/2+\underline{\gamma},2}$ (and the weight in the norm gives an extra factor of $\varepsilon^{2/3-\gamma}$ for these).

The term (III) the terms of (8.25) of the form $\langle ig_{\xi}.\Phi_{\tau},\varphi_{1}\rangle$ and $\langle ig_{\xi}.\varphi_{1},\Phi_{\tau}\rangle$ is can be bounded simply by pulling an $\varepsilon\|\eta\|_{3/2+\underline{\gamma}}$ out of the integral and using the weight on the $H^{1,+}_{\varepsilon}$ -norm involving $\mu$ . Finally, for the correction terms $\langle ig_{\xi}.\varphi_{1},\varphi_{1}\rangle$ , a bound by $\varepsilon^{11/12}\|\eta\|_{3/2+\underline{\gamma}}$ follows from the interpolation inequality in the upcoming Lemma 9.4. ∎

9.2. The Range Component of Deformations

The previous subsection provided bounds on the perturbation term in (9.2); this subsection bounds the main term $\text{d}\not{\mathbb{D}}(\eta,0)$ . The range component $(1-\Pi_{\tau})\text{d}\not{\mathbb{D}}(\eta,0)$ is the crucial term affected by the loss of regularity whose significance was discussed in Subsection 2.5.1 (the boxed term in 2.26).

The key bound is provided by the following lemma. The lemma shows that, due to the use of the mode-dependent deformations, the derivative $\text{d}\not{\mathbb{D}}(\eta,0)$ has effective support (Definition 4.6) equal to the support of the deformation $\eta$ in Fourier space.

Proposition 9.3.

Suppose that $\eta\in C^{\infty}(\mathcal{Z}_{\tau},N\mathcal{Z}_{\tau})$ satisfies the following property: there is an $M\in\mathbb{R}$ such that $\|\eta\|_{m+1/2,2}\leq CM^{m}\|\eta\|_{1/2,2}$ for all $m>0$ . Then deformation operator (6.3) of the universal Dirac operator satisfies

\|\text{d}\not{\mathbb{D}}_{h_{0}}(\eta,0)\|_{L^{2}_{-\nu}}\leq CM^{\nu}\|\eta\|_{1/2,2}

(9.11)

for any weight $\nu>0$ . In particular, it is effectively supported where $r=O(M^{-1})$ in the sense of Definition 4.6.

Proof.

By Lemma 4.5, $\Phi_{\tau}$ is polyhomogeneous. In particular, it satisfies the required bounds (6.22) uniformly in $\tau$ , so that each term (6.7) of $\text{d}_{h_{0}}\not{\mathbb{D}}(\eta,0)$ is a term of Type A with weight $w_{A}\leq 2$ in the sense of Lemma 6.12. The conclusion follows directly from applying Item (A) of that lemma with $\beta=-\nu$ and invoking the assumption on $\eta$ . ∎

9.3. Non-Linear Bounds

This subsection bounds the non-linear terms in Proposition 8.8.

The statement of the next lemma involves an auxiliary partition of unity defined as follows. With $\gamma<<1$ being the constant such that $\mathbb{1}^{+}$ is the indicator function of $\{r\leq\varepsilon^{2/3-\gamma}\}$ , let $\zeta^{\pm}$ be a partition of unity consisting of two logarithmic cut-off functions such that $|d\zeta^{\pm}|\leq Cr^{-1}$ , and $\zeta^{+}=1$ where $\{r\leq\varepsilon^{2/3-2\gamma}\}$ while $\text{supp}(\zeta^{+})\subseteq\{r\leq 2\varepsilon^{2/3-\gamma}\}$ . The purpose of introducing $\zeta^{+}$ is that it is equal to $1$ on a neighborhood larger than the support of $\mathbb{1}^{+}$ by a factor of $\varepsilon^{\gamma}$ ; this extra buffer zone allows the exponential decay of Lemma 7.13 to take effect on $\text{supp}(\zeta^{-})$ for configurations that decay away from the support of $\mathbb{1}^{+}$ (see the proof of Corollary 10.8 in Section 10).

Lemma 9.4.

Let $\mathbb{Q}=Q_{\text{SW}}+Q_{\Phi_{1}}+Q_{A_{1}}+Q_{a.\varphi}+Q_{\mu}$ be the non-linear terms from Proposition 8.8. Then these satisfy the following bounds.

(I)

$Q_{\text{SW}}$ satisfies

	$\displaystyle\ \ \ \ \ \\|Q_{\text{SW}}(\varphi,a)\zeta^{+}\\|\leq C\varepsilon^{1/3-\gamma}\\|(\varphi,a)\\|_{H^{1}_{\varepsilon}}^{2}\hskip 42.67912pt$		$\displaystyle\\|Q^{\text{Re}}_{\text{SW}}(\varphi,a)\zeta^{-}\\|_{L^{2}}\leq C\varepsilon^{1/4}\\|(\varphi^{\text{Im}},a)\\|^{2}_{H^{1}_{\varepsilon}}$
			$\displaystyle\\|Q^{\text{Im}}_{\text{SW}}(\varphi,a)\zeta^{-}\\|_{L^{2}}\leq C\varepsilon^{1/4}\\|\varphi^{\text{Re}}\\|_{H^{1}_{\varepsilon}}\\|(\varphi^{\text{Im}},a)\\|_{H^{1}_{\varepsilon}}.$

(II)

Provided $\|\xi\|_{3/2+\underline{\gamma}}\leq 1$ , then $Q_{\Phi_{1}},Q_{A_{1}}$ satisfy

	$\displaystyle\\|Q_{\Phi_{1}}(\eta,\varphi)\mathbb{1}^{+}\\|_{L^{2}}$	$\displaystyle\leq$	$\displaystyle C\varepsilon^{1-\gamma}\\|\eta\\|_{3/2+\underline{\gamma}}\left(\\|\varphi\\|_{H^{1}_{\varepsilon}}\ +\ \varepsilon^{1/3}\\|\eta\\|_{1}+\varepsilon\\|\eta\\|_{2}\right)+C\varepsilon^{5/3-\gamma}\\|\eta\\|_{5/2+\underline{\gamma}}\\|\varphi\\|_{H^{1}_{\varepsilon}}$
	$\displaystyle\\|Q_{\Phi_{1}}(\eta,\varphi)\mathbb{1}^{-}\\|_{L^{2}}$	$\displaystyle\leq$	$\displaystyle C\left(\varepsilon\\|\eta\\|_{3/2+\underline{\gamma}}\\|\varphi\\|_{H^{1}_{\varepsilon}}\ +\ \varepsilon\\|\eta\\|_{1/2}\\|\eta\\|_{3/2+\underline{\gamma}}\right),$

and identically for $Q_{A_{1}}$ with $\|a\|_{H^{1}_{\varepsilon}}$ in place of $\|\varphi\|_{H^{1}_{\varepsilon}}$ .

(III)

Provided $\|\xi\|_{3/2+\underline{\gamma}}\leq 1$ , then

	$\displaystyle\\|Q_{a.\varphi}(\eta,a,\varphi)\\|_{L^{2}}$	$\displaystyle\leq$	$\displaystyle C\varepsilon\\|\eta\\|_{3/2+\underline{\gamma}}\\|Q_{\text{SW}}(\varphi,a)\\|_{L^{2}}$
	$\displaystyle\\|Q_{\mu}(\eta,\varphi)\\|_{L^{2}}$	$\displaystyle\leq$	$\displaystyle C\varepsilon\\|\eta\\|_{3/2+\underline{\gamma}}\left(\\|\varphi\\|_{H^{1}_{e}}+\\|Q_{\text{SW}}(\varphi,a)\\|_{L^{2}}\right).$

Proof.

(I) Configurations on $Y$ satisfy the interpolation inequality

\|u\|_{L^{4}}\leq C\|u\|_{L^{2}}^{1/4}\|u\|_{L^{1,2}}^{3/4}.

(9.12)

Since $|\zeta^{\pm}|^{2}\leq|\zeta^{\pm}|\leq 1$ , and $|d\zeta^{\pm}|\leq CR_{\varepsilon}^{-1}$ is bounded by the weight on the $L^{2}$ -terms in the $H^{1}_{\varepsilon}$ -norm, applying this shows, e.g.

$\displaystyle\\|Q_{\text{SW}}(\zeta^{+}\varphi,\zeta^{+}a)\\|_{L^{2}}$	$\displaystyle\leq$	$\displaystyle C\\|\zeta^{+}\varphi\\|_{L^{2}}^{1/4}\ \\|\zeta^{+}\varphi\\|^{3/4}_{L^{1,2}}\ \\|\zeta^{+}a\\|^{1/4}_{L^{2}}\ \\|\zeta^{+}a\\|^{3/4}_{L^{1,2}}$
	$\displaystyle\leq$	$\displaystyle C\cdot\text{max}(\zeta^{+}R_{\varepsilon}^{1/2})\cdot\\|(\varphi,a)\\|^{2}_{H^{1}_{\varepsilon}}$
	$\displaystyle\leq$	$\displaystyle C\varepsilon^{1/3-\gamma}\\|(\varphi,a)\\|^{2}_{H^{1}_{\varepsilon}}.$

because the $H^{1}_{\varepsilon}$ -norm dominates the $L^{2}$ -norm with an extra weight of $R_{\varepsilon}$ . On the support of $\zeta^{-}$ , the proof is the same but the fact that $Q_{\text{SW}}$ has at most one factor in $S^{\text{Re}}$ means the weight gives an extra factor of $\varepsilon^{1/4}|\Phi_{\tau}|^{-1/4}R_{\varepsilon}^{1/4}\leq\varepsilon^{1/4}R_{\varepsilon}^{1/8}$ for $Q_{\text{SW}}^{\text{Im}}$ on $\text{supp}(\zeta^{-})$ and $\varepsilon^{1/2}|\Phi_{\tau}|^{-1/2}<\varepsilon^{1/4}$ for $Q_{\text{SW}}^{\text{Re}}$ .

(II) Follows from the same considerations as Lemmas 9.2 and 9.1 (with $(\varphi,a)$ replacing $(\varphi_{1},a_{1})$ in 9.5), and employing Lemma 6.14 in place of Lemma 6.12, then invoking the assumption that $\|\xi\|_{3/2+\underline{\gamma}}<1$ .

(III) Using Taylor’s theorem on the terms involving $\xi$ , then pulling out a factor of $\|\xi\|_{3/2+\underline{\gamma}}=\varepsilon\|\eta\|_{3/2+\underline{\gamma}}$ as in the proof of Lemma 6.14 reduces to terms of the form in (I) or of the form $\varepsilon^{-1}|\Phi_{\tau}||\varphi|$ which are bounded by the $H^{1}_{\varepsilon}$ -norm directly. ∎

10. Contraction Subspaces

This section and the next complete the bulk of the proof of Theorem 1.6. This is done by constructing three linear parametrices, to be denoted by $P_{\xi},P^{+},P^{-}$ , one each for the three steps of the cyclic iteration described in Section 2.4. These three parametrices are combined into a (non-linear) approximate inverse $\mathbb{A}:L^{2}\to\mathbb{H}^{1}_{\varepsilon}(\mathcal{E}_{\tau}\times\mathcal{X}_{\tau})$ of $\overline{\mathbb{SW}}$ as in (8.14), which is the non-linear analogue of (2.23). Then define

T=\text{Id}-\mathbb{A}\circ\overline{\mathbb{SW}}_{\Lambda},

(10.1)

where $\overline{\mathbb{SW}}_{\Lambda}=\overline{\mathbb{SW}}-\chi^{-}\varepsilon^{-1}{\Lambda(\tau)\Phi_{\tau}}$ , so that each successive application of $T$ carries out a cycle of the alternating iteration (cf. 2.14).

Proposition 10.1.

There exist closed subspaces $\mathcal{H}\subseteq T(\mathbb{H}^{1}_{\varepsilon}(\mathcal{E}_{\tau}\times\mathcal{X}_{\tau})$ , and $\mathfrak{L}\subseteq\mathbb{L}^{2}_{\varepsilon}$ , and a closed ball $\mathcal{C}\subseteq\mathcal{H}$ containing $0$ such that the following hold for $h\in\mathcal{C}$ and $N\in\mathbb{N}$ , provided $\varepsilon_{0},\tau_{0}$ are sufficiently small.

(A)

The restriction $T:\mathcal{C}\to\mathcal{C}$ is continuous and depends smoothly on $(\varepsilon,\tau)$ .
(B)

$\|\overline{\mathbb{SW}}_{\Lambda}(T^{N}h)\|_{\mathfrak{L}}\leq\delta^{N}\|\overline{\mathbb{SW}}_{\Lambda}(h)\|_{\mathfrak{L}},$
(C)

$\|Th_{1}-Th_{2}\|_{\mathcal{H}}\leq C\sqrt{\delta}\|h_{1}-h_{2}\|_{\mathcal{H}}$

where $\delta=\varepsilon^{1/48}$ . In particular, $T$ is a contraction for $\varepsilon$ sufficiently small.

Note the proposition implicitly uses the exponential map from Section 6.1 to conflate an open subset of $T\mathbb{H}^{1}_{\varepsilon}$ with $\mathbb{H}^{1}_{\varepsilon}$ so that (10.1) makes sense. The remainder of the current section constructs the spaces $\mathcal{H},\mathfrak{L}$ , and Section 11 constructs the three parametrices to complete the proof of the proposition. Theorem 1.6 is deduced as a consequence of Proposition 10.1 in Section 12.

10.1. Three Fourier Regimes

The main challenge in the proof of Proposition 10.1 is controlling the loss of regularity of the deformation operator $\underline{T}_{\Phi_{\tau}}$ from Section (6.2). This requires careful analysis of three regimes of Fourier modes, which are dealt with differently; these are associated to three different length scales on $Y$ .

There are two different spaces of sections on $\mathcal{Z}_{\tau}$ with Fourier decompositions, and these Fourier decompositions are linked to length scales on $Y$ by two distinct mechanisms. First, the intrinsic concentration property of the obstruction (recall Proposition 5.3 and Lemma 5.4) links the length scales to the Fourier modes in $\text{\bf Ob}(\mathcal{Z}_{\tau})\simeq L^{2}(\mathcal{Z}_{\tau};\mathcal{S}_{\tau})$ . Second, by fiat, the use of mode-dependent deformations (Section 6.3) links length scales on $Y$ to the Fourier modes of a deformation $\eta\in L^{2,2}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})$ .

The three length scales $r\leq c\varepsilon^{2/3-\gamma},c\varepsilon^{2/3-\gamma}\leq r\leq c\varepsilon^{1/2}$ and $r\geq c\varepsilon^{1/2}$ of the inside, the outside, and the neck regions respectively give rise to three corresponding Fourier regimes.

Definition 10.2.

For a fixed $\gamma<<1$ , set

\displaystyle L^{\text{low}}=\varepsilon^{-(1/2+\gamma)}

\displaystyle L^{\text{med}}=\varepsilon^{-2/3}

and for $\ell\in\mathbb{Z}$ define

	$\displaystyle{\pi}^{\text{low}}(e^{i\ell t})$	$\displaystyle=$	$\displaystyle\begin{cases}e^{i\ell t}\hskip 14.22636pt\|\ell\|\leq L^{\text{low}}\\ 0\hskip 17.07182pt\ \ \|\ell\|>L^{\text{low}}\end{cases}\hskip 28.45274pt\pi^{\text{med}}(e^{i\ell t})=\begin{cases}e^{i\ell t}\hskip 14.22636ptL^{\text{low}}<\|\ell\|\leq L^{\text{med}}\\ 0\hskip 22.76228pt\|\ell\|>L^{\text{med}},\|\ell\|\leq L^{\text{low}}\end{cases}$
	$\displaystyle\pi^{\text{high}}(e^{i\ell t})$	$\displaystyle=$	$\displaystyle(\text{Id}-\pi^{\text{med}}-\pi^{\text{low}})e^{i\ell t}.$

We write e.g. $\Psi^{\text{low}}:=\pi^{\text{low}}\circ\text{ob}_{\tau}^{-1}(\Psi)$ as shorthand for the projections of an obstruction element $\Psi\in\text{Ob}(\mathcal{Z}_{\tau})$ to the corresponding Fourier regimes.

Lemma 5.4 yields the following statements for the projection operators of Definition 10.2: for $\varepsilon$ sufficiently small, suppose that $\psi_{1},\psi_{2}$ satisfy $\text{dist}(\mathcal{Z}_{\tau},\text{supp}(\psi_{i}))\geq R_{i}$ for $i=1,2$ where $R_{1}=c\varepsilon^{1/2}$ and $R_{2}=c\varepsilon^{2/3-\gamma}$ . Then for any $M\in\mathbb{N}$

	$\displaystyle\\|(1-\pi^{\text{low}})\circ\text{ob}_{\tau}^{-1}\circ\Pi_{\tau}(\psi_{1})\\|_{L^{2}(\mathcal{Z}_{\tau})}$	$\displaystyle\leq$	$\displaystyle C_{M}\varepsilon^{M}\\|\psi_{1}\\|_{L^{2}(Y)}$		(10.2)
	$\displaystyle\\|\pi^{\text{high}}\circ\text{ob}_{\tau}^{-1}\circ\Pi_{\tau}(\psi_{2})\\|_{L^{2}(\mathcal{Z}_{\tau})}$	$\displaystyle\leq$	$\displaystyle C_{M}\varepsilon^{M}\\|\psi_{2}\\|_{L^{2}(Y)}.$		(10.3)

hold for constants $C_{M}$ . Specifically, these are obtained by applying Lemma 5.4 with taking $\gamma=\gamma_{1}$ in the statement of the lemma, where $\gamma_{1}$ satisfies $-\tfrac{2}{3}-\gamma=-\tfrac{2}{3}(\tfrac{1}{1-\gamma^{\prime}})$ for $\gamma$ as in Definition 10.2.

Sections 5.2 and 6.2 described respectively two invertible elliptic operators, either of which can be used to cancel the obstruction:

	$\displaystyle\underline{T}_{\Phi_{\tau}}:L^{1/2,2}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})$	$\displaystyle\longrightarrow$	$\displaystyle L^{2}(\mathcal{Z}_{\tau};S_{\tau})$		(10.4)
	$\displaystyle\text{ob}_{\tau}^{-1}\not{D}:\ \ \ \ \ \ \ \ \ \ \ \mathcal{X}_{\tau}\ \ \ \ \ \ \ \$	$\displaystyle\longrightarrow$	$\displaystyle L^{2}(\mathcal{Z}_{\tau};S_{\tau}).$		(10.5)

The first is an isomorphism by Corollary 6.10, the second by Lemma 5.6. The pre-images of the three Fourier regimes in the codomain give rise to three corresponding regimes in each $L^{1/2,2}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})$ and $\mathcal{X}_{\tau}$ .

(10.4) is used to cancel the low and medium modes, while (10.5) is used to cancel the high modes. There is an important and delicate balance that must be struck here. The higher the Fourier mode, the more extreme the loss of regularity of (10.4) is, thus there is an upper limit on the modes for which (10.4) may be used to cancel the obstruction, else the alternating iteration will not converge. Conversely, there is a lower limit to the modes which may be solved for using (10.5). Because solutions of (10.5) grow across the neck region rather than decay, the modes solved for this way must be smaller than the dominant error by at least that growth factor. (10.2) and (10.3) provide such estimates, but only in the case of spinors with support restricted away from $\mathcal{Z}_{\tau}$ . The support of error terms coming from the cut-off functions $d\chi^{\pm}$ in the alternating method therefore yield a lower bound on the range of modes where using (10.5) is permissible ³³3This lower bound is dictated by the somewhat arbitrary choice that $d\chi^{\pm}$ are supported where $r=(\varepsilon^{1/2})$ and $r=O(\varepsilon^{2/3-\gamma})$ . Different choices, however, also change the powers of $\varepsilon$ appearing elsewhere (e.g. Theorem 7.5). Different choices result in different ranges for the three regimes in Definition 10.2, but the central issue cannot be avoided. The key point is that together, the regions enclosed by these upper and lower bounds cover the entire spectrum in $L^{2}(\mathcal{Z}_{\tau};S_{\tau})$ .

Next, we will define a combined space and operator that cancels the obstruction using (10.4) for the low and medium modes and (10.5) in the high modes. First, however, one minor alteration is required. To achieve control on higher Sobolev norms, we replace (10.4) by an auxiliary operator $\underline{T}^{\circ}_{\tau}$ which is a small perturbation of it, constructed as follows. In Fermi coordinates (Definition 3.9) and the accompanying trivialization (Lemma 3.10) on $N_{r_{0}}(\mathcal{Z}_{\tau})$ , smooth objects may be decomposed using Fourier modes in the $t$ -direction, leading to families of Fourier series smoothly parameterized by the normal coordinates $(x,y)$ . Since $\text{d}{\mathbb{D}}_{h_{0}}(\eta,0)$ is supported in $N_{r_{0}}(\mathcal{Z}_{0})$ , we may define $\underline{T}_{\tau}^{\circ}$ by restricting these Fourier modes. Write $g_{\tau}=g_{0}+h(t,x,y)$ where $g_{0}$ is the product metric in Fermi coordinates, and set

g^{\circ}=g_{0}+\pi^{\text{low}}(h)\hskip 56.9055pt\Phi_{\tau}^{\circ}:=\pi^{\text{low}}(\Phi_{\tau})

(10.6)

where $\pi^{\text{low}}$ from Definition 10.2 applied for every fixed $(x,y)\in D_{r_{0}}$ . Let $\underline{T}_{\tau}^{\circ}$ be the deformation operator defined analogously to $\underline{T}_{\Phi_{\tau}}$ (Theorem 6.5) using (10.6) in place of $g_{\tau},\Phi_{\tau}$ . By construction, $\underline{T}^{\circ}_{\tau}$ obeys the hypotheses of Corollary (6.8) with $M=\varepsilon^{-1/2}$ , since $g_{\tau},\Phi_{\tau}$ are smooth in the $t$ -direction.

Corollary 10.3.

Let $\mathfrak{t}_{\tau}^{\circ}$ be such that $\underline{T}_{\Phi_{\tau}}=\underline{T}_{\tau}^{\circ}+\mathfrak{t}^{\circ}_{\tau}$ . For any $M>0$ , there is a constant $C_{M}$ depending on $M$ such that

\|\mathfrak{t}^{\circ}_{\tau}(\eta)\|_{2}\leq C_{M}\varepsilon^{M}\|\eta\|_{1/2,2}.

In particular for $\varepsilon_{0}$ sufficiently small, $\underline{T}_{\tau}^{\circ}$ is invertible, and the estimates (6.16) hold uniformly in $\tau$ . In this case, if $\underline{T}_{\tau}^{\circ}(\eta^{\text{low}})=\Psi^{\text{low}}$ and $\underline{T}_{\tau}^{\circ}(\eta^{\text{med}})=\Psi^{\text{med}}$ , then

\|\eta^{\text{low}}\|_{m+1/2,2}\leq C_{m}(L^{\text{low}})^{-m}\|\Psi^{\text{low}}\|_{L^{2}}\hskip 56.9055pt\|\eta^{\text{med}}\|_{m+1/2,2}\leq C_{m}(L^{\text{med}})^{-m}\|\Psi^{\text{med}}\|_{L^{2}}

(10.7)

uniformly in $\tau$ for any $m>0$ . ∎

We use $\eta^{\text{low}},u^{\text{low}}$ to denote deformations and spinors whose images under (10.4) and (10.5) fall in the corresponding regimes in $L^{2}(\mathcal{Z}_{\tau};S_{\tau})$ , and likewise for the medium and high regimes. Note that $\eta^{\text{low}}$ need not precisely have modes restricted to the low regime in $L^{2}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})$ since $\underline{T}_{\tau}^{\circ}$ does not preserve support in Fourier space; for the purposes of estimates, however, (10.7) says this is effectively true.

Definition 10.4.

Let $\mathfrak{W}_{\varepsilon,\tau}\subseteq L^{1/2,2}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})\oplus\mathcal{X}_{\tau}$ be the subspace defined as the image of the following composition.

Equip $\mathfrak{W}_{\varepsilon,\tau}$ with the norm

\|(\eta^{\text{low}},\eta^{\text{med}},u)\|_{\mathfrak{W}}:=\left(\|\eta^{\text{low}}\|_{1/2,2}^{2}\ +\ \varepsilon^{-1/3}\|\eta^{\text{med}}\|_{1/2,2}^{2}\ +\ \varepsilon^{-4/3}\|u\|_{H^{1}_{e}}\right)^{1/2}.

(10.8)

By construction, $(\underline{T}_{\tau}^{\circ},\text{ob}_{\tau}^{-1}\not{D}_{A_{\tau}}):\mathfrak{W}_{\varepsilon,\tau}\to L^{2}(\mathcal{Z}_{\tau};S_{\tau})$ is an isomorphism. The first two components give rise to a corresponding set of renormalized deformations $\xi=\varepsilon\eta$ .

Here, we may alter $\pi^{\text{med}}$ so that the projection smoothly interpolates between $0$ and $1$ on the mode with Fourier index $\left\lceil L^{\text{med}}\right\rceil$ . The same applies to the norm between the low and medium modes. It may therefore be arranged that the family $\mathfrak{W}_{\varepsilon,\tau}$ form a smooth vector bundle over pairs $(\varepsilon,\tau)$ .

10.2. Contraction Subspaces

This subsection defines the subspaces $\mathcal{H},\mathfrak{L}$ whose existence is asserted in Proposition 10.1. The definitions have two tasks: (1) finding a closed subspace $\mathcal{H}\subseteq T\mathbb{H}^{1}_{\varepsilon}(\mathcal{E}_{\tau}\times\mathcal{X}_{\tau})$ on which the linearization has index 0 (cf. Proposition 2.4), and (2) defining correctly weighted norms.

The upcoming weighted norms are chosen in hindsight and are necessary for the map $T$ to satisfy Items (B) and (C) of Proposition 10.1. The proof that a single application of $T$ reduces the error as in (B) relies the error having a certain effective support property (recall Definition 4.6): the majority of its effective support must be clustered where $d\chi^{+}\neq 0$ . The weights in the upcoming norms impose this property (without Difficulty II from Section 2.5, the weighted norms could simply weight the integral over support of $d\chi^{+}$ differently — the use of effective support should be thought of as imposing a slight generalization of this).

Recall that $T\mathbb{H}^{1}_{\varepsilon}(\mathcal{E}_{\tau}\times\mathcal{X}_{\tau})$ is isomorphic via $\underline{\Upsilon}$ (Lemma 8.2) to a trivial bundle with fiber $H^{1,+}_{\varepsilon}\oplus H^{1,-}_{\varepsilon}\oplus L^{2,2}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})\oplus\mathcal{X}_{\tau}$ . Notice that $\mathfrak{W}_{\varepsilon,\tau}\subseteq T\mathbb{H}^{1}_{\varepsilon}(\mathcal{E}_{\tau}\times\mathcal{X}_{\tau})$ despite the lower regularity in (10.8), since only a finite-dimensional subspace of deformations is included.

Definition 10.5.

Set

	$\displaystyle\mathcal{H}^{+}$	$\displaystyle:=$	$\displaystyle\left\{\mathcal{L}_{(\Phi_{1},A_{1})}^{-1}(g\mathbb{1}^{+})\ \ \Big{\|}\ \ g\in L^{2}(Y^{+})\right\}\subseteq H^{1,+}_{\varepsilon}$
	$\displaystyle\mathcal{H}^{-}$	$\displaystyle:=$	$\displaystyle\left\{\mathcal{L}_{(\Phi_{\tau},A_{\tau})}^{-1}((1-\Pi_{\tau})g\mathbb{1}^{-})\ \ \Big{\|}\ \ g\in L^{2}(Y^{-})\right\}\subseteq H^{1,-}_{\varepsilon}$

where $\mathcal{H}^{-}$ uses the solution in $H^{1,-}_{\varepsilon}$ that is $L^{2}$ -orthgonal to $\Phi_{\tau}$ on $Y^{-}$ . Equip these with the norms

	$\displaystyle\\|(\varphi,a)\\|_{\mathcal{H}^{+}}$	$\displaystyle=$	$\displaystyle\varepsilon^{-1/12-\gamma}\\|\mathcal{L}_{(\Phi_{1},A_{1})}(\varphi,a)\mathbb{1}^{+}\\|_{L^{2}}$		(10.9)
	$\displaystyle\\|(\psi,b)\\|_{\mathcal{H}^{-}}$	$\displaystyle=$	$\displaystyle\left(\\|\psi^{\text{Re}}\\|^{2}_{rH^{1}_{e}}\ +\ \varepsilon^{\nu}\\|\psi^{\text{Re}}\mathbb{1}^{-}\\|^{2}_{rH^{1}_{-\nu}}\ +\ \varepsilon^{-1/2}\\|(0,\psi^{\text{Im}},b)\\|_{H^{1,-}_{\varepsilon}}\right)^{1/2}.$		(10.10)

where $\nu=\nu^{-}=\tfrac{1}{2}-\gamma^{-}$ .

Definition 10.6.

Define

\displaystyle\mathcal{H}_{\varepsilon,\tau}

\displaystyle:=

\displaystyle\mathcal{H}^{+}\oplus\mathcal{H}^{-}\oplus\mathfrak{W}_{\varepsilon,\tau}\oplus\mathbb{R}\hskip 51.21504pt\mathfrak{L}_{\varepsilon,\tau}:=L^{2}(Y)

and equip these with the norms

$\displaystyle\\|(h_{1},h_{2},\eta,\mu)\\|_{\mathcal{H}}$	$\displaystyle:=$	$\displaystyle\left(\\|h_{1}\\|_{\mathcal{H}^{+}}^{2}\ +\ \\|h_{2}\\|_{\mathcal{H}^{-}}^{2}\ +\ \\|\eta\\|_{\mathfrak{W}}^{2}\ +\ \varepsilon^{-2}\|\mu\|^{2}\right)^{1/2}\vspace{.5cm}$	(10.11)
$\displaystyle\vskip 6.0pt plus 2.0pt minus 2.0pt\\|\mathfrak{e}\\|_{\mathfrak{L}}$	$\displaystyle:=$	$\displaystyle\Big{(}\varepsilon^{-2/12-2\gamma}\\|\mathfrak{e}\mathbb{1}^{+}\\|^{2}_{L^{2}}\ +\ \\|\mathfrak{e}^{\text{Re}}\mathbb{1}^{-}\\|^{2}_{L^{2}}\ +\ \varepsilon^{\nu}\\|\mathfrak{e}^{\text{Re}}\mathbb{1}^{-}\\|^{2}_{L^{2}_{-\nu}}\vskip 3.0pt plus 1.0pt minus 1.0pt$	(10.13)
		$\displaystyle\ \ \ +\ \ \varepsilon^{-1/3}\\|\mathfrak{e}^{\text{Im}}\mathbb{1}^{-}\\|^{2}_{L^{2}}\ \ +\ \ \varepsilon^{-1/3}\\|\pi^{\text{med}}\Pi_{\tau}(\mathfrak{e})\\|^{2}_{L^{2}}\Big{)}^{1/2}$	(10.13)

where $\mathfrak{e}=(\mathfrak{e}^{\text{Re}},\mathfrak{e}^{\text{Im}})\in L^{2}(S^{\text{Re}})\oplus L^{2}(S^{\text{Im}}\oplus\Omega)$ . These families of spaces form smooth Banach vector bundles over pairs $(\varepsilon,\tau)$ for $\varepsilon,\tau$ sufficiently small.

Note that the weighted terms dictate that the $\nu$ -weighted term in the norms is larger than the unweighted term by at most a factor of $\varepsilon^{-\nu/2}$ , despite the fact that $\sup r^{-\nu}=\varepsilon^{-\nu(2/3-\gamma)}$ on $\text{supp}(\mathbb{1}^{-})$ . This shows that configurations in $\mathfrak{L}$ have $\varepsilon^{1/2}$ -effective support. Note also that Theorem 7.5 implies that the pullback (10.9) is indeed a norm.

The following lemma shows the linearized Seiberg–Witten equations are bounded on the above spaces with operator norm independent of $\varepsilon$ (up to a factor of $\gamma)$ . This assertion is non-trivial: it means that the new norms control the loss of regularity of the deformation operator. Indeed, if one simply uses the $L^{1/2,2},H^{1,\pm}_{\varepsilon}$ and $L^{2}$ -norms, the operator norm is only bounded by $\varepsilon^{-4/3}$ .

Lemma 10.7.

There is a constant $C$ independent of $\varepsilon,\tau$ such that the linearization $\text{d}\overline{\mathbb{SW}}_{h_{1}}:\mathcal{H}\to\mathfrak{L}$ satisfies

\|\text{d}\overline{\mathbb{SW}}_{h_{1}}(h)\|_{\mathfrak{L}}\leq C\varepsilon^{-\gamma}\|h\|_{\mathcal{H}}

(10.14)

for some $\gamma<<1$ .

Proof.

By Proposition 8.7, the linearization on $h=(\xi,\varphi,a,\psi,b,\mu,u)$ where $\xi=\varepsilon\eta$ may be written in the trivialization of Lemma 8.2 as

	$\displaystyle\text{d}\overline{\mathbb{SW}}_{h_{1}}(h)$	$\displaystyle=$	$\displaystyle\tfrac{1}{\varepsilon}\underline{\mathcal{B}}_{\Phi_{1}}(\xi)\ +\ \mathcal{L}_{(\Phi_{1},A_{1})}(\mathfrak{p})\ +\ \mu\chi^{-}\frac{\Phi_{\tau}}{\varepsilon}$		(10.15)
		$\displaystyle=$	$\displaystyle\text{d}\not{\mathbb{D}}_{h_{0}}(\eta,0)+\Xi(\eta)+\mathcal{L}_{(\Phi_{1},A_{1})}(\mathfrak{p})\ +\ \mu\chi^{-}\frac{\Phi_{\tau}}{\varepsilon}.$		(10.16)

where $\Xi$ is as in (9.2), and $\mathfrak{p}=\chi^{+}(\varphi,a)+\chi^{-}(\psi+u,b)$ where $(\varphi,a)\in H^{1,+}_{\varepsilon}$ , $(\psi,b)\in H^{1,-}_{\varepsilon}$ and $u\in\mathcal{X}_{\tau}$ . The proof now consists of four steps.

Step 1 ( $(\varphi,a)$ terms): Abbreviating $\mathcal{L}_{(\Phi_{1},A_{1})}=\mathcal{L}$ , and using the definition of $\mathcal{H}^{+}$ ,

$\displaystyle\\|\mathcal{L}(\chi^{+}(\varphi,a))\\|_{\mathfrak{L}}$	$\displaystyle=$	$\displaystyle\varepsilon^{-1/12-\gamma}\\|\mathcal{L}(\varphi,a)\mathbb{1}^{+}\\|_{L^{2}}+\\|d\chi^{+}\varphi^{\text{Re}}\\|_{L^{2}}+\ \varepsilon^{\nu}\\|d\chi^{+}\varphi^{\text{Re}}\\|_{L^{2}_{-\nu}}$
		$\displaystyle\ +\ \varepsilon^{-1/3}\\|d\chi^{+}(\varphi^{\text{Im}},a)\\|_{L^{2}}\ +\ \varepsilon^{-1/3}\\|\pi^{\text{med}}\Pi_{\tau}(d\chi^{+}\varphi^{\text{Re}})\\|_{L^{2}}\vskip 6.0pt plus 2.0pt minus 2.0pt$
	$\displaystyle\leq$	$\displaystyle\\|(\varphi,a)\\|_{\mathcal{H}^{+}}\ +\varepsilon^{1/24-\gamma}\\|\mathcal{L}(\varphi,a)\mathbb{1}^{+}\\|_{L^{2}}\ +\ \\|d\chi^{+}\varphi^{\text{Re}}\\|_{L^{2}}$
		$\displaystyle O(\varepsilon^{-3}e^{\varepsilon^{-\gamma}})\\|(\varphi,a)\\|_{H^{1,+}}\ +\ O(\varepsilon^{M})\\|(\varphi,a)\\|_{L^{2}}$
	$\displaystyle\leq$	$\displaystyle\\|(\varphi,a)\\|_{\mathcal{H}^{+}}.$

where the five terms are bounded respectively as follows. The first by the definition (10.9) of the $\mathcal{H}^{+}$ -norm, the second is reduced to the case of the first by Lemma 7.12, and the third is identical to the second since $d\chi^{+}$ is supported where $r=O(\varepsilon^{1/2})$ . The fourth is exponentially small by Lemma (7.13) and Theorem 7.5 because the latter shows $\|-\|_{L^{2}}\leq\|-\|_{H^{1,+}_{\varepsilon}}\leq\|-\|_{\mathcal{H}^{+}}$ , and the fifth is polynomially small by (10.2).

Step 2 ( $(\psi,b)$ terms): Write $\mathcal{L}_{(\Phi_{1},A_{1})}=\mathcal{L}_{(\Phi_{0},A_{0})}+K_{1}$ . The boundedness of the $\mathcal{L}_{(\Phi_{0},A_{0})}(\chi^{-}(\psi+u,b))$ terms now follows similarly to the above, using the decay result of Lemma 4.7 in place of Lemma 7.12, and the boundedness of (7.11) and (4.11) for both $\nu=0,\nu^{-}$ . In this, the bound $\|\psi^{\text{Re}}+u\|_{rH^{1}_{\varepsilon}}\leq 2\|h\|_{\mathcal{H}}$ is used, which comes from the $\varepsilon^{-4/3}$ -weight in (10.8),,

The only minor difference from Step 1 is for the $\pi^{\text{med}}$ -term in the $\mathfrak{L}$ -norm: this is zero on $(\psi,b)$ by definition, and on $u$ is dominated by the factor of $\varepsilon^{-4/3}$ in the $\|u\|_{\mathfrak{W}}$ norm. Finally, $K_{1}$ is exponentially small on $\text{supp}(\mathbb{1}^{+})$ by Lemma 7.13 (applied with $K_{\varepsilon}=\{r\geq\varepsilon^{2/3-\gamma/2}\}$ ).

Step 3 ( $\mu$ term): The term $\chi^{-}\varepsilon^{-1}\mu\Phi_{\xi,\tau}$ is obviously bounded by $\|\mu\|_{\mathcal{H}}$ because the $\varepsilon^{-2}$ weight cancels the $\varepsilon$ in the denominator, and $\Phi_{\tau}\in L^{2}\cap L^{2}_{-\nu}$ . The $\pi^{\text{med}}$ -term is smaller, because $\|\pi^{\text{med}}(\chi^{-}\Phi_{\tau})\|_{L^{2}}=\|\pi^{\text{med}}((1-\chi^{-})\Phi_{\tau})\|_{L^{2}}\leq C\varepsilon$ using Lemma 4.5.

Step 4 (deformation terms): Proceeding now to the terms involving $\eta=\varepsilon^{-1}\xi$ , Corollary 10.3 provides bounds on the higher Sobolev norms of $\eta\in\mathfrak{W}$ . Using these, Proposition 9.3 applied with $\nu=0,\nu^{-}$ and Proposition 9.1 show that

$\displaystyle\\|\text{d}\not{\mathbb{D}}_{h_{0}}(\eta,0)\mathbb{1}^{-}\\|_{L^{2}}\$	$\displaystyle\leq$	$\displaystyle C\\|\eta\\|_{1/2}$	(10.17)
$\displaystyle\\|\text{d}\not{\mathbb{D}}_{h_{0}}(\eta,0)\mathbb{1}^{-}\\|_{L^{2}_{-\nu}}$	$\displaystyle\leq$	$\displaystyle C(\varepsilon^{-\nu(1/2+\gamma)}\\|\eta^{\text{low}}\\|_{1/2}+C\varepsilon^{-2\nu/3}\\|\eta^{\text{med}}\\|_{1/2})\leq C\varepsilon^{-\nu/2-\gamma}\\|\eta\\|_{\mathfrak{W}}$	(10.18)
$\displaystyle\\|\Xi^{-}(\eta)\\|_{L^{2}}$	$\displaystyle\leq$	$\displaystyle C\varepsilon^{11/12-\gamma}(\varepsilon^{-1/2}\\|\eta^{\text{low}}\\|_{1/2}+\varepsilon^{-2/3}\\|\eta^{\text{med}}\\|_{1/2})\leq C\varepsilon^{5/12-\gamma}\\|\eta\\|_{\mathfrak{W}}.$	(10.19)

Moreover, since $r^{-\nu}\leq\varepsilon^{-2\nu/3}\leq\varepsilon^{-\nu/2}\varepsilon^{-\nu/6}$ on $\text{supp}(\mathbb{1}^{-})$ , (10.19) shows that

\varepsilon^{\nu/2}\|\Xi^{-}(\eta)\|_{L^{2}_{-\nu}}\leq C\varepsilon^{4/12-\gamma}\|\eta\|_{\mathfrak{W}}

(10.20)

as well. Next, Proposition 9.2 (and the bounds of Corollary 10.3 again) shows that

$\displaystyle\\|\text{d}\not{\mathbb{D}}_{h_{0}}(\eta,0)\mathbb{1}^{+}+\Xi^{+}(\eta)\\|_{L^{2}}$	$\displaystyle\leq$	$\displaystyle C\varepsilon^{-\gamma}\Big{(}\varepsilon^{1/3}\\|\eta\\|_{1}\ +\ \varepsilon^{11/12}\\|\eta\\|_{3/2+\underline{\gamma}}\ +\ \ \varepsilon\\|\eta\\|_{2}\ +\ \varepsilon^{19/12}\\|\eta\\|_{5/2+\underline{\gamma}}\Big{)}$	(10.21)
	$\displaystyle\leq$	$\displaystyle C\varepsilon^{1/12-\gamma}\\|\eta^{\text{low}}\\|_{1/2}+C\\|\eta^{\text{med}}\\|_{1/2}$	(10.21)
	$\displaystyle\leq$	$\displaystyle C\varepsilon^{1/12-\gamma}\\|\eta\\|_{\mathfrak{W}}.$	(10.22)

Together, (10.17)–(10.22) show that all but the $\pi^{\text{med}}$ -term in the $\mathfrak{L}$ -norm are bounded for $\text{d}\not{\mathbb{D}}_{h_{0}}+\Xi$ .

For this final term, one has

\|\pi^{\text{med}}\Pi_{\tau}\text{d}_{h_{0}}\not{\mathbb{D}}(\eta,0)\mathbb{1}^{-}\|_{L^{2}}\ \leq\ \|\pi^{\text{med}}\Pi_{\tau}\text{d}_{h_{0}}\not{\mathbb{D}}(\eta,0)\|_{L^{2}}\ +\ \|\pi^{\text{med}}\Pi_{\tau}\text{d}_{h_{0}}\not{\mathbb{D}}(\eta,0)\mathbb{1}^{+}\|_{L^{2}}

(10.23)

by the triangle inequality, and we bound each of these terms separately.

For the first term of (10.23),

	$\displaystyle\varepsilon^{-1/6}\\|\pi^{\text{med}}\Pi_{\tau}\text{d}_{h_{0}}\not{\mathbb{D}}(\eta,0)\\|_{L^{2}}$	$\displaystyle\leq$	$\displaystyle\varepsilon^{-1/6}\\|\pi^{\text{med}}\underline{T}_{\tau}^{\circ}(\eta)\\|_{L^{2}}\ +\ \varepsilon^{-1/6}\\|\mathfrak{t}_{\tau}^{\circ}(\eta)\\|_{L^{2}}$
		$\displaystyle\leq$	$\displaystyle C\\|\eta^{\text{med}}\\|_{\mathfrak{W}}+O(\varepsilon^{M})\\|\eta\\|_{\mathfrak{W}}$

by Definition (10.4) and Lemma 10.3.

For the second term of (10.23), split $\eta=\eta^{\text{low}}+\eta^{\text{med}}$ ; the portion involving $\eta^{\text{med}}$ is obviously bounded because the same $\varepsilon^{-1/6}$ weight appears in (10.8) and in the $\mathfrak{L}$ -norm. Thus it suffices to consider the $\eta^{\text{low}}$ terms. Write $\text{d}\not{\mathbb{D}}_{h_{0}}=\text{d}^{\circ}\not{\mathbb{D}}+\mathfrak{d}^{\circ}$ where the latter are formed using the metrics of Lemma 10.3. Just as in that lemma, $\|\mathfrak{d}\|\leq C\varepsilon^{M}$ and so the terms involving the latter are negligible. Recalling Proposition 5.3 now, $\pi^{\text{med}}$ is calculated by the sequence of inner-products $\langle\Psi_{\ell},\text{d}^{\circ}\not{\mathbb{D}}(\eta^{\text{low}})\mathbb{1}^{+}\rangle_{{\mathbb{C}}}$ for $L^{\text{low}}<|\ell|\leq L^{\text{med}}$ . The second and third bullet points of Proposition 5.3 show that the $\zeta_{\ell},\xi_{\ell}$ terms of $\Psi_{\ell}$ are smaller by a factor of $|\ell|^{-1}>\varepsilon^{-1/2}$ , which dominates the $\varepsilon^{-1/6}$ weight. For the leading order terms $\chi\Psi_{\ell}^{\circ}$ , the restriction of Fourier modes on $\text{d}^{\circ}\not{\mathbb{D}}$ and Cauchy-Schwartz imply

$\displaystyle\langle\chi\Psi_{\ell}^{\circ},\text{d}^{\circ}\not{\mathbb{D}}(\eta^{\text{low}})\mathbb{1}^{+}\rangle_{{\mathbb{C}}}$	$\displaystyle=$	$\displaystyle 0\hskip 187.78836pt\|\ell\|\geq 2L^{\text{low}}$
$\displaystyle\langle\chi\Psi_{\ell}^{\circ},\text{d}^{\circ}\not{\mathbb{D}}(\eta^{\text{low}})\mathbb{1}^{+}\rangle_{{\mathbb{C}}}$	$\displaystyle\leq$	$\displaystyle C\\|\Psi_{\ell}^{\circ}\\|_{L^{2}(\mathbb{1}^{+})}\\|\text{d}^{\circ}\not{\mathbb{D}}(\eta^{\text{low}})\\|_{L^{2}(\mathbb{1}^{+})}\hskip 28.45274ptL^{\text{low}}\leq\|\ell\|\leq 2L^{\text{low}}$
	$\displaystyle\leq$	$\displaystyle C\varepsilon^{1/6-2\gamma}\varepsilon^{1/12-\gamma}\\|\eta^{\text{low}}\\|_{1/2}$

where the final line directly integrates (5.1) over $\text{supp}(\mathbb{1}^{+})$ , and invokes the second bound on the second term of (9.9) in Proposition 9.2, which applies equally well to $\text{d}^{\circ}\not{\mathbb{D}}$ . Returning to (10.23), the above combine to show

\|\pi^{\text{med}}\Pi_{\tau}\text{d}\not{\mathbb{D}}_{h_{0}}(\eta,0)\mathbb{1}^{-}\|_{L^{2}}\leq C\|\eta\|_{\mathfrak{W}}

as desired. ∎

Lemma 9.4 may also be used to bound the nonlinear terms in terms of the $\mathcal{H},\mathfrak{L}$ -norms.

Corollary 10.8.

Suppose that $h_{N}\in\mathcal{H}$ satisfies $\|h_{N}\|_{\mathcal{H}}\leq C\varepsilon^{-1/20}$ . Then there is a $C>0$ such that

(A)

$\|\mathbb{Q}(h)\|_{\mathfrak{L}}\leq C\varepsilon^{2/12-\gamma}\|h\|_{\mathcal{H}}^{2}$ ,
(B)

$\|\text{d}\mathbb{Q}_{h_{N}}(h)\|_{\mathfrak{L}}\leq C\varepsilon^{1/12-\gamma}\|h\|_{\mathcal{H}}$
(C)

If $\|h_{1}\|_{\mathcal{H}},\|h_{2}\|_{\mathcal{H}}\leq C\varepsilon^{-1/20}$ , then $\|\mathbb{Q}(h_{1})-\mathbb{Q}(h_{2})\|_{\mathfrak{L}}\leq C\varepsilon^{1/12-\gamma}\|h_{1}-h_{2}\|_{\mathcal{H}}$

hold uniformly in $\varepsilon,\tau$ .

Proof.

Let $h=(\xi,\varphi,a,\psi,b,\mu,u)$ . Using that $\|\psi^{\text{Re}}+u\|_{rH^{1}}\leq 2\|h\|_{\mathcal{H}}$ as in the proof of the previous lemma, it suffices to prove the corollary with $\psi$ tacitly standing in for $\psi+u$ .

(A) Let $\mathcal{Q}=Q_{\Phi}+Q_{A}+Q_{a.\varphi}+Q_{\mu}$ be the latter four nonlinear terms in Proposition (8.8). Definition (10.8) means $\varepsilon\|\eta\|_{3/2+\underline{\gamma}}\leq C\varepsilon^{1/2}\|\eta\|_{\mathfrak{W}}$ , and Definition 10.5 and Theorem 7.5 imply $\|(\varphi,a)\|_{H^{1}_{\varepsilon}}\leq\|(\varphi,a)\|_{\mathcal{H}}$ . That the same holds for $(\psi,b)$ is immediate from Definition 10.5. Item (II) and (III) of Lemma 9.4 therefore imply

\|\mathcal{Q}(h)\|_{L^{2}}\leq C\varepsilon^{1/2-\gamma}\|h\|_{\mathcal{H}}^{2}.

and because the weights in the $\mathfrak{L}$ -norm are larger than the $L^{2}$ -norm by at most $\varepsilon^{-1/6}$ , it follows that

\|\mathcal{Q}(h)\|_{\mathfrak{L}}\leq C\varepsilon^{3/12-\gamma}\|h\|_{\mathcal{H}}^{2}.

Proceeding now to $Q_{\text{SW}}$ , there are four sub-terms coming from $(\chi^{+})^{2}Q_{\text{SW}}(\varphi,a)$ , $(\chi^{-})^{2}Q_{\text{SW}}(\psi,b)$ and the cross-terms. We bound two pieces of each coming from the partition of unity $\zeta^{\pm}$ in Proposition 9.4. By Item (I) of Proposition 9.4 and the above,

\|Q_{\text{SW}}(\varphi,a)\zeta^{+}\|_{\mathfrak{L}}\leq\varepsilon^{-1/6}\|\zeta^{+}Q_{\text{SW}}(\varphi,a)\zeta^{+}\|_{L^{2}}\leq C\varepsilon^{-1/6}\varepsilon^{4/12-\gamma}\|(\varphi,a)\|_{H^{1,+}_{\varepsilon}}^{2}\leq C\varepsilon^{2/12-\gamma}\|h\|_{\mathcal{H}}^{2}

and likewise for the other three sub-terms where $\zeta^{+}>0$ .

Where $\zeta^{-}>0$ , the $S^{\text{Re}}\oplus\Omega$ -components of $(\varphi,a)$ are exponentially small by Lemma 7.13 applied with compact sets $K_{\varepsilon}$ whose boundary lies halfway between $\text{supp}(\mathbb{1}^{+})$ and $\text{supp}(d\zeta^{+})$ . In this same region, the $S^{\text{Im}}\oplus\Omega$ -components of $(\psi,b)$ are smaller by a factor of $\varepsilon^{1/6}$ by (10.10). Thus in this region, the right colum of Item (I) of Lemma 9.4 provdies the necessary bounds (in fact with $\varepsilon^{3/12-\gamma})$ .

(B) The derivative $\text{d}_{h_{N}}\mathbb{Q}(h)$ is given by the five terms of Proposition 8.8 viewed as multilinear functions of the arguments, with precisely one argument being chosen from the components of $h$ . The proof of the bound in (A) applies equally well in the bilinear setting to show that

\|\text{d}\mathbb{Q}_{h_{N}}(h)\|_{\mathfrak{L}}\leq C\varepsilon^{2/12-\gamma}\|h_{N}\|_{\mathcal{H}}\|h\|_{\mathcal{H}}

and the assertion follows. More specifically, the requirement that $\|h_{N}\|\leq C\varepsilon^{-1/20}$ , means $\varepsilon\|\eta_{N}\|_{3/2+\underline{\gamma}}\leq C$ , hence the assumptions of Items (II) and (III) in Lemma 9.4 are satisfied, and the terms of $\text{d}\mathbb{Q}_{h_{N}}$ multi-linear in $h_{N}$ can be bounded just as in that lemma.

11. The Alternating Iteration

This section proves Proposition 10.1 by carrying out the cyclic iteration outlined in Section 2.4. This is done by constructing three parametrices $P_{\xi},P^{+},P^{-}$ , one corresponding to each of the three stages of the iteration (2.17).

11.1. The Deformation Step

This subsection constructs the deformation parametrix $P_{\xi}$ , and establishes the first of the three induction steps in the cyclic iteration (2.17). This step (which corresponds to the horizontal arrow in the diagram below (2.17)) shows the singular set $\mathcal{Z}_{\xi,\tau}$ can be adjusted to effectively cancel the obstruction components of an error term, without the error term growing much larger (in particular, this requires controlling the key term discussed in Section 2.5.1).

The following proposition is applied to the error terms inductively, beginning with the error $\mathfrak{e}_{1}$ of the initial approximate solutions $(\Phi_{1},A_{1})$ in Theorem 7.5. For the remainder of Section 11, we fix, once and for all, a choice of $(\varepsilon,\tau)\in(0,\varepsilon_{0})\times(-\tau_{0},\tau_{0})$ and omit this dependence from the subscripts in the notation where no confusion will arise.

Let $\Pi^{\perp}:L^{2}\to\text{Ob}^{\perp}(\mathcal{Z}_{\tau})$ be the $L^{2}$ -orthogonal projection, so that $\Pi_{\tau}=(\Pi^{\perp},\pi_{\tau})$ in the orthogonal decomposition in Definition 5.1 where $\pi_{\tau}$ is the $L^{2}$ -orthogonal projection to the span of $\Phi_{\tau}$ . Define the deformation parametrix

P_{\xi}:\mathfrak{L}\longrightarrow\mathfrak{W}\hskip 56.9055ptP_{\xi}:=\left(\underline{T}_{\tau}^{\circ},\not{D}_{A_{\tau}}\right)^{-1}\circ\Pi^{\perp}\circ\mathbb{1}^{-}

(11.1)

where $(\underline{T}_{\tau}^{\circ},\not{D}_{A_{\tau}})$ is the map from Definition 10.4, with the map ob now kept implicit in the notation. In the following proposition, $\overline{\mathbb{SW}}_{\Lambda}$ is as in (11.30).

Proposition 11.1.

$P_{\xi}$ is a linear operator uniformly bounded in $\varepsilon,\tau$ and satisfies the following property. If for $N\in\mathbb{N}$ , $h_{N}\in\mathcal{H}$ is a configuration with

(I)

$\|h_{N}\|_{\mathcal{H}}\leq C\varepsilon^{-\gamma}\|\mathfrak{e}_{1}\|_{\mathfrak{L}}$
(II)

$\overline{\mathbb{SW}}_{\Lambda}(h_{N})=\mathfrak{e}_{N}$ (resp. $\text{d}\overline{\mathbb{SW}}_{h_{1}}(h_{N})=\mathfrak{e}_{N}$ ) where $\|\mathfrak{e}_{N}\|_{\mathfrak{L}}\leq C\delta^{N-1}\|\mathfrak{e}_{1}\|_{\mathfrak{L}}$ ,

then the updated configuration

h_{N}^{\prime}=(\text{Id}-P_{\xi}\circ\overline{\mathbb{SW}}_{\Lambda})h_{N}

(11.2)

satisfies

\overline{\mathbb{SW}}_{\Lambda}(h^{\prime}_{N})=(1-\Pi_{\tau})\mathfrak{e}_{N}^{\prime}\ +\ \mathfrak{g}^{\prime}_{N}\ +\ \lambda\varepsilon^{-1}\Phi_{\tau}\ +\ \mathfrak{e}_{N+1}

(resp. the same for $\text{d}\overline{\mathbb{SW}}_{h_{1}}$ ), where $\mathfrak{e}_{N}^{\prime},\mathfrak{e}_{N+1}\in\mathfrak{L}$ , and $\lambda\in\mathbb{R}$ obey

(1)

$\|\mathfrak{e}_{N}^{\prime}\|_{\mathfrak{L}}\leq C\varepsilon^{-\gamma}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$ , and $\Pi(\mathfrak{e}^{\prime}_{N})=0$ .
(2)

$\|\mathfrak{g}_{N}^{\prime}\|_{\mathfrak{L}}\leq C\varepsilon^{-\gamma}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$ and $\mathfrak{g}_{N}^{\prime}=\mathfrak{g}_{N}^{\prime}\mathbb{1}^{+}$ .
(3)

$\|\mathfrak{e}_{N+1}\|_{\mathfrak{L}}\leq C\varepsilon^{1/48}\delta\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$
(4)

$|\lambda|\leq C\varepsilon^{1-\gamma}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$ .

Moreover, $h_{N}^{\prime}$ continues to satisfy (I).

Proof.

The error term may be written $\mathfrak{e}_{N}=e_{N}+f_{N}+g_{N}+\Psi_{N}$ where

	$\displaystyle g_{N}$	$\displaystyle:=$	$\displaystyle\mathfrak{e}_{N}\mathbb{1}^{+}\hskip 105.2751pt\ f_{N}:=\mathfrak{e}_{N}^{\text{Im}}\mathbb{1}^{-}+\pi^{\text{med}}\Pi(\mathfrak{e}^{\text{Re}}_{N}\mathbb{1}^{-})$
	$\displaystyle\Psi_{N}$	$\displaystyle:=$	$\displaystyle\pi^{\text{high}}\Pi(\mathfrak{e}^{\text{Re}}_{N}\mathbb{1}^{-})\hskip 71.13188pte_{N}:=(1-\Pi)\mathfrak{e}_{N}^{\text{Re}}\mathbb{1}^{-}+\pi^{\text{low}}\Pi(\mathfrak{e}_{N}^{\text{Re}}\mathbb{1}^{-})+\pi_{\tau}(\mathfrak{e}_{N}^{\text{Re}}\mathbb{1}^{-}).$

The definition of the $\mathfrak{L}$ -norm in (10.5) implies that $\|g_{N}\|_{L^{2}}\leq C\varepsilon^{1/12-\gamma}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$ , $\|f_{N}\|_{L^{2}}\leq C\varepsilon^{1/6}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$ and $\|\Pi(e_{N}+f_{N}+\Psi_{N})\|_{L^{2}_{-\nu}}\leq C\varepsilon^{-\nu/2}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$ . Moreover, (10.3) implies $\|\Psi_{N}\|_{L^{2}}\leq C\varepsilon^{10}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$ .

Define

(\eta,u):=P_{\xi}(\Pi^{\perp}(e_{N}+f_{N}+\Psi_{N}))

so that

$\displaystyle\underline{T}^{\circ}(\eta^{\text{low}})\$	$\displaystyle=$	$\displaystyle\Pi^{\perp}(e_{N})\hskip 2.84544pt\ =\ \pi^{\text{low}}\Pi^{\perp}(\mathfrak{e}^{\text{Re}}_{N}\mathbb{1}^{-})$	(11.3)
$\displaystyle\underline{T}^{\circ}(\eta^{\text{med}})$	$\displaystyle=$	$\displaystyle\Pi^{\perp}(f_{N})\hskip 2.84544pt\ =\ \pi^{\text{med}}\Pi^{\perp}(\mathfrak{e}^{\text{Re}}_{N}\mathbb{1}^{-})$	(11.4)
$\displaystyle\not{D}_{A_{\tau}}u$	$\displaystyle=$	$\displaystyle\Pi^{\perp}(\Psi_{N})\ =\ \pi^{\text{high}}\Pi^{\perp}(\mathfrak{e}^{\text{Re}}_{N}\mathbb{1}^{-})$	(11.5)

where $\eta=\eta^{\text{low}}+\eta^{\text{med}}$ . The above estimates and the uniform bounds on $(\underline{T}^{\circ},\not{D})^{-1}$ coming from Corollary 6.8 (the version in Corollary 6.10) and Corollary 5.6 show that $\eta,u$ satisfy $\|\eta\|_{\mathfrak{W}}\leq C\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$ and $\|u\|_{\mathfrak{W}}\leq C\varepsilon^{8}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$ .

We now proceed to calculate $\overline{\mathbb{SW}}_{\Lambda}(h^{\prime}_{N})$ , where $h_{N}^{\prime}=h_{N}-(\eta,u)$ as in (11.2). First, with $\xi=\varepsilon\eta$ as previously, we compute:

$\displaystyle\text{d}\overline{\mathbb{SW}}_{h_{1}}(\xi,0,0,0,0)$	$\displaystyle=$	$\displaystyle\text{d}\not{\mathbb{D}}(\eta,0)+\Xi^{+}(\eta)+\Xi^{-}(\eta)$	(11.6)
	$\displaystyle=$	$\displaystyle\underline{T}^{\circ}(\eta)+\mathfrak{t}^{\circ}(\eta)+(1-\Pi)\text{d}\not{\mathbb{D}}_{h_{0}}(\eta)+\pi_{\tau}\text{d}\not{\mathbb{D}}_{h_{0}}(\eta)+\Xi^{+}(\eta)+\Xi^{-}(\eta)$	(11.6)
$\displaystyle\text{d}\overline{\mathbb{SW}}_{h_{1}}(0,0,0,0,u)$	$\displaystyle=$	$\displaystyle\not{D}_{A_{\tau}}(\chi^{-}u)+K_{1}$	(11.7)
	$\displaystyle=$	$\displaystyle\not{D}_{A_{\tau}}u-(1-\chi^{-})\not{D}_{A_{\tau}}u+d\chi^{-}.u+K_{1}(\chi^{-}u)$	(11.7)

where $K_{1}$ is as in Step 2 of the proof of Lemma 10.14.

With these, we now compute:

$\displaystyle\overline{\mathbb{SW}}_{\Lambda}(h_{N}^{\prime})$	$\displaystyle=$	$\displaystyle\overline{\mathbb{SW}}(h_{1})+\text{d}_{h_{1}}\mathbb{SW}(h_{N})\ -\ \text{d}_{h_{1}}\mathbb{SW}(\xi,u)\ +\ \mathbb{Q}(h_{N}^{\prime})\ -\ (\mu+\Lambda)\chi^{-}\frac{\Phi_{\tau}}{\varepsilon}$	(11.9)
	$\displaystyle=$	$\displaystyle\ \overline{\mathbb{SW}}_{\Lambda}(h_{N})\ -\ \text{d}_{h_{1}}\mathbb{SW}(\xi,u)\ +\ \mathbb{Q}(h^{\prime}_{N})-\mathbb{Q}(h_{N})$
	$\displaystyle=$	$\displaystyle\mathfrak{e}_{N}\ -\ \underline{T}^{\circ}(\eta^{\text{low}})\ -\ \underline{T}^{\circ}(\eta^{\text{med}})\ -\ \not{D}_{A_{\tau}}u\ -\ (1-\Pi)\text{d}\not{\mathbb{D}}_{h_{0}}(\eta)-\pi_{\tau}\text{d}\not{\mathbb{D}}_{h_{0}}(\eta)$
		$\displaystyle\ +\ \Xi^{+}(\eta)\ +\ \text{{Zh}}(\eta,u)$
	$\displaystyle=$	$\displaystyle g_{N}\ +\ \Xi^{+}(\eta)\ +\ \mathfrak{e}^{\text{Im}}_{N}\mathbb{1}^{-}\ +\ (1-\Pi)(\mathfrak{e}_{N}^{\text{Re}}\mathbb{1}^{-}+\text{d}\not{\mathbb{D}}_{h_{0}}(\eta))+\pi_{\tau}(\mathfrak{e}_{N}^{\text{Re}}\mathbb{1}^{-}+\text{d}\not{\mathbb{D}}_{h_{0}}(\eta))$
		$\displaystyle\ +\ \text{{Zh}}(\eta,u)$

where (11.9) is obtained by substituting (11.6) and (11.7) with

\text{{Zh}}(\eta,u):=\left(\mathbb{Q}(h^{\prime}_{N})-\mathbb{Q}(h_{N})\right)\ -\ \Xi^{-}(\eta)\ -\ \mathfrak{t}^{\circ}(\eta)\ +\ (1-\chi^{-})\not{D}_{A_{\tau}}u\ -\ d\chi^{-}.u\ -\ K_{1}(\chi^{-}u)

(11.10)

and (11.9) from substituting (11.4)–(11.5).

Splitting up the terms of (11.9), define

$\displaystyle\mathfrak{e}_{N}^{\prime}$	$\displaystyle:=$	$\displaystyle\mathfrak{e}^{\text{Im}}_{N}\mathbb{1}^{-}\ +\ (1-\Pi)(\mathfrak{e}_{N}^{\text{Re}}\mathbb{1}^{-}+\text{d}\not{\mathbb{D}}_{h_{0}}(\eta))$
$\displaystyle\mathfrak{g}_{N}^{\prime}$	$\displaystyle=$	$\displaystyle g_{N}\ +\ \Xi^{+}(\eta)$
$\displaystyle\lambda$	$\displaystyle:=$	$\displaystyle\varepsilon\langle\pi_{\tau}(\mathfrak{e}_{N}^{\text{Re}}\mathbb{1}^{-}+\text{d}\not{\mathbb{D}}_{h_{0}}(\eta))\ ,\ \Phi_{\tau}\rangle_{L^{2}}$
$\displaystyle\mathfrak{e}_{N+1}$	$\displaystyle:=$	$\displaystyle\text{{Zh}}(\eta,u).$

It now suffices to show that these satisfy the conclusions (1)–(4) of the proposition.

Beginning with (3), we have that

\|\text{{Zh}}(\eta,u)\|_{\mathfrak{L}}\leq C\varepsilon^{1/48}\delta\|\mathfrak{e}_{N}\|_{\mathfrak{L}}.

(11.11)

By the bound $\|u\|_{\mathfrak{W}}\leq C\varepsilon^{8}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$ from above and Corollary 10.3, all the terms of (11.10) except those involving $\mathbb{Q},\Xi^{+},\Xi^{-}$ are bounded by, say, $\varepsilon^{5}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$ and may be safely ignored. Corollary 9.4, and repeating the argument that obtained (10.19) for $\Xi^{-}$ shows that $\|\text{{Zh}}(\eta,u)\|_{\mathfrak{L}}\leq C\varepsilon^{1/12-\gamma}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$ . Since $\gamma<<1$ and $\varepsilon^{1/48}\delta=\varepsilon^{1/24}$ , (11.11) follows, which is conclusion (2).

(4) follows directly from the definition of $\lambda$ and Cauchy-Schwartz, since $\|\mathfrak{e}_{N}^{\text{Re}}\mathbb{1}^{-}\|_{L^{2}}\leq\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$ and

\|\text{d}\not{\mathbb{D}}_{h_{0}}(\eta)\|_{L^{2}}\leq C\|\eta\|_{1/2}\leq C\|\eta\|_{\mathfrak{W}}

by Proposition 9.3.

For (1)–(2), the fact the $\Pi(\mathfrak{e}_{N}^{\prime})=0$ and $\mathfrak{g}^{\prime}_{N}=\mathfrak{g}_{N}^{\prime}\mathbb{1}^{+}$ are immediate from the definitions. It remains to show that the asserted bounds hold. To re-iterate the cancellation that led to (11.9), adding and subtracting (11.4),(11.5), $\Psi_{N}$ and $\lambda\varepsilon^{-1}\Phi_{\tau}$ , then using $\Pi(\text{d}\not{\mathbb{D}})=T^{\circ}+\mathfrak{t}^{\circ}$ yields

\mathfrak{e}_{N}^{\prime}=\mathfrak{e}_{N}^{\text{Im}}\mathbb{1}^{-}\ +\ e_{N}^{\text{Re}}\mathbb{1}^{-}\ +\ \text{d}\not{\mathbb{D}}_{h_{0}}(\eta)\ -\mathfrak{t}^{\circ}(\eta)\ -\ \Psi_{N}\ -\ \lambda\varepsilon^{-1}\Phi_{\tau}

Because $\mathfrak{t}^{\circ}(\eta),\Psi_{N}$ are $O(\varepsilon^{8})$ these may be safely ignored as in the proof of (3). Since $\mathfrak{e}^{\text{Im}}_{N}$ is unchanged, and $\pi^{\text{med}}\Pi(\mathfrak{e}_{N}^{\prime})=0$ , it suffices to bound the three terms on the top line in the norm (10.13).

$\displaystyle\\|\text{d}\not{\mathbb{D}}_{h_{0}}(\eta)\mathbb{1}^{+}\\|_{L^{2}}\$	$\displaystyle\leq$	$\displaystyle\ \ \ C\varepsilon^{1/12-\gamma}\\|\eta\\|_{\mathfrak{W}}\ \hskip 73.97733pt\leq\ C\varepsilon^{1/12-\gamma}\\|\mathfrak{e}_{N}\\|_{\mathfrak{L}}$
$\displaystyle\\|(\mathfrak{e}_{N}^{\text{Re}}+\ \text{d}\not{\mathbb{D}}_{h_{0}}(\eta))\mathbb{1}^{-}\\|_{L^{2}}\$	$\displaystyle\leq$	$\displaystyle\ \ \ C\\|\mathfrak{e}_{N}\\|_{\mathfrak{L}}\ \ \ \ +\ \ \ \ C\\|\eta\\|_{\mathfrak{W}}\ \ \ \ \ \ \ \ \ \ \ \leq\ \ C\\|\mathfrak{e}_{N}\\|_{\mathfrak{L}}$
$\displaystyle\\|(\mathfrak{e}_{N}^{\text{Re}}+\ \text{d}\not{\mathbb{D}}_{h_{0}}(\eta))\mathbb{1}^{-}\\|_{L^{2}_{-\nu}}$	$\displaystyle\leq$	$\displaystyle\ C\varepsilon^{-\nu/2}\\|\mathfrak{e}_{N}\\|_{\mathfrak{L}}\ +C\varepsilon^{-\nu/2-\gamma}\\|\eta\\|_{\mathfrak{W}}\ \ \ \leq\ \ C\varepsilon^{-\gamma}\varepsilon^{-\nu/2}\\|\mathfrak{e}_{N}\\|_{\mathfrak{L}}$
$\displaystyle\\|g_{N}+\Xi^{+}(\eta)\\|_{L^{2}}\$	$\displaystyle\leq$	$\displaystyle C\varepsilon^{1/12-\gamma}\\|\mathfrak{e}_{N}\\|_{\mathfrak{L}}\ +\ C\varepsilon^{1/12-\gamma}\\|\eta\\|_{\mathfrak{W}}\ \leq\ C\varepsilon^{1/12-\gamma}\\|\mathfrak{e}_{N}\\|_{\mathfrak{L}}$

where each line, the definition of the $\mathfrak{L}$ -norm is used in conjunction with, (10.22), (10.17), and (10.18). For the $\lambda\varepsilon^{-1}\Phi_{\tau}$ , the same bounds hold by (2) and the fact that $\Phi_{\tau}\in L^{2}\cap L^{2}_{-\nu}$ . Conclusion (1) follows from the first three lines, and (2) from the fourth.

That (I) holds for $h_{N}^{\prime}$ is immediate from the bounds $\|\eta\|_{\mathfrak{W}}\ +\ \|u\|_{\mathcal{X}}\leq C\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$ , and the fact that it holds for $h_{N}$ . Finally, the proof of the (resp. $\text{d}\overline{\mathbb{SW}}$ ) statements is identical, omitting any mention of $\mathbb{Q}$ and $\Lambda$ . ∎

11.2. The Outside Step

This subsection covers the second of the three stages of the cyclic induction (2.17). Now that the error terms are essentially orthogonal to the obstruction, solving in the outside can proceed using Lemma 4.4 and Proposition 7.11.

Define the outside parametrix $P^{-}$ by

\displaystyle P^{-}:\mathfrak{L}\longrightarrow\mathcal{H}^{-}\oplus\mathbb{R}\hskip 56.9055ptP^{-}

\displaystyle:=

\displaystyle\left(\mathcal{L}^{-1}_{(\Phi_{\tau},A_{\tau})}(1-\Pi^{\perp})\ ,\ -\varepsilon\pi_{\tau}\right)\circ\mathbb{1}^{-}

(11.12)

where $\pi_{\tau}:\mathfrak{L}\to\mathbb{R}$ is understood to mean the coefficient of $\Phi_{\tau}$ in $\mathbb{R}\Phi_{\tau}$ . Notice that the first component indeed lands in $\mathcal{H}^{-}$ by Definition (10.5).

Proposition 11.2.

$P\!^{-}$ is a linear operator uniformly bounded in $\varepsilon,\tau$ and satisfies the following property. If for $N\in\mathbb{N}$ , $h^{\prime}_{N}\in\mathcal{H}$ is a configuration satisfying the conclusions of Proposition 11.1, then the updated configuration

h_{N}^{\prime\prime}=(\text{Id}-P^{-}\circ\overline{\mathbb{SW}}_{\Lambda})h_{N}^{\prime}

(11.13)

satisfies

\overline{\mathbb{SW}}_{\Lambda}(h^{\prime\prime}_{N})=\mathfrak{e}_{N}^{\prime\prime}\mathbb{1}^{+}\ +\ \mathfrak{e}_{N+1}

(resp. the same for $\text{d}_{h_{1}}\overline{\mathbb{SW}}$ ), where $\mathfrak{e}_{N}^{\prime\prime},\mathfrak{e}_{N+1}\in\mathfrak{L}$ obey

(1’)

$\|\mathfrak{e}_{N}^{\prime\prime}\|_{\mathfrak{L}}\leq C\varepsilon^{-\gamma}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$
(2’)

Item (2) from Proposition 11.1 continues to hold.

Moreover, $h_{N}^{\prime\prime}$ continues to satisfy (I) from Proposition 11.1.

Proof.

As in the proof of Proposition 11.1, write $\mathfrak{e}_{N}^{\prime}+\mathfrak{g}_{N}^{\prime}=e^{\prime}_{N}+f^{\prime}_{N}+g^{\prime}_{N}$ , where

g_{N}^{\prime}:=\mathfrak{g}_{N}^{\prime}\mathbb{1}^{+}\hskip 56.9055ptf_{N}^{\prime}:=(\mathfrak{e}^{\prime}_{N})^{\text{Im}}\mathbb{1}^{-}\hskip 56.9055pte_{N}^{\prime}:=(\mathfrak{e}_{N}^{\prime})^{\text{Re}}.

(11.14)

Conclusions (1)–(4) of Proposition 11.1 mean that $\Pi(e_{N}^{\prime})=0$ , that $\|e_{N}\|_{L^{2}_{-\nu}}\leq C\varepsilon^{-\gamma}\varepsilon^{-\nu/2}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$ , for both $\nu=0,\nu^{-}$ , and that $\|f_{N}\|_{L^{2}}\leq C\varepsilon^{1/6}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$ and $\|g_{N}\|_{L^{2}}\leq C\varepsilon^{1/12-\gamma}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$ , where $\mathfrak{e}_{N}$ is the original error from Proposition 11.1.

Set

(\psi,b,\mu):=P^{-}(e^{\prime}_{N}+f^{\prime}_{N}+\lambda\varepsilon^{-1}\Phi_{\tau})

so that

$\displaystyle\not{D}_{A_{\tau}}\psi^{\text{Re}}$	$\displaystyle=$	$\displaystyle e_{N}^{\prime}$
$\displaystyle\mathcal{L}_{(\Phi_{\tau},A_{\tau})}^{\text{Im}}(\psi^{\text{Im}},b)$	$\displaystyle=$	$\displaystyle f_{N}^{\prime}$
$\displaystyle\mu$	$\displaystyle=$	$\displaystyle-\varepsilon\pi(\lambda\varepsilon^{-1}\Phi_{\tau})=-\lambda$

where $\psi=(\psi^{\text{Re}},\psi^{\text{Im}})\in S^{\text{Re}}\oplus S^{\text{Im}}$ .

We now show that $P^{-}$ is uniformly bounded. Lemma 4.4 and Proposition 7.11 and the above bounds on $e^{\prime}_{N},f^{\prime}_{N}$ show that these unique solutions (where $\pi_{\tau}(\psi^{\text{Re}})=0$ ), satisfy

$\displaystyle\\|\psi^{\text{Re}}\\|_{rH^{1}_{e}}$	$\displaystyle\leq$	$\displaystyle C\\|e_{N}^{\prime}\\|_{L^{2}}\ \ \ \ \ \ \ \ \ \ \leq\ C\varepsilon^{-\gamma}\\|\mathfrak{e}_{N}\\|_{\mathfrak{L}}$	(11.15)
$\displaystyle\varepsilon^{\nu/2}\\|\psi^{\text{Re}}\\|_{r^{1+\nu}H^{1}_{e}}$	$\displaystyle\leq$	$\displaystyle C\varepsilon^{\nu/2}\\|e_{N}^{\prime}\\|_{L^{2}_{-\nu}}\ \ \hskip 5.12128pt\leq\ C\varepsilon^{-\gamma}\\|\mathfrak{e}_{N}\\|_{\mathfrak{L}}$	(11.16)
$\displaystyle\varepsilon^{-1/6}\\|(\psi^{\text{Im}},b)\\|_{\mathcal{H}^{1,-}_{\varepsilon}}$	$\displaystyle\leq$	$\displaystyle C\varepsilon^{-1/6}\\|f_{N}^{\prime}\\|_{L^{2}}\ \ \ \leq\ C\varepsilon^{-\gamma}\\|\mathfrak{e}_{N}\\|_{\mathfrak{L}}$	(11.17)
$\displaystyle\varepsilon^{-1}\|\mu\|$	$\displaystyle\leq$	$\displaystyle C\\|\lambda\varepsilon^{-1}\Phi_{\tau}\\|_{L^{2}}\ \hskip 7.96674pt\leq\ C\varepsilon^{-\gamma}\\|\mathfrak{e}_{N}\\|_{\mathfrak{L}}.$	(11.18)

To explain further how to obtain the bounds in the middle column, the condition that $\pi_{\tau}(\psi^{\text{Re}})=0$ means that the $\nu=0$ version of Lemma 4.4 applies without the projection term, which yields (11.15). In turn, (11.15) and Cauchy-Schwartz show that $\pi_{\nu}(\psi^{\text{Re}})\leq C\|\psi\|_{L^{2}}\leq C\|e^{\prime}_{N}\|_{L^{2}}$ since $\Phi_{\tau}\in L^{2}_{-2\nu}$ , after which (11.16) follows from Lemma 4.4 taking $\nu=\nu^{-}$ . (11.17) and (11.18) are immediate from Proposition 7.11 and the definition of $\mu$ . The final column follows immediately from the bounds on (11.14) and conclusion (4) of Proposition 11.1.

To show that $P^{-}$ is uniformly bounded, it must be shown that $\|\mathfrak{e}_{N}^{\prime}+\lambda\varepsilon^{-1}\Phi_{\tau}\|_{\mathfrak{L}}$ dominates each term in the middle column (since $P^{-}$ ignores $\mathfrak{g}_{N}^{\prime}$ ). Indeed, $\mathfrak{e}_{N}^{\prime}$ is, by its construction in the proof of Proposition 11.1, $L^{2}$ -orthogonal to $\Phi_{\tau}$ . Therefore,

	$\displaystyle\\|e_{N}^{\prime}\\|_{L^{2}}+\\|\lambda\varepsilon^{-1}\Phi_{\tau}\\|_{L^{2}}$	$\displaystyle\leq$	$\displaystyle\ \ \ \ \ \ \\|\mathfrak{e}_{N}^{\prime}+\lambda\varepsilon^{-1}\Phi_{\tau}\\|_{L^{2}}\hskip 93.89418pt\ \leq C\\|\mathfrak{e}_{N}^{\prime}+\lambda\varepsilon^{-1}\Phi_{\tau}\\|_{\mathfrak{L}}\ \ \ \ \ \ \$		(11.19)
	$\displaystyle\\|e_{N}^{\prime}\\|_{L^{2}_{-\nu}}$	$\displaystyle\leq$	$\displaystyle\\|(e_{N}^{\prime}+\lambda\varepsilon^{-1}\Phi_{\tau})\mathbb{1}^{-}\\|_{L^{2}_{-\nu}}+\\|(\lambda\varepsilon^{-1}\Phi_{\tau})\mathbb{1}^{-}\\|_{L^{2}_{-\nu}}\leq C\\|\mathfrak{e}_{N}^{\prime}+\lambda\varepsilon^{-1}\Phi_{\tau}\\|_{\mathfrak{L}}.$		(11.20)

The first of these is simply orthogonality along with fact that the $\mathfrak{L}$ -norm dominates the $L^{2}$ -norm. The second is the triangle inequality, then invoking the first one along with the fact that the $L^{2}_{-\nu}$ and $L^{2}$ -norms are equivalent on the 1-dimensional span of $\Phi_{\tau}$ . That $\varepsilon^{-1/6}\|f_{N}^{\prime}\|_{L^{2}}$ in (11.17) is likewise bounded by the right side of (11.19) is immediate from the Definition of the $\mathfrak{L}$ -norm. This completes the claim that $P^{-}$ is uniformly bounded.

We now proceed to calculate $\overline{\mathbb{SW}}_{\Lambda}(h^{\prime\prime}_{N})$ where $h_{N}^{\prime\prime}=h_{N}-(\psi,b,\mu)$ as in (11.13). First,

$\displaystyle\text{d}\overline{\mathbb{SW}}_{h_{1}}(\psi,b,\mu)$	$\displaystyle=$	$\displaystyle\mathcal{L}_{(\Phi_{\tau},A_{\tau})}\chi^{-}(\psi,b)\ +\ \mathcal{K}_{1}(\psi,b)\ -\ \mu\varepsilon^{-1}\chi^{-}\Phi_{\tau}$	(11.21)
	$\displaystyle=$	$\displaystyle\not{D}_{A_{\tau}}\psi^{\text{Re}}\ +\ \mathcal{L}_{(\Phi_{\tau},A_{\tau})}^{\text{Im}}(\psi^{\text{Im}},b)\ -\ \mu\varepsilon^{-1}\Phi_{\tau}$
		$\displaystyle\ +\ d\chi^{-}(\psi,b)\ +\ K_{1}(\psi,b)\ +\ (1-\chi^{-})\mu\varepsilon^{-1}\Phi_{\tau},$

where we have used the fact that $\chi^{-}=1$ on the support of $\mathbb{1}^{-}$ . Using the conclusion of Proposition 11.1 and the definitions (11.14), then substituting (11.21) yields

$\displaystyle\overline{\mathbb{SW}}_{\Lambda}(h_{N}^{\prime\prime})$	$\displaystyle=$	$\displaystyle\text{d}_{h_{1}}\mathbb{SW}(h_{N}^{\prime})\ -\ \text{d}_{h_{1}}\mathbb{SW}(\psi,b,\mu)\ +\ \mathbb{Q}(h_{N}^{\prime\prime})\ -\ (\mu_{N}+\mu+\Lambda)\chi^{-}\frac{\Phi_{\tau}}{\varepsilon}$
	$\displaystyle=$	$\displaystyle\ \overline{\mathbb{SW}}_{\Lambda}(h_{N}^{\prime})\ -\ \text{d}_{h_{1}}\mathbb{SW}(\psi,b,\mu)\ +\ \mathbb{Q}(h_{N}^{\prime\prime})-\mathbb{Q}(h_{N}^{\prime})$
	$\displaystyle=$	$\displaystyle e_{N}^{\prime}\ +\ f_{N}^{\prime}\ +\ g_{N}^{\prime}\ +\ \lambda\varepsilon^{-1}\Phi_{\tau}\ -\ \not{D}_{A_{\tau}}\psi^{\text{Re}}\ -\ \mathcal{L}_{(\Phi_{\tau},A_{\tau})}^{\text{Im}}(\psi^{\text{Im}},b)\ +\ \mu\varepsilon^{-1}\Phi_{\tau}\ \ \$
		$\displaystyle\ +\ \left(\mathbb{Q}(h_{N}^{\prime\prime})-\mathbb{Q}(h_{N}^{\prime})\right)\ +\ d\chi^{-}(\psi,b)\ +\ K_{1}(\psi,b)\ +\ (1-\chi^{-})\mu\varepsilon^{-1}\Phi_{\tau}\ +\ \mathfrak{e}_{N+1}$
	$\displaystyle=$	$\displaystyle g_{N}^{\prime}+\ \left(\mathbb{Q}(h_{N}^{\prime\prime})-\mathbb{Q}(h_{N}^{\prime})\right)\ +\ d\chi^{-}(\psi,b)\ +\ K_{1}(\psi,b)\ +\ (1-\chi^{-})\mu\varepsilon^{-1}\Phi_{\tau}\ +\ \mathfrak{e}_{N+1}.$

Then (re)-define

	$\displaystyle\mathfrak{e}_{N}^{\prime\prime}$	$\displaystyle:=$	$\displaystyle g_{N}^{\prime}\ +\ d\chi^{-}(\psi,b)\ +\ (1-\chi^{-})\mu\varepsilon^{-1}\Phi_{\tau}$		(11.22)
	$\displaystyle\mathfrak{e}_{N+1}$	$\displaystyle\mapsto$	$\displaystyle\mathfrak{e}_{N+1}\ +\ \left(\mathbb{Q}(h_{N}^{\prime})-\mathbb{Q}(h_{N}^{\prime\prime})\right)\ +\ K_{1}(\psi,b).$		(11.23)

Notice that $\mathfrak{e}_{N}^{\prime\prime}$ now includes the crucial alternating error term $d\chi^{-}(\psi,b)$ .

To conclude, we show $\mathfrak{e}_{N}^{\prime\prime}$ and $\mathfrak{e}_{N+1}$ satisfy conclusions (1)–(2). That (1) holds for the first term of $\mathfrak{e}_{N}^{\prime\prime}$ in (11.22) is immediate from the bound on $g^{\prime}_{N}$ from 11.14. The bounds on $e_{N}^{\prime}$ in 11.14 also imply that $e_{N}^{\prime}$ has $\varepsilon^{1/2}$ -effective support, hence Lemma 4.7 applies to show that

\|d\chi^{-}.\psi^{\text{Re}}\|_{L^{2}}\leq C\varepsilon^{1/12-\gamma}\|e_{N}^{\prime}\|_{L^{2}}\leq C\varepsilon^{1/12-2\gamma}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}.

(11.24)

Note that the $(1-\chi^{-})\mathfrak{e}_{N}^{\prime}$ term in Lemma 4.7 vanishes because $\chi^{-}=1$ on $\text{supp}(\mathbb{1}^{-})$ . (11.24) also holds for $(\psi^{\text{Im}},b)$ simply because of the $\varepsilon^{1/6}$ weight on these components in (11.17). Finally, for the third term of $\mathfrak{e}_{N}^{\prime\prime}$ in (11.22) , notice that direct integration using the fact that $|\Phi_{\tau}|\leq Cr^{1/2}$ shows that $\|(1-\chi^{-})\Phi_{\tau}\|_{L^{2}}\leq C\varepsilon^{1-\gamma}$ , and (11.18) therefore shows the third term of $\mathfrak{e}_{N}^{\prime\prime}$ is bounded by the right hand side of (11.24) with an extra factor of $\varepsilon^{11/12-\gamma}$ . Combining these three shows that

\|\mathfrak{e}_{N}^{\prime\prime}\|_{\mathfrak{L}}\leq C\varepsilon^{-\gamma}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}

and $\mathfrak{e}_{N}^{\prime\prime}=\mathfrak{e}_{N}^{\prime\prime}\mathbb{1}^{-}$ because $\chi^{-}=1$ outside $\text{supp}(\mathbb{1}^{+})$ .

Using the bounds (11.15)–(11.17), the terms involving $\mathbb{Q}$ can be bounded identically to in the proof of Proposition 11.1. As in the proof of Lemma 10.14, $K_{1}$ is exponentially small, hence negligible. Conclusion (2) follows after increasing $\gamma$ slightly.

Finally, that (I) continues to hold for $h_{N}^{\prime\prime}$ and of the (resp. $\text{d}\overline{\mathbb{SW}}_{h_{1}}$ ) statements follows identically to in Proposition 11.1 using the uniform boundedness of $P^{-}$ . ∎

11.3. The Inside Step

The section completes the third stage of the cyclic induction by constructing $P^{+}$ . Define

\displaystyle P^{+}:\mathfrak{L}\longrightarrow\mathcal{H}^{+}\hskip 56.9055ptP^{+}

\displaystyle:=

\displaystyle\mathcal{L}^{-1}_{(\Phi_{1},A_{1})}\mathbb{1}^{+}.

(11.25)

Definition 10.5 ensures that $P^{+}$ lands in $\mathcal{H}^{+}$ .

Proposition 11.3.

$P^{+}$ is a linear operator uniformly bounded in $\varepsilon,\tau$ and satisfies the following property. If for $N\in\mathbb{N}$ , $h^{\prime\prime}_{N}\in\mathcal{H}$ is a configuration satisfying the conclusions of Proposition 11.2, then the updated configuration

h_{N+1}=(\text{Id}-P^{-}\circ\overline{\mathbb{SW}}_{\Lambda})h_{N}^{\prime\prime}

(11.26)

(resp. the same for $\text{d}\overline{\mathbb{SW}}_{h_{1}})$ satisfies the hypotheses of Proposition 11.1 with $N^{\prime}=N+1$ .

Proof.

Uniform boundedness is immediate from the definition of the $\mathcal{H}^{+}$ -norm in Definition 10.5. Let $\mathfrak{e}_{N}^{\prime\prime}$ be as in the conclusion of Proposition 11.2, so that we may write

g_{N}^{\prime\prime}:=\mathfrak{e}_{N}^{\prime\prime}\mathbb{1}^{+}=\mathfrak{e}_{N}^{\prime\prime}

where $\|g_{N}^{\prime\prime}\|_{L^{2}}\leq C\varepsilon^{1/12-\gamma}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$ .

Set

(\varphi,a):=P^{+}(g_{N}^{\prime\prime})

so that

\mathcal{L}_{(\Phi_{1},A_{1})}(\varphi,a)=g_{N}^{\prime\prime},

and

\|(\varphi,a)\|_{H^{1,+}_{\varepsilon}}\ \leq\ \|(\varphi,a)\|_{\mathcal{H}^{+}}=C\varepsilon^{-1/12-\gamma}\|g_{N}^{\prime\prime}\|\ \leq\ C\varepsilon^{-\gamma}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}.

(11.27)

We now proceed to calculate $\overline{\mathbb{SW}}_{\Lambda}(h_{N+1})$ where $h_{N+1}=h_{N}-(\varphi,a)$ as in (11.13).

$\displaystyle\overline{\mathbb{SW}}_{\Lambda}(h_{N+1})$	$\displaystyle=$	$\displaystyle\text{d}_{h_{1}}\mathbb{SW}(h_{N}^{\prime\prime})\ -\ \text{d}_{h_{1}}\mathbb{SW}(\varphi,a)\ +\ \mathbb{Q}(h_{N+1})\ -\ (\mu_{N}+\mu+\Lambda)\chi^{-}\frac{\Phi_{\tau}}{\varepsilon}$	(11.28)
	$\displaystyle=$	$\displaystyle\ \overline{\mathbb{SW}}_{\Lambda}(h_{N}^{\prime})\ -\ \text{d}_{h_{1}}\mathbb{SW}(\psi,b,\mu)\ +\ \mathbb{Q}(h_{N+1})-\mathbb{Q}(h_{N}^{\prime\prime})$
	$\displaystyle=$	$\displaystyle g_{N}^{\prime\prime}\ -\ \mathcal{L}_{(\Phi_{1},A_{1})}(\varphi,a)\ +\ \left(\mathbb{Q}(h_{N+1})-\mathbb{Q}(h_{N}^{\prime\prime})\right)\ +\ d\chi^{+}(\varphi,a)\ +\ \mathfrak{e}_{N+1}$
	$\displaystyle=$	$\displaystyle\left(\mathbb{Q}(h_{N+1})-\mathbb{Q}(h_{N}^{\prime\prime})\right)\ +\ d\chi^{+}(\varphi,a)\ +\ \mathfrak{e}_{N+1}$

The alternating error from $d\chi^{+}$ has now been shifted back to the outside region.

Re-defining $\mathfrak{e}_{N+1}$ to include all three terms of (11.28), we claim that it satisfies (II) of Proposition 11.1 with $N^{\prime}=N+1$ . In fact, one has the slightly stronger bound

\|\mathfrak{e}_{N+1}\|_{\mathfrak{L}}\leq C\varepsilon^{1/48-\gamma}\delta\|\mathfrak{e}_{N}\|_{\mathfrak{L}}.

(11.29)

Indeed, the terms involving $\mathbb{Q}$ in (11.28) may be bounded as in the proof of Proposition 11.1. Then, since $d\chi^{+}$ is supported where $r=O(\varepsilon^{1/2})$ , Lemma 7.12 implies

	$\displaystyle\\|d\chi^{+}(\varphi,a)\\|_{\mathfrak{L}}$	$\displaystyle\leq$	$\displaystyle\\|d\chi^{+}(\varphi,a)\\|_{L^{2}}+\varepsilon^{\nu}\\|d\chi^{+}(\varphi,a)\\|_{L^{2}_{-\nu}}$
		$\displaystyle\leq$	$\displaystyle C\\|d\chi^{+}(\varphi,a)\\|_{L^{2}}\leq C\varepsilon^{-1/24-\gamma}\\|g_{N}^{\prime\prime}\\|_{L^{2}}\leq C\varepsilon^{1/24-\gamma}\\|\mathfrak{e}_{N}\\|_{\mathfrak{L}}.$

Finally, the original $\mathfrak{e}_{N+1}$ in (11.28) already satisfies (11.29) by virtue of (2’) in Proposition 11.2. (11.29) follows.

Over the course of the proofs of (a single cycle of) Propositions 11.1, 11.2, and 11.3, the constants $C,\gamma$ have been increased a finite number of times. Let $C_{0}$ being the original constant in Item (II) of Proposition 11.1 and $C_{1},\gamma_{1}$ be the final versions of the constants appearing in (11.29). Since $\gamma_{1}<<1$ still, we may assume that

C_{1}\varepsilon^{1/48-\gamma_{1}}\leq C_{0}

once $\varepsilon$ is sufficiently small, which reduces (11.29) to hypothesis (II) of Proposition 11.1 with $N^{\prime}=N+1$ as desired. The proof of (I) and of the (resp. $\text{d}\overline{\mathbb{SW}}_{h_{1}}$ ) statements again follows as in Propositions 11.1 and 11.2 using (11.27). ∎

11.4. Proof of Proposition 10.1

This section completes the proof of Proposition 10.1.

Analogously to (2.23), define $\mathbb{A}$ and $\mathbb{P}_{1}=\text{d}\mathbb{A}$ by

	$\displaystyle\mathbb{A}$	$\displaystyle=$	$\displaystyle P_{\xi}\ +\ P^{-}(\text{Id}-\ \overline{\mathbb{SW}}_{\Lambda}P_{\xi})\ \ +\ \ P^{+}\left(\text{Id}-\ \overline{\mathbb{SW}}_{\Lambda}P_{\xi}\ -\ \ \overline{\mathbb{SW}}_{\Lambda}P^{-}(\text{Id}\ -\ \overline{\mathbb{SW}}_{\Lambda}P_{\xi})\right),$		(11.30)
	$\displaystyle\mathbb{P}_{1}$	$\displaystyle=$	$\displaystyle P_{\xi}\ +\ P^{-}(\text{Id}-\text{d}\overline{\mathbb{SW}}_{h_{1}}P_{\xi})\ +\ P^{+}\left(\text{Id}-\text{d}\overline{\mathbb{SW}}_{h_{1}}P_{\xi}-\ \text{d}\overline{\mathbb{SW}}_{h_{1}}P^{-}(\text{Id}-\text{d}\overline{\mathbb{SW}}_{h_{1}}P_{\xi})\right),$		(11.31)

So that (analogously to 2.24),

	$\displaystyle T=\text{Id}-\mathbb{A}\circ\ \overline{\mathbb{SW}}_{\Lambda}\ \$	$\displaystyle=$	$\displaystyle(\text{Id}-P^{+}\ \overline{\mathbb{SW}}_{\Lambda}\ )(\text{Id}-P^{-}\overline{\mathbb{SW}}_{\Lambda}\ )(\text{Id}-P_{\xi}\ \overline{\mathbb{SW}}_{\Lambda}\ ),$		(11.32)
	$\displaystyle\text{dT}=\text{Id}-\mathbb{P}_{1}\circ\text{d}\overline{\mathbb{SW}}_{h_{1}}$	$\displaystyle=$	$\displaystyle(\text{Id}-P^{+}\text{d}\overline{\mathbb{SW}}_{h_{1}})(\text{Id}-P^{-}\text{d}\overline{\mathbb{SW}}_{h_{1}})(\text{Id}-P_{\xi}\text{d}\overline{\mathbb{SW}}_{h_{1}}).$		(11.33)

Thus applying $T$ carries one complete cycle of the iteration (2.17), and $\text{d}T$ one complete cycle of the linearized iteration (the resp. statements in Propositions 11.1–11.3).

The next lemma is the analogue of Proposition 2.4.

Lemma 11.4.

The linearization $\text{d}\overline{\mathbb{SW}}_{h_{1}}:\mathcal{H}\to\mathfrak{L}$ is invertible, and there is a constant $C$ independent of $\varepsilon,\gamma$ such that

\|h\|_{\mathcal{H}}\leq C\varepsilon^{-\gamma}\|\text{d}\overline{\mathbb{SW}}_{h_{1}}(h)\|_{\mathfrak{L}}

(11.34)

holds.

Proof.

Let $\mathbb{P}_{0}:\mathfrak{L}\to\mathcal{H}$ be defined by

\mathbb{P}_{0}:=P^{+}\oplus\begin{pmatrix}P_{\xi}&0\\ *&P^{-}\end{pmatrix}

where $*=-P^{-}(1-\Pi^{\perp})\text{d}\not{\mathbb{D}}_{h_{0}}P_{\xi}$ . Since the map $\beta:\mathfrak{L}(Y)\to\mathfrak{L}(Y^{+})\oplus\mathfrak{L}(Y^{-})$ defined by $\beta(\mathfrak{e}):=(\mathfrak{e}\mathbb{1}^{+},\mathfrak{e}\mathbb{1}^{-})$ , which is built into the definitions of $P_{\xi},P^{\pm}$ is an isomorphism, and so is each of $P_{\xi},P^{-},P^{+}$ , it follows that $\mathbb{P}_{0}$ is an isomorphism hence Fredholm with index 0.

Then, the calculations (11.6), (11.7), and (11.21) and the subsequent bounds (e.g. 11.11) in the proofs of Propositions 11.1 and 11.2 show that

	$\displaystyle\mathbb{P}_{1}-\mathbb{P}_{0}$	$\displaystyle=$	$\displaystyle-(P^{+}+P^{-})(\Xi+\mathfrak{t}^{\circ})P_{\xi}\ -\ P^{+}(d\chi^{-}P^{-}(\text{Id}-\text{d}\overline{\mathbb{SW}}_{h_{1}}P_{\xi}))\ +\ \star$		(11.35)
			$\displaystyle-(P^{+}+P^{-})O(\varepsilon^{8}).$		(11.35)

where $\star$ consists of terms involving the 1-dimensional span of $\Phi_{\tau}$ , and the $O(\varepsilon^{8})$ accounts for terms involving $u$ and $K_{1}$ . The terms in the top line of (11.35) involving are compact, because they factor through either the compact inclusion $\mathcal{H}^{1,-}\hookrightarrow L^{2}$ or through the finite-dimensional spaces spanned by $\eta\in\mathfrak{W}$ and $\Phi_{\tau}$ respectively. Since $\|\mathbb{P}_{0}^{-1}\|\leq C\varepsilon^{-\gamma}$ , the $O(\varepsilon^{8})$ terms may be safely ignored. It follows that $\mathbb{P}_{1}$ is Fredholm of index 0 once $\varepsilon$ is sufficiently small.

The same argument shows that $\mathbb{P}_{N}$ defined by

\text{Id}-\mathbb{P}_{N}\circ\text{d}\overline{\mathbb{SW}}_{h_{1}}=\text{d}T^{N}

is likewise Fredholm of Index 0. By construction (cf 2.21–2.24), $\mathbb{P}_{N}$ applies $N$ stages of the (resp. $\text{d}\overline{\mathbb{SW}}_{h_{1}}$ ) alternating iteration from Propositions 11.1–11.3 starting from $\mathfrak{e}_{1}=\text{Id}-\text{d}\overline{\mathbb{SW}}_{h_{1}}\mathbb{P}_{0}$ . We conclude that

\text{d}\overline{\mathbb{SW}}_{h_{1}}\mathbb{P}_{N}=\text{Id}+O(\delta^{N-1})

It follows as in the proof of Proposition 2.4 that $\text{d}\overline{\mathbb{SW}}_{h_{1}}$ is an isomorphism. (11.34) follows from the bound $\|\mathbb{P}_{0}^{-1}\|\leq C\varepsilon^{-\gamma}$ above, and statement (I) in Propositions 11.1–11.3 with $h_{N}$ being the correction from $\mathbb{P}_{N}-\mathbb{P}_{0}$ , since $\|\mathfrak{e}_{1}\|_{\mathfrak{L}}\leq C\varepsilon^{-\gamma}$ . ∎

Proof of Proposition 10.1.

Let $\mathcal{C}=B_{r}(0)\subseteq\mathcal{H}$ of radius $r=\varepsilon^{-1/20}$ , and let $T$ be given by (10.1) with $\mathbb{A}$ defined by (11.30).

(A) is deduced assuming (C) as follows. Let $h\in\mathcal{C}$ , then since $\|\overline{\mathbb{SW}}_{\Lambda}(0)\|_{\mathfrak{L}}\leq C\varepsilon^{-1/24-\gamma}$ by Theorem 7.5,

\|T(h)\|_{\mathcal{H}}\leq\|T(h)-T(0)\|_{\mathcal{H}}+\|T(0)\|_{\mathcal{H}}\leq C\sqrt{\delta}\|h\|_{\mathcal{H}}+C\varepsilon^{-1/24-2\gamma}\leq r,

where the bound on $T(0)$ is a consequence of (I) in Propositions 11.1–11.3 with $\mathfrak{e}_{1}=\overline{\mathbb{SW}}_{\Lambda}(0)$ . Thus $T:\mathcal{C}\to\mathcal{C}$ preserves $\mathcal{C}$ . Continuity of $T$ is immediate from (C). Smooth dependence on $(\varepsilon,\tau)$ is immediate from the smooth dependence of the Seiberg–Witten equations on $p_{\tau}$ , and the smooth dependence of $(\Phi_{1},A_{1})$ and $(\Phi_{\tau},A_{\tau})$ and the linearized equations at these and their inverses used to construct $T$ .

(B) By (11.32), applying $T$ constitutes a full cycle of the three-stage iteration carried out by Propositions 11.1–11.3. The conclusion follows from applying these inductively.

T(h)=\text{d}T(h)+\mathfrak{q}(h).

where $\text{d}T$ is as in (11.33). The same argument as (B), now using the (resp. $\text{d}\overline{\mathbb{SW}}_{h_{1}}$ ) statements in Propositions 11.1–11.3 shows that

\|\text{d}\overline{\mathbb{SW}}_{h_{1}}(\text{d}T(h))\|_{\mathfrak{L}}\leq C\delta\|\text{d}\overline{\mathbb{SW}}_{h_{1}}(h)\|_{\mathfrak{L}}.

Therefore since $\text{d}T$ is linear, Lemmas 10.14 and 11.4 shows

$\displaystyle\\|T(h_{1})-T(h_{2})\\|_{\mathcal{H}}$	$\displaystyle\leq$	$\displaystyle C\varepsilon^{-\gamma}\\|\text{d}\overline{\mathbb{SW}}_{h_{1}}(T(h_{1})-T(h_{2}))\\|_{\mathfrak{L}}$
	$\displaystyle\leq$	$\displaystyle C\varepsilon^{-\gamma}\\|\text{d}\overline{\mathbb{SW}}_{h_{1}}(\text{d}T(h_{1})-\text{d}T(h_{2}))\\|_{\mathfrak{L}}\ +\ C\varepsilon^{-\gamma}\\|\text{d}\overline{\mathbb{SW}}_{h_{1}}(\mathfrak{q}(h_{1})-\mathfrak{q}(h_{2}))\\|_{\mathfrak{L}}$
	$\displaystyle\leq$	$\displaystyle C\varepsilon^{-\gamma}\delta\\|\text{d}\overline{\mathbb{SW}}_{h_{1}}(h_{1}-h_{2})\\|_{\mathfrak{L}}\ +\ C\varepsilon^{-\gamma}\\|\text{d}\overline{\mathbb{SW}}_{h_{1}}(\mathfrak{q}(h_{1})-\mathfrak{q}(h_{2}))\\|_{\mathfrak{L}}$
	$\displaystyle\leq$	$\displaystyle C\varepsilon^{-2\gamma}\delta\\|h_{1}-h_{2}\\|_{\mathcal{H}}\ +\ C\varepsilon^{-2\gamma}\\|\mathfrak{q}(h_{1})-\mathfrak{q}(h_{2})\\|_{\mathcal{H}}.$

Since $C\varepsilon^{-2\gamma}\delta<\sqrt{\delta}$ once $\varepsilon$ is sufficiently, small, the following bound completes (C):

\|\mathfrak{q}(h_{1})-\mathfrak{q}(h_{2})\|_{\mathcal{H}}\leq C\varepsilon^{1/12-6\gamma}\|h_{1}-h_{2}\|_{\mathcal{H}}.

(11.36)

To prove (11.36), we calculate $\mathfrak{q}=T-\text{d}T$ by expanding (11.32) and (11.33), with $\overline{\mathbb{SW}}_{\Lambda}=\text{d}\overline{\mathbb{SW}}_{h_{1}}+\mathbb{Q}+\Lambda$ where the latter is shorthand for $\Lambda=\Lambda(\tau)\varepsilon^{-1}\chi^{-}\Phi_{\tau}$ . Writing $\mathbb{L}=\text{d}\overline{\mathbb{SW}}_{h_{1}},$ this shows

$\displaystyle\mathfrak{q}$	$\displaystyle=$	$\displaystyle P^{-}\mathbb{Q}\ +\ P_{\xi}\mathbb{Q}\ +\ P^{+}\mathbb{Q}\ +\ P^{-}\mathbb{Q}P_{\xi}\mathbb{L}\ +\ P^{-}\mathbb{Q}P_{\xi}\mathbb{Q}\ +\ P^{-}\mathbb{L}P_{\xi}\mathbb{Q}$
	$\displaystyle+$	$\displaystyle P^{+}\mathbb{Q}P^{-}\mathbb{L}\ +\ P^{+}\mathbb{Q}P_{\xi}\mathbb{L}\ +\ P^{+}\mathbb{Q}P^{-}\mathbb{Q}\ +\ P^{+}\mathbb{Q}P_{\xi}\mathbb{Q}\ +\ P^{+}\mathbb{L}P^{-}\mathbb{Q}\ +\ P^{+}\mathbb{L}P_{\xi}\mathbb{Q}$
	$\displaystyle+$	$\displaystyle P^{+}\mathbb{L}P^{-}\mathbb{Q}P_{\xi}\mathbb{L}\ +\ P^{+}\mathbb{L}P^{-}\mathbb{L}P_{\xi}\mathbb{Q}+P^{+}\mathbb{Q}P^{-}\mathbb{L}P_{\xi}\mathbb{L}\ +\ P^{+}\mathbb{L}P^{-}\mathbb{Q}P_{\xi}\mathbb{Q}$
	$\displaystyle+$	$\displaystyle P^{+}\mathbb{Q}P^{-}\mathbb{Q}P_{\xi}\mathbb{L}\ +\ P^{+}\mathbb{Q}P^{-}\mathbb{L}P_{\xi}\mathbb{Q}\ +\ P^{+}\mathbb{Q}P^{-}\mathbb{Q}P_{\xi}\mathbb{Q}$
	$\displaystyle+$	$\displaystyle\mathfrak{q}_{\Lambda}$

where $\mathfrak{q}_{\Lambda}$ is the same collection of terms replacing each instance of $\mathbb{Q}$ with $\Lambda$ . Since $\Lambda$ is constant, $q_{\Lambda}(h_{1})-q_{\Lambda}(h_{2})=0$ . Because each remaining term has at least one factor of $\mathbb{Q}$ , (11.36) now follows from the boundedness of $P_{\xi},P^{\pm}$ in Propositions 11.1–11.3, the boundedness of $\mathbb{L}$ from Lemma 10.14, along with Items (A) and (C) of Corollary 10.8. ∎

12. Gluing

This final section concludes the proof Theorem 1.6 using the cyclic iteration from Section 9. The iteration leads to glued solutions of a “Seiberg–Witten eigenvector” equation for every pair of parameters $(\varepsilon,\tau)$ . These fail to solve the true Seiberg–Witten equations by a 1-dimensional obstruction coming from a multiple of the eigenvector. The condition that this obstruction vanish defines $\tau(\varepsilon)$ implicitly as a function of $\varepsilon$ , completing the proof of the theorem.

12.1. Glued Configurations

Proposition 10.1 and the Banach fixed-point theorem (with smooth dependence on parameters) immediately imply the following.

Corollary 12.1.

For every pair $(\varepsilon,\tau)\in(-\varepsilon_{0},\varepsilon_{0})\times(-\tau_{0},\tau_{0})$ with $\varepsilon_{0},\tau_{0}$ sufficiently small, there exist tuples $(\mathcal{Z},\Phi,A,\mu)$ depending smoothly on $\varepsilon,\tau$ where

(1)

$\mathcal{Z}(\varepsilon,\tau)=\mathcal{Z}_{\tau,\xi(\varepsilon,\tau)}$ is the singular set arising from a linearized deformation $\xi(\varepsilon,\tau)\in C^{\infty}(\mathcal{Z}_{\tau},N\mathcal{Z}_{\tau})$ ,
(2)

$\mu(\varepsilon,\tau)\in\mathbb{R}$ , and
(3)

$(\Phi(\varepsilon,\tau),A(\varepsilon,\tau))\in C^{\infty}(Y;S_{E}\oplus\Omega)$

that satisfy

\text{SW}\left(\frac{\Phi(\varepsilon,\tau)}{\varepsilon},A(\varepsilon,\tau)\right)=(\Lambda(\tau)+\mu(\varepsilon,\tau)\chi^{-}_{\varepsilon}\cdot\frac{\Phi_{\tau,\xi(\varepsilon,\tau)}}{\varepsilon}

(12.1)

where $\Phi_{\tau,\xi(\varepsilon,\tau)}$ is as in Definition 8.3, and $\chi^{-}_{\varepsilon}=\chi^{-}_{\varepsilon,\tau,\xi(\varepsilon,\tau)}$ cut-off function (8.10).

Proof.

Part (C) of Proposition 10.1 and the Banach fixed-point theorem give a smoothly varying family of fixed points $h=(\xi,\varphi,a,\psi,b,\mu,u)\in\mathbb{H}^{1}_{e}\times\mathcal{X}$ . By (B) of Proposition 10.1, these satisfy, $\overline{\mathbb{SW}}_{\Lambda}(h)=0$ . Setting

(\Phi(\varepsilon,\tau),A(\varepsilon,\tau))=(\Phi_{1}({\varepsilon,\tau,\xi}),A_{1}({\varepsilon,\tau,\xi}))\ +\ \chi_{\varepsilon}^{+}(\varphi,a)\ +\ \chi_{\varepsilon}^{-}(\psi+u,b),

where $\xi=\xi(\varepsilon,\tau)$ , the conclusion follows the Definition (8.12) of $\overline{\mathbb{SW}}$ and Corollary 8.5 with right-hand side $g=\varepsilon^{-1}\chi_{\varepsilon}^{-}\cdot\Lambda(\tau)\Phi_{\tau,\xi(\varepsilon,\tau)}$ , which satisfies $g\in C^{\infty}(Y)$ as in the proof of that corollary. ∎

12.2. The One-Dimensional Obstruction

The configurations (12.1) solve the Seiberg–Witten equations if and only if

\Lambda(\tau)+\mu(\varepsilon,\tau)=0

(12.2)

is satisfied. The next lemma shows that the assumption of transverse spectral crossing (Definition 1.5) means the condition (12.2) defines $\tau$ implicitly as a function of $\varepsilon$ .

Lemma 12.2.

The solutions $(\Phi(\varepsilon,\tau),A(\varepsilon,\tau),\mu(\varepsilon,\tau))$ from Proposition 12.1 depend smoothly on $(\varepsilon,\tau)\in(0,\varepsilon_{0})\times(-\tau_{0},\tau_{0})$ . Moreover,

|\mu(\varepsilon,\tau)|\ +\ |\partial_{\tau}\mu(\varepsilon,\tau)|\leq C\varepsilon^{2/3}.

(12.3)

holds uniformly.

Proof.

The standard proof of the Banach fixed point theorem with smooth dependence on parameters shows that a smooth family of fixed points $h_{\tau}$ satisfy

\|\partial_{\tau}h(\tau)\|_{\mathcal{H}}\leq C\|(\partial_{\tau}T_{\tau})h(\tau)\|_{\mathcal{H}}.

Because of the weight on $\mu$ in (10.6), it therefore suffices to show that

\|(\partial_{\tau}T_{\tau})h(\tau)\|_{\mathcal{H}}\leq C\varepsilon^{-1/4}.

(12.4)

By Theorem 7.5 Item (I), Lemma 7.11, the expression Proposition 8.7 (using Item (II) of Theorem 7.5), and the proofs of Propositions 9.1, 9.2, and 9.3 (which may be repeated equally well with $(\partial_{\tau}\Phi_{1},\partial_{\tau}A_{1})$ ) show that

\|\partial_{\tau}\mathcal{L}_{(\Phi_{1},A_{1})}^{+}\|\ +\ \|\partial_{\tau}\mathcal{L}_{(\Phi_{1},A_{1})}^{-}\|+\|\partial_{\tau}\text{d}\overline{\mathbb{SW}}_{h_{1}}(\xi)\|\leq C\varepsilon^{-\gamma}

are uniformly bounded as maps the $H^{1,\pm}_{\varepsilon}\to L^{2}$ and $\mathfrak{W}\to L^{2}$ . A similar argument applies to $\partial_{\tau}\mathbb{Q}$ . Together with, Item (I) of Theorem 7.5, these show that $\partial_{\tau}\overline{\mathbb{SW}}_{\Lambda}$ is bounded by $C\varepsilon^{-\gamma}$ at $h\in\mathcal{C}$ . Differentiating $\text{Id}=P_{\tau}^{-1}P_{\tau}$ for all three parametrices and using the bounds from Propositions 11.1–11.3 yields bounds on $\partial_{\tau}P_{\xi},\partial_{\tau}P^{\pm}$ by at most $C\varepsilon^{-1/12-\gamma}$ . Using the product rule on (11.32) and combining these yields (12.4). ∎

The proof of the following lemma is essentially identical the treatment of [60, Eq. (10.6)].

Lemma 12.3.

If the family of parameters $p_{\tau}=(g_{\tau},B_{\tau})$ has a transverse spectral crossing, then (12.2) implicitly defines a function $\tau(\varepsilon)$ so that

\Lambda(\tau(\varepsilon))+\mu(\varepsilon,\tau(\varepsilon))=0

for $\varepsilon\in(0,\varepsilon_{0})$ , and either $\tau(\varepsilon)>0$ or $\tau(\varepsilon)<0$ .

Proof.

The assumption that $\tau=0$ is a transverse spectral crossing means that $\Lambda(0)=0$ and $\dot{\Lambda}(0)\neq 0$ . By the inverse function theorem, there is an inverse $\Lambda^{-1}$ defined on an open neighborhood of $\tau=0$ . Set $\Gamma(\varepsilon)=\Lambda\circ\tau(\varepsilon)$ , so that (12.2) becomes the condition that

\Gamma(\varepsilon)+\mu(\varepsilon,\Lambda^{-1}\circ\Gamma(\varepsilon))=0.

This is an equation for a real number $\Gamma$ depending on a parameter $\varepsilon$ . (12.3) implies that this equation can be solved for $\Gamma\in\mathbb{R}$ using the Inverse Function Theorem with smooth dependence on the parameter $\varepsilon$ ; thus $\tau(\varepsilon)=\Lambda^{-1}\circ\Gamma(\varepsilon)$ solves (12.2).

To see the sign of $\tau(\varepsilon)$ , we expand (12.2) in series. From the proof of Proposition 10.1 implies that $\mu$ is a sum $\mu(\varepsilon,\tau)=\mu_{1}(\varepsilon,\tau)+\delta\mu_{2}(\varepsilon,\tau)+\delta^{2}\mu_{3}(\varepsilon,\tau)+\ldots$ . Meanwhile $\Lambda(\tau)$ is smooth and may be expanded in Taylor series.

	$\displaystyle 0$	$\displaystyle=$	$\displaystyle\Lambda(\tau)+\mu(\varepsilon,\tau)$
		$\displaystyle=$	$\displaystyle\dot{\Lambda}(0)\tau+\varepsilon^{2/3}\overline{\mu}_{1}(\varepsilon,0)+O(\tau^{2})+O(\varepsilon^{2/3}\delta)+O(\varepsilon^{2/3}\tau),$

where $\mu_{i}=\varepsilon^{2/3}\overline{\mu}_{i}$ . It follows that for $\varepsilon$ sufficiently small, $\tau$ has the opposite sign of $\frac{\overline{\mu}_{1}(\varepsilon,0)}{\dot{\Lambda}(0)}$ . ∎

Proof of Theorem 1.6.

Part (1) follows directly from Lemma 3.8. Along the family of parameters $\tau(\varepsilon)$ satisfying (12.2) constructed in Lemma 12.3, the glued configurations of Proposition 12.1 satisfy the (extended) Seiberg–Witten equations. Integrating by parts shows the $0$ -form component $a_{0}$ vanishes, and setting $(\Psi_{\varepsilon},A_{\varepsilon})=(\varepsilon^{-1}{\Phi(\varepsilon,\tau(\varepsilon))}\ ,\ A(\varepsilon,\tau(\varepsilon)))$ yields the solutions in Part (2).

The glued configurations have $\|\Psi_{\varepsilon}\|_{L^{2}}=\varepsilon^{-1}+O(\varepsilon^{-1/24-\gamma})$ , and this norm depends smoothly on $\varepsilon$ . By re-parameterizing, it may be assumed that $\|\varepsilon\Psi_{\varepsilon}\|_{L^{2}}=1$ as in (1.13). As $\varepsilon\to 0$ , Theorem 3.2 shows that after renormalization, $(\Psi_{\varepsilon},A_{\varepsilon})$ converges in $C^{\infty}_{loc}$ to a $\mathbb{Z}_{2}$ -harmonic spinor for the parameter $p_{0}=(g_{0},B_{0})$ and that $\|\varepsilon\Psi_{\varepsilon}\|\to|\Phi_{0}|$ in $C^{0,\alpha}$ . Since $(\mathcal{Z}_{0},A_{0},\Phi_{0})$ is regular, it is the unique $\mathbb{Z}_{2}$ -harmonic spinor for this parameter and must therefore be the limit. This establishes Part (3). ∎

Remark 12.4.

Recalling the construction of $\mu$ from Step 5 of the proof of Proposition 11.2 with $N=1$ , the leading order obstruction in the proof of Lemma 12.3 appears to be

\varepsilon^{2/3}\overline{\mu}_{1}(\varepsilon,0)=\varepsilon\langle\text{d}_{h_{0}}\not{\mathbb{D}}(\eta_{1},0)\ +\ d\chi^{+}.\varphi_{1},\Phi_{0}\rangle,

(12.5)

where $\varphi_{1}$ is as in Item (II) of Theorem 7.5 for $\tau=0$ . In fact, since $\Phi_{0}=O(r^{1/2})$ , simple integration shows that the second term is smaller by a factor of $\varepsilon^{1/2}$ than the bound on $\text{d}_{h_{0}}\not{\mathbb{D}}(\eta_{1},0)$ obtained in the proof of Proposition 11.1. The latter bound, however, might not be optimal.

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$\displaystyle\Big{\\|}\underline{\chi}[\eta^{\prime\prime}]\sigma_{j}\Phi\Big{\\|}^{2}_{L^{2}(N_{r_{0}})}$	$\displaystyle=$	$\displaystyle\int_{D_{\mathbb{r}_{0}}}\int_{S^{1}}\|\underline{\chi}[\eta^{\prime\prime}]\sigma_{j}\Phi\|^{2}dt\ rd\theta dr$
	$\displaystyle\leq$	$\displaystyle\int_{D_{r_{0}}}\|C_{0}r^{\alpha}\|^{2}\cdot\int_{S^{1}}\|\underline{\chi}[\eta^{\prime\prime}]\|^{2}dt\ rd\theta dr$
	$\displaystyle\leq$	$\displaystyle C_{1}^{2}\int_{D_{r_{0}}}\|\sum_{p\in\mathbb{Z}}\eta_{p}(ip)^{2}e^{ipt}\chi_{p}(r)\|^{2}r^{2\alpha+1}d\theta drdt$

$\displaystyle\Big{\\|}\underline{\chi}[\eta^{\prime\prime}]\sigma_{j}\psi r^{\beta}\Big{\\|}^{2}_{L^{2}(N_{r_{0}})}$	$\displaystyle=$	$\displaystyle\sum_{n\geq 1}\int_{A_{n}}\int_{S^{1}}\|\underline{\chi}[\eta^{\prime\prime}]\sigma_{j}\psi\|^{2}r^{2\beta}dt\ rd\theta dr$
	$\displaystyle\leq$	$\displaystyle\sum_{n\geq 1}\sup_{A_{n}}\\|\underline{\chi}(\eta^{\prime\prime})\\|_{C^{0}(S^{1})}\int_{A_{n}}\int_{S^{1}}\|\psi\|^{2}r^{2\beta}dt\ rd\theta dr$
	$\displaystyle\leq$	$\displaystyle\sum_{n\geq 1}\sup_{A_{n}}\\|\underline{\chi}(\eta^{\prime\prime})\\|^{2}_{C^{0}(S^{1})}\sup_{A_{n}}r^{2+2\beta}\\|\tfrac{\psi}{r}\\|_{L^{2}(A_{n})}.$

	$\displaystyle\Big{\\|}\underline{\chi^{\prime}}[\eta]\underline{\chi^{\prime}}[\xi]\nabla\Phi\Big{\\|}_{L^{2}(N_{r_{0}})}$	$\displaystyle\leq$	$\displaystyle C\\|\eta\\|_{3/2+\underline{\gamma}}\cdot\\|\xi\\|_{1-\alpha}$
	$\displaystyle\Big{\\|}\underline{\chi^{\prime\prime}}[\eta]\underline{\chi^{\prime}}[\xi]\Phi\Big{\\|}_{L^{2}(N_{r_{0}})}+\Big{\\|}\underline{\chi^{\prime}}[\eta]\underline{\chi^{\prime\prime}}[\xi]\Phi\Big{\\|}_{L^{2}(N_{r_{0}})}$	$\displaystyle\leq$	$\displaystyle C(\\|\eta\\|_{3/2+\underline{\gamma}}\cdot\\|\xi\\|_{1-\alpha}+\\|\eta\\|_{1-\alpha}\cdot\\|\xi\\|_{3/2+\underline{\gamma}})$

	$\displaystyle\Big{\\|}\underline{\chi^{\prime}}[\eta]\underline{\chi^{\prime}}[\xi]r^{\beta}\nabla\psi\Big{\\|}_{L^{2}(N_{r_{0}})}$	$\displaystyle\leq$	$\displaystyle C\\|\eta\\|_{3/2+\underline{\gamma}-\beta}\\|\xi\\|_{3/2+\underline{\gamma}}\\|\tfrac{\psi}{r}\\|_{L^{2}}$
	$\displaystyle\Big{\\|}\underline{\chi^{\prime\prime}}[\eta]\underline{\chi^{\prime}}[\xi]r^{\beta}\psi\Big{\\|}_{L^{2}(N_{r_{0}})}+\Big{\\|}\underline{\chi^{\prime}}[\eta]\underline{\chi^{\prime\prime}}[\xi]r^{\beta}\psi\Big{\\|}_{L^{2}(N_{r_{0}})}$	$\displaystyle\leq$	$\displaystyle C(\\|\xi\\|_{3/2+\underline{\gamma}}\\|\eta\\|_{3/2+\underline{\gamma}-\beta}+\\|\eta\\|_{3/2+\underline{\gamma}}\\|\xi\\|_{3/2+\underline{\gamma}-\beta})\\|\tfrac{\psi}{r}\\|_{L^{2}}$

	$\displaystyle\\|(\varphi_{1},\varphi_{2},a)\\|_{H^{1,-}_{\varepsilon,\nu}}$	$\displaystyle:=$	$\displaystyle\left(\\|\varphi_{1}\\|_{r^{1-\nu}H^{1}_{e}(Y)}^{2}\ +\ \int_{Y^{-}}\left(\|\nabla\varphi_{2}\|^{2}+\|\nabla a\|^{2}+\frac{\|\varphi_{2}\|^{2}\|\Phi\|^{2}}{\varepsilon^{2}}+\frac{\|a\|^{2}\|\Phi\|^{2}}{\varepsilon^{2}}\right)R_{\varepsilon}^{2\nu}\ dV\right)^{1/2}$
	$\displaystyle\\|(\varphi_{1},\varphi_{2},a)\\|_{L^{2,-}_{\varepsilon,\nu}}$	$\displaystyle:=$	$\displaystyle\left(\\|\varphi_{1}\\|^{2}_{r^{-\nu}L^{2}(Y)}\ +\ \int_{Y^{-}}\left(\|\varphi\|^{2}+\|a\|^{2}\right)R_{\varepsilon}^{2\nu}\ dV\right)^{1/2}$

Gluing ℤ2\mathbb{Z}_{2}-Harmonic Spinors and Seiberg–Witten Monopoles on 3-Manifolds

Abstract.

1. Introduction

1.1. The Seiberg–Witten Equations

1.2. ℤ2\mathbb{Z}_{2}-Harmonic spinors

1.3. The Gluing Problem

1.4. Main Results

Definition 1.1.

Theorem 1.2.

Definition 1.3.

Theorem 1.4.

Definition 1.5.

Theorem 1.6.

Remark 1.7.

Remark 1.8.

Remark 1.9.

1.5. Wall-Crossing Formulas

Theorem 1.10.

Remark 1.11.

1.6. Outline

Acknowledgements

2. Gluing by the Alternating Method

2.1. The Structure of Gluing Problems

Example 2.1.

2.2. The Alternating Method

Remark 2.2.

2.3. Gluing as Non-linear Excision

Proposition 2.3 (Excision).

Proof.

Proposition 2.4.

Proof.

Remark 2.5.

2.4. The Semi-Fredholm Alternating Method

2.5. Key Technical Difficulties

2.5.1. Loss of Regularity

2.5.2. Non-local Deformation Operator

2.5.3. Non-Uniform Invertibility

Appendix of Gluing Parameters

Important Remark.

3. ℤ2\mathbb{Z}_{2}-Harmonic Spinors and Compactness

3.1. Compactness Theorem

Definition 3.1.

Theorem 3.2.

Remark 3.3.

Remark 3.4.

3.2. The Haydys Corresondence

Lemma 3.5.

Corollary 3.6.

Proof.

Remark 3.7.

3.3. Recovering Spinc\text{Spin}^{c} Structures

Lemma 3.8.

3.4. Adapted Coordinates

Definition 3.9.

Lemma 3.10.

4. The Singular Linearization

4.1. Singular Linearization

Lemma 4.1.

Lemma 4.2.

Proof.

4.2. The Singular Dirac Operator

Definition 4.3.

Lemma 4.4.

Lemma 4.5.

4.3. Polynomial Decay

Definition 4.6.

Lemma 4.7.

Proof.

Remark 4.8.

5. The Obstruction Bundle

5.1. The Obstruction Basis

Definition 5.1.

Proposition 5.2.

Proof.

Proposition 5.3.

Lemma 5.4.

Proof.

5.2. The Surjective Weights

Lemma 5.5.

Proof.

Gluing $\mathbb{Z}_{2}$ -Harmonic Spinors and Seiberg–Witten Monopoles on 3-Manifolds

1.2. $\mathbb{Z}_{2}$ -Harmonic spinors

3. $\mathbb{Z}_{2}$ -Harmonic Spinors and Compactness

3.3. Recovering $\text{Spin}^{c}$ Structures