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

Gregory J. Parker  Department of Mathematics, Stanford University
<EMAIL_ADDRESS>
###### 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.

###### Contents

  1. 1 Introduction
  2. 2 Gluing by the Alternating Method
  3. 3 $\mathbb{Z}_{2}$-Harmonic Spinors and Compactness
  4. 4 The Singular Linearization
  5. 5 The Obstruction Bundle
  6. 6 Deformations of Singular Sets
  7. 7 Concentrating Local Solutions
  8. 8 Universal Seiberg–Witten Equations
  9. 9 Relating Deformation Operators
  10. 10 Contraction Subspaces
  11. 11 The Alternating Iteration
  12. 12 Gluing

##  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 Spinc 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:

  1. (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.

  2. (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.

  3. (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:

$\left\\{\begin{matrix}\text{error with \ }\\\ \text{supp. on
}Y^{-}\end{matrix}\right\\}$$\left\\{\begin{matrix}\text{error on }Y^{-}\\\
\perp\text{obstruction}\end{matrix}\right\\}$$\left\\{\begin{matrix}\text{error
with \ }\\\ \text{supp. on }Y^{+}\end{matrix}\right\\}.$deform
$\mathcal{Z}$$\begin{matrix}\text{solve on $Y^{+}$}\end{matrix}$solve on
$Y^{-}$ (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,

$\Phi_{N}\ \ $$\ \ \Phi_{N+1}:=\Phi_{N}+\delta\Phi_{N}$correct

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:

$\left\\{\begin{matrix}\mathfrak{F}(\Phi_{N})=\mathfrak{e}_{N}\\\
\text{supp}(\mathfrak{e}_{N})\subseteq
Y^{-}\end{matrix}\right\\}$$\left\\{\begin{matrix}\mathfrak{F}(\Psi_{N})=\mathfrak{e}^{\prime}_{N}\\\
\text{supp}(\mathfrak{e}^{\prime}_{N})\subseteq
Y^{+}\end{matrix}\right\\}$$\begin{matrix}\text{solve on $Y^{+}$}\\\ N\mapsto
N+1,\end{matrix}$solve on $Y^{-}$

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. (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. (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. (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. (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 $\mathfrak{F}(\Phi_{0})=0$.

$\chi^{+}$$\chi^{-}$$Y^{+}$$Y^{-}$$\mathfrak{e}_{N}$$\mathfrak{e}^{\prime}_{N}$$\mathfrak{e}_{N+1}$
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

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

by

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

  1. (I’)

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

  2. (II’)

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

  3. (III’)

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

  4. (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:

$\left\\{\begin{matrix}\mathfrak{F}(\xi_{N},\Phi_{N})=\mathfrak{e}_{N}\\\
\text{supp}(\mathfrak{e}_{N})\subseteq
Y^{-}\end{matrix}\right\\}$$\left\\{\begin{matrix}\mathfrak{F}(\xi_{N+1},\Phi_{N})=\mathfrak{e}^{\prime}_{N}\\\
\Pi_{0}(\mathfrak{e}_{N}^{\prime})=0\end{matrix}\right\\}$$\left\\{\begin{matrix}\mathfrak{F}(\xi_{N+1},\Psi_{N})=\mathfrak{e}^{\prime\prime}_{N}\\\
\text{supp}(\mathfrak{e}_{N}^{\prime\prime})\subseteq
Y^{+}\end{matrix}\right\\}.$$\text{solve on }Y^{+}$adjust
$\xi$$\begin{matrix}N\mapsto N+1\end{matrix}\ \ \ $solve on $Y^{-}$ (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 $\mathcal{L}^{-}$.

  1. (0)

Let $\xi$ denote the unique solution of

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

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

  2. (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. (1)

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

  2. (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. (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. (1)

The first chern class satisfies

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

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

  2. (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 $-\tfrac{1}{2}<\nu<\tfrac{1}{2}$,

  1. (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.)

  2. (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.

  3. (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.

  1. (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 $N\geq 2$.

  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. (1)

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

  2. (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. (3)