# Concentrating Dirac Operators and Generalized Seiberg-Witten Equations

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

This article studies a class of Dirac operators of the form
$D_{\varepsilon}=D+\varepsilon^{-1}\mathcal{A}$, where $\mathcal{A}$ is a
zeroth order perturbation vanishing on a subbundle. When $\mathcal{A}$
satisfies certain additional assumptions, solutions of the Dirac equation have
a concentration property in the limit $\varepsilon\to 0$: components of the
solution orthogonal to $\ker(\mathcal{A})$ decay exponentially away from the
locus $\mathcal{Z}$ where the rank of $\ker(\mathcal{A})$ jumps up. These
results are extended to a class of non-linear Dirac equations.

This framework is then applied to study the compactness properties of moduli
spaces of solutions to generalized Seiberg-Witten equations. In particular, it
is shown that for sequences of solutions which converge weakly to a
$\mathbb{Z}_{2}$-harmonic spinor, certain components of the solutions
concentrate exponentially around the singular set of the
$\mathbb{Z}_{2}$-harmonic spinor. Using these results, the weak convergence to
$\mathbb{Z}_{2}$-harmonic spinors proved in existing convergence theorems
([38, 37, 16, 40, 46]) is improved to $C^{\infty}_{loc}$.

###### Contents

  1. 1 Introduction
  2. 2 Concentrating Dirac Operators
  3. 3 Non-Linear Concentrating Dirac Equations
  4. 4 Generalized Seiberg-Witten Equations
  5. 5 Concentration Properties of Generalized Seiberg-Witten Equations
  6. 6 Bootstrapping
  7. A An Extension in $n=3$ Dimensions
  8. B Estimates for the Green’s Function

##  1\. Introduction

Let $(Y,g)$ be a Riemannian manifold of dimension $n\geq 3$, and
$D:\Gamma(E)\to\Gamma(E)$ a Dirac operator on sections of a Clifford module
$E\to Y$, i.e. a first-order elliptic operator whose principal symbol
satisfies $\sigma_{D}^{2}=-\text{Id}$. A 1-parameter family of Dirac operators
displaying a concentration property or, more succinctly, a Concentrating Dirac
Operator (sometimes called a localizing Dirac operator) is a parameterized
perturbation

$D_{\varepsilon}=D+\tfrac{1}{\varepsilon}\mathcal{A}$ (1.1)

of $D$ by positive scalings of a zeroth order operator
$\mathcal{A}\in\text{End}(E)$ such that the support of solutions concentrates
along a distinguished collection of submanifolds $\mathcal{Z}\subset Y$ as
$\varepsilon\to 0$. Concentrating Dirac operators were introduced to the
mathematical literature by Witten’s celebrated work on Morse theory [47],
although they were familiar to physicists for decades prior to this. Since
then, concentrating Dirac operators have been employed to give geometric
proofs of many results in index theory [18, 19, 28, 23, 6, 13, 51], and
geometric quantization [9, 11, 10, 44].

Concentrating Dirac operators have also played a significant role in Seiberg-
Witten gauge theory. It is an observation due to C. Taubes that the
linearization of the Seiberg-Witten equations behaves as a concentrating Dirac
operator in certain limits. This perspective is central to Taubes’s celebrated
work “SW=Gr” relating the Seiberg-Witten and Gromov invariants of symplectic
4-manifolds [31, 33, 32, 35, 34], and to his resolution of the Weinstein
Conjecture [36]. In these situations, the role of the perturbation
$\mathcal{A}$ is occupied by a large multiple of the symplectic or contact
form, and the solutions concentrate along submanifolds of codimension 2 which
Taubes proves are the pseudo-holomorphic curves enumerated by the Gromov
invariant or the Reeb orbits whose existence was postulated by Weinstein in
the two cases respectively.

Concentrating Dirac operators are also relevant in more recent work of Taubes
and others on the compactness of moduli spaces for generalized Seiberg-Witten
theories. For a specified compact Lie group $G$, a system of generalized
Seiberg-Witten equations on a 3 or 4-dimensional manifold is a system of
first-order non-linear PDEs for a pair $(\Psi,A)$ of a spinor $\Psi$ and a
connection $A$ on a principal $G$-bundle which has the schematic form

$\displaystyle\not{D}_{A}\Psi$ $\displaystyle=$ $\displaystyle 0$ (1.2)
$\displaystyle\star F_{A}$ $\displaystyle=$
$\displaystyle-\tfrac{1}{2}\mu(\Psi,\Psi)$ (1.3)

where $\not{D}_{A}$ (resp. $\not{D}_{A}^{+}$ on a 4-manifold) is the Dirac
operator twisted by the connection $A$, $F_{A}$ (resp. $F_{A}^{+}$) is the
curvature of $A$ (resp. the self-dual component thereof), and $\mu$ is a
pointwise quadratic map 111Note that my convention for the sign of $\mu$
differs from that used by many authors. That is to say, I denote by $-\mu$
what others denote by $\mu$; the equations (thus their relevant compactness
properties) are the same.. Examples include the Vafa-Witten equations [41, 30,
29], the Kapustin-Witten equations [42, 21, 22, 48], the
$\text{SL}(2,\mathbb{C})$ anti-Self-Dual Yang-Mills equations [38, 37], the
Seiberg-Witten equations with multiple spinors [16, 40], and the ADHM Seiberg-
Witten Equations [46]. The reader is referred to [49, 5, 2, 15, 45] for
discussions of conjectures relating these equations to the geometry of
manifolds and to other gauge theories. The main barrier to progress on all of
these conjectures is the lack of a well-understood compactification for the
moduli space of solutions: unlike for the standard Seiberg-Witten equations,
these more general theories do not admit compact moduli space of solutions and
there instead may be sequences of solutions for which the $L^{2}$-norm of the
spinor diverges. For such sequences, a renormalized (sub)sequence must
converge to a $\mathbb{Z}_{2}$-harmonic spinor or more general Fueter section
(proved for each respective equation in the above references).

A promising approach to constructing well-understood compactifications for
these moduli spaces is to attach boundary strata consisting of
$\mathbb{Z}_{2}$-harmonic spinors or Fueter sections. A necessary step in
showing the suitability of any putative compactification formed in this way is
to construct boundary charts for the moduli space. Constructing these charts
requires gluing results (see [3, 25, 26, 27] for progress in this direction).
Even after showing appropriate gluing results, however, the existence of the
desired charts does not follow immediately since it is not a priori clear that
any sequence approaching a given boundary point necessary arises from such a
gluing—in other words, the gluing may only construct a subset of the desired
chart. In order for the compactifications to be well-behaved, extraneous
sequences not captured by gluing constructions must be ruled out: this problem
is known as the surjectivity of gluing. The convergence to
$\mathbb{Z}_{2}$-harmonic spinors and Fueter sections proved in extant
convergence results is rather weak (see Section 4.2 for a precise statement);
in particular it leaves open the possibility for the existence of sequences
converging to a $\mathbb{Z}_{2}$-harmonic spinor or Fueter section in a space
of low regularity that would necessarily elude gluing constructions (which
automatically have convergence in higher-regularity spaces).

As explained in Section 6, attempts to bootstrap the convergence to
$\mathbb{Z}_{2}$-harmonic spinors using standard methods are doomed to fail by
the accumulation of powers of the spinor’s diverging $L^{2}$-norm in the
relevant estimates, and more robust techniques are required. The theory of
concentrating Dirac operators supplies these techniques: for a sequence of
solutions to (1.2–1.3) converging to a $\mathbb{Z}_{2}$-harmonic spinor, the
linearization of the equations behaves as a particular type of concentrating
Dirac operator, with the diverging $L^{2}$-norm of the spinor occupying the
role of $\varepsilon^{-1}$ in the expression (1.1). Although the perspective
and philosophy of concentrating Dirac operators implicitly informs the
approach of [38, 37, 16, 40, 46, 41, 42], there is more to be gained by making
the connection precise.

This article is best viewed as consisting of three parts. First, Sections 2–3
extend results about the behavior of solutions to concentrating Dirac
operators to a larger class of operators than previously studied, and to Dirac
operators with certain types of non-linearities. Second, Sections 4–5, show
that the linearization of generalized Seiberg-Witten equations, by design,
fall into this new class of operators. Finally, Section 6 uses these results
to improve convergence results for sequences of solutions to generalized
Seiberg-Witten equations in [38, 37, 16, 40, 46] to the $C^{\infty}_{loc}$
topology. Although this bootstrapping is the main application given here, the
exponential convergence results obtained from the results of Sections 2–3 are
far stronger than is necessary simply for bootstrapping: they additionally
provide a more precise geometric picture of how the convergence occurs. In
particular, as discussed in Appendix A, they imply that there is an expected
invariant length scale for the concentration of curvature along the singular
set. The results and techniques developed here may therefore be helpful in
addressing questions related to gluing and the surjectivity of gluing (in
fact, the present work grew out of the necessity for the full strength of
these results in the gluing construction of [25, 27]).

###  1.1. Main Results

The key property that leads to concentration as $\varepsilon\to 0$ is a
commutation relation of $\mathcal{A}$ and the principal symbol $\sigma_{D}$ of
the unperturbed operator $D$. These are required to satisfy

$\mathcal{A}^{\star}\sigma_{D}(\xi)=\sigma_{D}^{\star}(\xi)\mathcal{A}$

for any $\xi\in T^{\star}Y$. In previous work on concentrating Dirac
operators, it has typically been assumed that $\mathcal{A}$ is invertible on
an open dense subset of $Y$; here this assumption is weakened to include the
case that $\mathcal{A}$ vanishes identically along a subbundle of $E$. More
precisely, let $r$ denote the maximal rank of $\mathcal{A}$, and set
$\mathcal{Z}=\\{y\ |\ \text{rank}(\mathcal{A}(y))<r\\}\subseteq Y$. Assume
that there is a parallel decomposition
$E|_{Y-\mathcal{Z}}=\mathfrak{N}\oplus\mathfrak{H}$ of the Clifford module’s
restriction to $Y-\mathcal{Z}$, in which $\mathcal{A}$ takes the form

$\mathcal{A}=\begin{pmatrix}0&0\\\ 0&A_{\mathfrak{H}}\end{pmatrix}.$ (1.4)

Dirac operators satisfying these two assumptions will be referred to as
concentrating Dirac operators with fixed degeneracy (see Definition 2.1 for a
more precise definition).

The first main result shows that the
$\mathfrak{H}=\ker(\mathcal{A})^{\perp}$-components of a solution to the Dirac
equation concentrate along $\mathcal{Z}$ and decay exponentially away from it:

######  Theorem 1.1.

Suppose that $D_{\varepsilon}$ is a concentrating Dirac operator with fixed
degeneracy, and that $\mathfrak{q}\in\Gamma(E)$ is a solution i.e.

$(D+\tfrac{1}{\varepsilon}\mathcal{A})\mathfrak{q}=0.$

For any compact subset $K\Subset Y-\mathcal{Z}$, let
$R_{K}=\text{dist}(K,\mathcal{Z})$. Then, there exists a compact subset
$K^{\prime}$ with $K\Subset K^{\prime}\Subset Y-\mathcal{Z}$, and constants
$C,c$ independent of $K$ such that for $\varepsilon$ sufficiently small, the
components of $\mathfrak{q}$ in the subbundle $\mathfrak{H}$ obey

$\|\pi_{\mathfrak{H}}(\mathfrak{q})\|_{C^{0}(K)}\leq\frac{C}{R_{K}^{n/2}}\
\text{Exp}\left(-\frac{c\Lambda_{K}}{\varepsilon}R_{K}\right)\|\mathfrak{q}\|_{L^{1,2}(K^{\prime})}$
(1.5)

where $\Lambda_{K}=\underset{y\in
K,v\in\mathfrak{H}_{y}}{\inf}\frac{\|\mathcal{A}v\|}{\|v\|}$, i.e.
$\Lambda_{K}$ is the minimum fiberwise norm of $A_{\mathfrak{H}}^{-1}$ over
$K$.

The proof of the above result occupies Section 2. It is worth remarking that
the proof is easily adapted to show that the conclusion of Theorem 1.1 also
holds if $\mathfrak{q}$ is an eigenspinor of $D_{\varepsilon}$ with eigenvalue
$\lambda<\tfrac{c_{K}}{\varepsilon}$ for a small constant $c_{K}$ depending
depending on the compact set $K$, though this extension is not needed here and
the details of the proof are omitted.

The arguments in the proof of Theorem 1.1 can be extended to the following
class of non-linear concentrating Dirac operators. Suppose that
$Q:\Gamma(E)\to\Gamma(E)$ is a pointwise non-linear bundle map that takes the
form $Q(\mathfrak{q})=Q_{1}(\mathfrak{q})\pi_{\mathfrak{H}}(\mathfrak{q})$
where $Q_{1}$ is again a pointwise non-linear bundle map and
$\pi_{\mathfrak{H}}$ the projection onto the subbundle $\mathfrak{H}$. Said
more simply, it is assumed that the non-linearity has at least a linear factor
in the $\mathfrak{H}$ components. Additionally, assume that
$Q_{1}(\mathfrak{q})$ obeys the same commutation relation as $\mathcal{A}$:
that is,
$Q_{1}(\mathfrak{q})^{\star}\sigma_{D}=\sigma_{D}^{\star}Q_{1}(\mathfrak{q})$.
The second main result is the following corollary, whose proof occupies
Section 3.

######  Corollary 1.2.

If $\mathfrak{q}$ solves the non-linear equation

$(D+\tfrac{1}{\varepsilon}\mathcal{A})\mathfrak{q}+Q(\mathfrak{q})=f$ (1.6)

where $D,\mathcal{A}$ are as in Theorem 1.1,
$f\in\Gamma(\mathfrak{H}^{\perp})$, and where
$Q(\mathfrak{q})=Q_{1}(\mathfrak{q})\pi_{\mathfrak{H}}(q)$ is of the above
form and satisfies

$\varepsilon\|Q_{1}(\mathfrak{q})\|_{L^{1,n}(K^{\prime})}\to 0\hskip
56.9055pt\varepsilon\|Q_{1}(\mathfrak{q})\|_{C^{0}(K^{\prime})}\to 0.$ (1.7)

Then the conclusion (1.5) of Theorem 1.1 holds.

The motivation for and application of the above results come from
investigations of the compactness of moduli spaces of solutions to generalized
Seiberg-Witten equations (see Sections Sections 4.1–4.2 below). As explained
in the introduction, it is known in many cases of interest that sequences of
solutions to (1.2–1.3) lacking bounded $L^{2}$ subsequences subconverge after
renormalization to a type of $\mathbb{Z}_{2}$-harmonic spinor. There are four
cases of interest:

Case (I): The two-spinor Seiberg-Witten equations on a compact 3-manifold $Y$
[16]

Case (II): The equations for a flat $\text{SL}(2,\mathbb{C})$ connection on a
compact 3-manifold $Y$[38, 46]

Case (III): The two-spinor Seiberg-Witten equations on a compact 4-manifold
$X$ [40]

Case (IV): The $\text{SL}(2,\mathbb{C})$ ASD equations on a compact 4-manifold
$X$ [37]

The precise statements of the known compactness results for these cases is
amalgamated in Theorem 4.10 in Section 4.2. A precise definition of
$\mathbb{Z}_{2}$-harmonic spinors is given in the subsequent Definition 4.11.

######  Theorem 1.3.

Suppose that a sequence $(\Phi_{i},A_{i},\varepsilon_{i})$ of re-normalized
solutions to a generalized Seiberg-Witten equation converge to a
$\mathbb{Z}_{2}$-harmonic spinor $(\mathcal{Z}_{0},A_{0},\Phi_{0})$ in the
sense of Theorem 4.10 for one of Cases (I)–(IV) above. Then, in the limit
$\varepsilon_{i}\to 0$

Case (I)–(III): The equations (1.2–1.3) behave as a non-linear concentrating
Dirac operator satisfying the assumptions of Corollary 1.2.

Case (IV) : If, in addition to the conclusion of Theorem 4.10, the sequence
$(\Phi_{i},A_{i})$ satisfies $A_{i}\to A_{0}$ in $L^{1,p}_{loc}$ and
$\Phi_{i}\to\Phi_{0}$ in $L^{2,p}_{loc}$ for any $p>2$, then the same
conclusion holds.

Additionally, in these situations

$\Phi_{i}\overset{C^{\infty}_{\text{loc}}}{\longrightarrow}\Phi_{0}\hskip
56.9055ptA_{i}\overset{C^{\infty}_{\text{loc}}}{\longrightarrow}A_{0}$

where local convergence mean on compact subsets $K\subseteq Y-\mathcal{Z}_{0}$
(resp. $X-\mathcal{Z}_{0}$ in Cases (III)–(IV)). Moreover, in all cases the
limiting connection satisfies $F_{A_{0}}=0$ (resp. $F_{A_{0}}^{+}=0$).

######  Remark 1.4.

In Cases (I)–(III), Theorem 1.3 applies directly to strengthen the results of
[38, 16, 40, 46]. In Case (IV), the additional requirement beyond the
conclusions of [37] that $A_{i}\to A_{0}$ in $L^{1,p}_{loc}$ and
$\Phi_{i}\to\Phi_{0}$ in $L^{2,p}_{loc}$ for some $p>2$ is necessary. This is
needed to overcome the quadratic non-linearity in $F_{A}^{+}$ which is
borderline for the relevant Sobolev embeddings in dimension 4 (this assumption
is not necessary in Case (III), which is abelian). It is unclear if this
slightly stronger convergence result than Theorem 4.10 can be proved by
extending the techniques of [37, 41], or if a new approach is required to
bridge the gap from the results therein to the point where Theorem 1.3
applies.

######  Remark 1.5.

[41] proves a version of Theorem 4.10 for the Vafa-Witten equations as well,
though this case is notably absent from Theorem 1.3. Although the equations
have a similar form to the equations of Cases (I)–(IV), it is not expected for
the Vafa-Witten equations that the sequence of connections $A_{i}$ should
converge. In particular, for these equations there exist sequences of
solutions such that the $L^{2}$-norm of the curvature over any compact set
with non-empty interior diverges. Solutions with this behavior are the subject
of forthcoming work of C. Taubes [43], and were known to E. Witten and others
prior to the work [41]. For sequences along which the connections $A_{i}$ do
happen to converge in $L^{1,p}_{loc}$ for some $p>2$, then the conclusions of
Theorem 1.3 apply—the proof is trivially different from Case (IV) and is
omitted.

######  Remark 1.6.

Also absent from the cases enumerated in Theorem 1.3 are the compactness
theorems for the $\text{ADHM}_{1,2}$ Seiberg-Witten equations established in
[46] and for Seiberg-Witten equations with $r>2$ spinors from [40, 16]. The
approach of Theorem 1.3 fails in these cases, because the
$\mathfrak{H}$-components of the equations are too strongly coupled to the
other components. This occurs in two slightly different ways: for the
$\text{ADHM}_{1,2}$ Seiberg-Witten equations, the linearized equations
properly fit into the framework of Theorem 1.1, but the non-linear terms do
not satisfy the hypotheses of Corollary 1.2. For the Seiberg-Witten equations
with $r>2$ spinors, the reverse occurs: the non-linear terms have the desired
form, but the splitting 1.4 fails to be parallel and the
$\mathfrak{H}$-components are too strongly coupled to the other components for
the linearized equations. It is an interesting task to investigate whether
Theorem 1.1 may be extended to the case where the splitting is not parallel.

## Acknowledgements

This work grew out of the authors Ph.D. thesis. The author is grateful to his
advisor Clifford Taubes for his support and suggestions. This work also
benefitted from conversations with Rafe Mazzeo, Aleksander Doan, and Thomas
Walpuski, and was supported by a National Science Foundation Graduate Research
Fellowship and by National Science Foundation Grant No. 2105512.

##  2\. Concentrating Dirac Operators

Let $(Y,g)$ be an n-dimensional Riemannian manifold (not necessarily compact),
and $E\to Y$ a real Clifford module. That is to say, $E$ is a real vector
bundle equipped with i) an inner product, denoted $\langle-,-\rangle$, ii) a
metric-compatible connection denoted $\nabla$, and iii) a Clifford
multiplication $\sigma_{D}:T^{*}Y\to\text{End}(E)$ obeying the Clifford
relation

$\sigma_{D}(\xi)\sigma_{D}(\eta)+\sigma_{D}(\xi)\sigma_{D}(\eta)=-2\langle\xi,\eta\rangle.$
(2.1)

If $n$ is even, we assume that there is a splitting $E=E^{+}\oplus E^{-}$ with
respect to which $\sigma_{D}$ is off-diagonal. Consider an
$\varepsilon$-parameterized family of Dirac operators
$D_{\varepsilon}:\Gamma(E)\to\Gamma(E)$ of the form

$D_{\varepsilon}\mathfrak{q}=\left(D+\frac{1}{\varepsilon}\mathcal{A}\right)\mathfrak{q}$
(2.2)

where $D$ is the Dirac operator $\sigma_{D}\circ\nabla$, and $\mathcal{A}:E\to
E$ is a bundle map. In the case that $n$ is even, we denote by the same symbol
the Dirac operator $D:\Gamma(E^{+})\to\Gamma(E^{-})$ and assume that
$\mathcal{A}:E^{+}\to E^{-}$. The operators $D_{\varepsilon}$ and
$\mathcal{A}$ are only assumed to be $\mathbb{R}$-linear, even in the case
that $E$ possesses a complex structure.

Let $r$ denote the maximal rank of $\mathcal{A}$, and define the singular set
of $\mathcal{A}$ by

$\mathcal{Z}:=\\{y\in Y\ |\ \text{rank}(\mathcal{A}(y))<r\\}.$

That is, the set where $\mathcal{A}$ has strictly lower than its maximal rank.
We assume that $\mathcal{A}$ is continuous, hence $\mathcal{Z}$ is closed. We
additionally assume, for convenience, that $\mathcal{Z}$ is non-empty (else
all assertions are vacuous). In the case of the Seiberg-Witten equations
linearized at a $\mathbb{Z}_{2}$-harmonic spinor $\Phi_{0}$, this set
coincides with the singular set $\mathcal{Z}=|\Phi_{0}|^{-1}(0)$ of the
$\mathbb{Z}_{2}$-harmonic spinor. We do NOT assume that $\mathcal{A}$ is
smooth (and indeed this is not the case in the setting of
$\mathbb{Z}_{2}$-harmonic spinors), nor do we assume that $\mathcal{Z}$ is a
submanifold or a union of submanifolds.

The focus will be on the following class of operators.

######  Definition 2.1.

A perturbed Dirac operator

$D_{\varepsilon}=D+\tfrac{1}{\varepsilon}\mathcal{A}$

is said to be a concentrating Dirac operator with fixed degeneracy if it obeys
the following two properties:

  1. 1).

(Concentration property) The principal symbol $\sigma_{D}$ of $D$ obeys

$\mathcal{A}^{\star}\sigma_{D}(\xi)=\sigma_{D}(\xi)^{\star}\mathcal{A}\hskip
56.9055pt\forall\ \xi\in T^{\star}Y$ (2.3)

  2. 2).

(Fixed Degeneracy): On $Y-\mathcal{Z}$, there is a bundle splitting
$E|_{Y-\mathcal{Z}}=\mathfrak{N}\oplus\mathfrak{H}$ (resp.
$E^{\pm}|_{Y-\mathcal{Z}}=\mathfrak{N}^{\pm}\oplus\mathfrak{H}^{\pm}$ if $n$
is even) which is parallel with respect to $\nabla$ and preserved by
$\sigma_{D}$, and in this splitting $\mathcal{A}$ takes the form

$\mathcal{A}=\begin{pmatrix}0&0\\\ 0&A_{\mathfrak{H}}\end{pmatrix}$ (2.4)

where $A_{\mathfrak{H}}:\mathfrak{H}\to\mathfrak{H}$ (resp.
$A_{\mathfrak{H}}:\mathfrak{H}^{+}\to\mathfrak{H}^{-}$).

Note that item 2) implies that $\mathfrak{N}=\ker(\mathcal{A})$. By taking
$\mathfrak{N}$ to be empty, this definition generalizes the class of operators
previously considered in [18, 19, 28] where $\mathcal{A}$ is invertible on an
open dense set. In Section 4, it is shown that generalized Seiberg-Witten
equations provide a rich class of examples for which the degeneracy is non-
trivial.

The remainder of this section establishes Theorem 1.1. To begin, we prove a
general Weitzenböck formula for concentrating Dirac operators with fixed
degeneracy.

######  Lemma 2.2.

A concentrating Dirac operator $D_{\varepsilon}$ satisfies

$D_{\varepsilon}^{\star}D_{\varepsilon}=D^{\star}D+\frac{1}{\varepsilon^{2}}\mathcal{A}^{\star}\mathcal{A}+\frac{1}{\varepsilon}\mathfrak{B}$

where $\mathfrak{B}$ is a zeroeth order term.

###### Proof.

The key point is that the concentration property (2.3) ensures that the first-
order cross terms cancel.

$\displaystyle D_{\varepsilon}^{\star}D_{\varepsilon}\mathfrak{q}$
$\displaystyle=$ $\displaystyle
D^{\star}D\mathfrak{q}+\tfrac{1}{\varepsilon^{2}}\mathcal{A}^{\star}\mathcal{A}\mathfrak{q}+\tfrac{1}{\varepsilon}\left(D^{\star}\mathcal{A}+\mathcal{A}^{\star}D\right)\mathfrak{q}$
$\displaystyle=$ $\displaystyle
D^{\star}D\mathfrak{q}+\tfrac{1}{\varepsilon^{2}}\mathcal{A}^{\star}\mathcal{A}\mathfrak{q}+\tfrac{1}{\varepsilon}\left(\sigma_{D}(e^{j})^{\star}\nabla_{j}^{\star}(\mathcal{A}\mathfrak{q})+\mathcal{A}^{\star}\sigma_{D}(e^{j})\nabla_{j}\mathfrak{q}\right)$
$\displaystyle=$ $\displaystyle
D^{\star}D\mathfrak{q}+\tfrac{1}{\varepsilon^{2}}\mathcal{A}^{\star}\mathcal{A}\mathfrak{q}+\tfrac{1}{\varepsilon}\big{(}-\sigma_{D}(e^{j})^{\star}(\nabla_{j}\mathcal{A})\mathfrak{q}+{\cancel{[-\sigma_{D}(e^{j})^{\star}\mathcal{A}+\mathcal{A}^{\star}\sigma_{D}(e^{j})]}}\nabla_{j}\mathfrak{q}\big{)}$
$\displaystyle=$ $\displaystyle
D^{\star}D\mathfrak{q}+\tfrac{1}{\varepsilon^{2}}\mathcal{A}^{\star}\mathcal{A}\mathfrak{q}+\tfrac{1}{\varepsilon}\left(-\sigma_{D}(e^{j})^{\star}(\nabla_{j}\mathcal{A})\mathfrak{q}\right)$
$\displaystyle=$ $\displaystyle
D^{\star}D\mathfrak{q}+\tfrac{1}{\varepsilon^{2}}\mathcal{A}^{\star}\mathcal{A}\mathfrak{q}+\tfrac{1}{\varepsilon}\mathfrak{B}\mathfrak{q}$

where the last line is taken as the definition of $\mathfrak{B}$. ∎

We now prove Theorem 1.1. The proof requires two brief lemmas. For these, we
denote the components of a spinor $\mathfrak{q}$ in the splitting of (2.4) by

$\mathfrak{q}=(q_{0},q_{1})\in\mathfrak{N}\oplus\mathfrak{H}.$

Additionally, we fix a compact subset $K\Subset Y-\mathcal{Z}$. To avoid
separating cases, it is understood in the above expression, and in the
discussion that follows that in the even-dimensional case the bundles
$\mathfrak{N}\oplus\mathfrak{H}$ undecorated by superscripts refer to
$\mathfrak{N}^{+}$ and $\mathfrak{H}^{+}$.

######  Lemma 2.3.

If $D_{\varepsilon}\mathfrak{q}=0$, then for sufficiently small $\varepsilon$
the scalar quantity $|q_{1}|^{2}$ satisfies the differential inequality

$d^{\star}d|q_{1}|^{2}+\frac{1}{\varepsilon^{2}}|\mathcal{A}q_{1}|^{2}\leq 0.$
(2.5)

###### Proof.

For the $\varepsilon$-independent operator $D$, one has a Weitzenböck formula

$D^{\star}D=\nabla^{\star}\nabla+\mathcal{R}$ (2.6)

where $\mathcal{R}$ is Clifford multiplication by a curvature term which is
bounded in $C^{0}$ by a constant independent of $\varepsilon$. The fixed
degeneracy assumption (2.4) implies that $(0,q_{1})\in\ker(D_{\varepsilon})$.
In a slight abuse of notation, we denote this pair simply by $q_{1}$, so that

$D_{\varepsilon}^{\star}D_{\varepsilon}q_{1}=0.$ (2.7)

Using (2.6), (2.7), and Lemma 2.2, for $\varepsilon$ sufficiently small we
have

$\displaystyle-\frac{1}{2}d^{\star}d|q_{1}|^{2}$ $\displaystyle=$
$\displaystyle\langle\nabla q_{1},\nabla q_{1}\rangle+\langle
q_{1},-\nabla^{\star}\nabla q_{1}\rangle$ $\displaystyle=$
$\displaystyle|\nabla q_{1}|^{2}+\langle q_{1},\mathcal{R}q_{1}\rangle+\langle
q_{1},-D^{\star}Dq_{1}\rangle$ $\displaystyle=$ $\displaystyle|\nabla
q_{1}|^{2}+\langle
q_{1},\mathcal{R}q_{1}\rangle+\tfrac{1}{\varepsilon^{2}}|\mathcal{A}q_{1}|^{2}+\tfrac{1}{\varepsilon}\langle
q_{1},\mathfrak{B}q_{1}\rangle$ $\displaystyle\geq$
$\displaystyle\tfrac{1}{2\varepsilon^{2}}|\mathcal{A}q_{1}|^{2}.$

In the last line, we have used the fact that $\mathcal{R}$ and $\mathfrak{B}$
are bounded independent of $\varepsilon$, so can be absorbed for sufficiently
small $\varepsilon$ since the injectivity of $\mathcal{A}$ on
$\mathfrak{H}|_{Y-\mathcal{Z}_{0}}$ implies that $|\mathcal{A}q_{1}|>c|q_{1}|$
holds for some constant $c$ on a compact subset $K\Subset Y-\mathcal{Z}_{0}$.
∎

The next lemma is an abstract result about scalar functions satisfying
differential inequalities of the form (2.5). The statement uses the following
notation: for each $y\in Y-\mathcal{Z}$, let $\Lambda(y)=\inf_{\|v\|=1}\|Av\|$
where $v\in\mathfrak{H}_{y}$. The fixed degeneracy hypothesis (2.4) ensures
$A_{\mathfrak{H}}$ is invertible on $Y-\mathcal{Z}$, hence $\Lambda(y)>0$.
Then, for a fixed compact subset $K\Subset Y-\mathcal{Z}$ as in the statement
of Theorem 1.1, let $\Lambda_{K}:=\inf_{y\in K}\Lambda(y)$ so that

$|\mathcal{A}\mathfrak{q}(y)|^{2}\geq\Lambda(y)^{2}|q_{1}(y)|^{2}\geq\Lambda^{2}_{K}|q_{1}(y)|^{2}$

holds for $y\in K$. As in the statement of Theorem 1.1,
$R_{K}=\text{dist}(K,\mathcal{Z})$. Additionally, in what follows,
$\Delta_{g}=d^{\star}d$ denotes the positive definite Laplacian defined by the
Riemannian metric $g$.

######  Lemma 2.4.

Suppose that $u:K\to\mathbb{R}$ satisfies $u\geq 0$ and

$\Delta_{g}u+\frac{\Lambda(y)^{2}}{\varepsilon^{2}}u\leq 0.$ (2.8)

Then there exist constants $C,c>0$ such that at a point $y_{0}\in K$ one has

$u(y_{0})\leq\frac{C}{R_{K}^{n}}\text{Exp}\left(-\frac{\Lambda(y_{0})c}{\varepsilon}R_{K}\right)\int_{K^{\prime}}|du|+{|u|}\
dV$ (2.9)

where $K^{\prime}$ is a compact set with $\overset{\circ}{K}\Subset
K^{\prime}\Subset Y-\mathcal{Z}$.

###### Proof.

Recall first that the Green’s function of $\Delta+{m^{2}}$ on $\mathbb{R}^{n}$
is

$G(x,x_{0})=\frac{C(n)}{|x-x_{0}|^{n-2}}\text{Exp}\left({-{m}|x-x_{0}|}\right)$
(2.10)

where $C(n)$ is a constant depending only on the dimension. Let
$K^{\prime}\Subset Y-\mathcal{Z}$ be a compact set whose interior contains
$K$, and take $1>c_{0}>0$ to be a small number such that the following three
conditions hold for $R_{0}=c_{0}R_{K}$:

  1. (i)

$y\in K\Rightarrow B_{R_{0}}\subseteq K^{\prime}$

  2. (ii)

$y\in B_{R_{0}}(y_{0})\Rightarrow\Lambda(y)^{2}\geq\tfrac{\Lambda_{0}^{2}}{2}$
where $\Lambda_{0}=\Lambda(y_{0})$.

  3. (iii)

The Green’s function of
$-\Delta_{g}-\tfrac{\Lambda_{0}^{2}}{2\varepsilon^{2}}$ on $B_{2R_{0}}$ with
Dirichlet boundary conditions satisfies

$\displaystyle G(y,y_{0})$ $\displaystyle\leq$
$\displaystyle\frac{c_{1}}{|y-y_{0}|^{n-2}}\text{Exp}\left({-\frac{\Lambda_{0}}{2\varepsilon}|y-y_{0}|}\right)$

uniformly on $Y$ (resp. $X$) once $\varepsilon$ is sufficiently small
(depending on $K$), and where $c_{1}$ depends only on the dimension.

The first of these is possible by the compactness of $K$, the second by the
fact that $\mathcal{A}$ is $C^{1}$. The third follows by using the comparison
principle on $B_{2R_{0}}$ comparing with the Green’s function (2.10) on
Euclidean space. The details are provided in Appendix B.

Now let $y_{0}\in K$ and consider the ball $B_{0}=B_{R_{0}}(y_{0})\subseteq
K$, where $R_{0}$ is as above. Green’s identity on $B_{0}$ for functions
$\eta,\psi$ states that

$\int_{B_{0}}\eta(-\Delta_{g}\psi)-\psi(-\Delta_{g}\eta)\
dV_{g}=\int_{\partial B_{0}}\eta\partial_{\nu}\psi-\psi\partial_{\nu}\eta\
dS_{g}$

and is derived by integrating the quantity $0=\int\langle
d\eta,d\psi\rangle-\langle d\psi,d\eta\rangle$ by parts. Here, $dV_{g},dS_{g}$
denote the volume forms arising from the Riemannian metric on the ball and the
sphere respectively. Now set

$M:={\frac{\Lambda_{0}}{\sqrt{2}\varepsilon}}.$

Adding and subtracting $M^{2}\eta\psi$ on the left-hand side yields

$\int_{B_{0}}\eta(-\Delta_{g}-M^{2})\psi+\psi(\Delta_{g}+M^{2})\eta\
dV_{g}=\int_{\partial B_{0}}\eta\partial_{\nu}\psi-\psi\partial_{\nu}\eta\
dS_{g}.$ (2.11)

Next, we let $\beta(y)$ denote a cutoff function equal to $1$ on
$B_{R_{0}/2}(y_{0})$ and supported in the interior of $B_{0}$ satisfying

$0\leq\beta\leq 1\hskip 56.9055pt|\nabla\beta|\leq\frac{C}{R_{0}}\hskip
56.9055pt|\nabla^{2}\beta|\leq\frac{C}{R_{0}^{2}}$ (2.12)

and apply the identity 2.11 with $\eta=u(y)\beta(y)$ and $\psi=G(y,y_{0})$.
The boundary term on the right hand side vanishes by the choice of $\beta$.
The first term becomes $-u(y_{0})$ since, by definition, the Green’s function
satisfies $(\Delta_{g}+M^{2})G=\delta_{y_{0}}$. Meanwhile for the second term,
the assumption that $u$ satisfies(2.8) implies

$(\Delta_{g}+M^{2})u=\left(\Delta_{g}+\frac{\Lambda_{0}^{2}}{2\varepsilon^{2}}\right)u\leq\left(\Delta_{g}+\frac{\Lambda(y)^{2}}{\varepsilon^{2}}\right)u\leq
0$ (2.13)

hence

$(\Delta_{g}+M^{2})\beta u=\beta(\Delta_{g}+M^{2})u-2\langle
d\beta,du\rangle+(\Delta_{g}\beta)u\leq
C\left(\frac{1}{R_{0}}|du|+\frac{1}{R_{0}^{2}}|u|\right)\chi_{A}$

where $\chi_{A}$ is the characteristic function equal to 1 on the outer
annulus $A=\\{R_{0}/2\leq r\leq R_{0}\\}$ and vanishing elsewhere.

Thus the identity (2.11) becomes the inequality

$\displaystyle u(y_{0})$ $\displaystyle\leq$ $\displaystyle
C\int_{A}G(y,y_{0})\left(\frac{1}{R_{0}}|du|+\frac{1}{R_{0}^{2}}|u|\right)\
dV_{g}$ $\displaystyle\leq$
$\displaystyle\frac{C}{R^{n-1}_{0}}\text{Exp}\left(-\frac{\Lambda_{0}}{8\varepsilon}R_{0}\right)\int_{A}|du|+\frac{|u|}{R_{0}}.$

Substituting the definition $R_{0}=c_{0}R_{K}$ yields the bound (2.9). ∎

###### Proof of Theorem 1.1.

Apply Lemma 2.9 to $u=|q_{1}|^{2}$. Kato’s inequality $d|q|\leq|\nabla q|$
shows that

$d|q_{1}|^{2}\leq 2|q_{1}|d|q_{1}|\leq 2|q_{1}||\nabla
q_{1}|\leq|q_{1}|^{2}+|\nabla q_{1}|^{2}.$

Applying this to the right side of (2.9) and setting $R_{K}=c_{0}R_{0}$ yields

$\displaystyle|q_{1}(y_{0})|^{2}$ $\displaystyle\leq$
$\displaystyle\frac{C}{c_{0}^{n}R_{K}^{n}}\text{Exp}\left(-\frac{\Lambda_{0}c_{0}}{\varepsilon}R_{K}\right)\|q_{1}\|^{2}_{L^{1,2}(K^{\prime})}.$
(2.14)

Taking the square root and decreasing constants by a factor of $2$ to replace
$\Lambda_{0}$ by $\Lambda_{K}$ shows the desired estimate (1.5).

∎

##  3\. Non-Linear Concentrating Dirac Equations

This section extends the proof of Theorem 1.1 to the case of the non-linear
equation (5.16) of Corollary 1.2.

Let us clarify the subtlety in deducing the estimate (1.5) for a non-linear
Dirac equation. Equation (5.16) can be written as the solution of a
concentrating Dirac operator as follows: let
$A_{\varepsilon}(q_{1})=\varepsilon Q_{1}(\mathfrak{q})q_{1}$ with $Q_{1}$ as
in the statement of Corollary 1.2. Then

$\displaystyle D_{\varepsilon}\mathfrak{q}+Q(\mathfrak{q})$ $\displaystyle=$
$\displaystyle 0$ (3.1)
$\displaystyle\left(D+\tfrac{1}{\varepsilon}\mathcal{A}+Q_{1}(\mathfrak{q})\right)\mathfrak{q}$
$\displaystyle=$ $\displaystyle 0$ (3.2)
$\displaystyle\left(D+\tfrac{1}{\varepsilon}\mathcal{(}A_{\mathfrak{H}}+A_{\varepsilon}(\mathfrak{q}))\right)\mathfrak{q}$
$\displaystyle=$ $\displaystyle 0$ (3.3)

i.e. $\mathfrak{q}$ is the solves a concentrating Dirac equation, but the
zeroth order perturbation now depends on $\mathfrak{q}$. In this case, the
proof of Theorem 1.1 fails in general because it is no longer clear we can
absorb the $\mathfrak{B}=\nabla\mathcal{A}$ term. Indeed, this would require a
$C^{1}$-bound on $A_{\varepsilon}$. The naive bootstrapping, however, leads
only to bounds with powers of $\varepsilon^{-1}$ larger than can be absorbed
(see also Section 6). The proof of Corollary 1.2 relies on stronger
differential inequalities obtained by not discarding the
$|\nabla\mathfrak{q}_{1}|^{2}$ term in Lemma 2.5.

Corollary 1.2 is deduced from the following proposition by setting
$A_{\varepsilon}(-)=\varepsilon Q_{1}(\mathfrak{q},-)$. The additional
assumption on $\varepsilon Q_{1}$ now manifests as the requirement that the
concentration property
$A_{\varepsilon}^{\star}\sigma_{D}-\sigma_{D}^{\star}A_{\varepsilon}$ is
satisfied for this term as well.

######  Proposition 3.1.

The conclusion (1.5) of Theorem 1.1 continues to hold if the zeroth order term
is given by

$\mathcal{A}=\begin{pmatrix}0&0\\\
0&A_{\mathfrak{H}}\end{pmatrix}+A_{\varepsilon}.$ (3.4)

where $A_{\mathfrak{H}}$ is smooth with all derivatives bounded independent of
$\varepsilon$ on $K$ and $\mathfrak{q},A_{\varepsilon}$ satisfy the relation
$A_{\varepsilon}^{\star}\sigma_{D}(\xi)=\sigma_{D}^{\star}(\xi)A_{\varepsilon}=$
for $\xi\in T^{\star}Y$ and

$\|A_{\varepsilon}\|_{L^{1,n}(K^{\prime})}\to 0\hskip
56.9055pt\|A_{\varepsilon}\|_{C^{0}(K^{\prime})}\to 0.$

More generally, the same conclusion holds for the inhomogeneous version of
equation $(\ref{nonlinearQ12})$,

$\left(D+\tfrac{1}{\varepsilon}\mathcal{(}A_{\mathfrak{H}}+A_{\varepsilon}(\mathfrak{q}))\right)\mathfrak{q}=f$

where $f$ satisfies $\pi_{\mathfrak{H}}(f)=0$.

###### Proof.

Retaining the notation of Lemma 2.9, let $\Lambda_{K}=\inf_{y\in
K}\Lambda(y_{0})$ where $\Lambda(y)=\inf_{\|v\|=1}\|Av\|$ for
$v\in\mathfrak{H}_{y}$ be as before (thus it has no dependence on
$A_{\varepsilon}$). Additionally, write

$\mathfrak{B}=\nabla A_{\mathfrak{H}}+\nabla A_{\varepsilon}$

where $\nabla$ is shorthand for the operator
$\sigma_{D}(e^{j})^{\star}\nabla_{j}$ appearing in the proof of Lemma 2.2.

Proceeding as in the proof of Lemma 2.5, we have the following differential
inequality. Note here that the inner product with $q_{1}$ and the assumption
that $\pi_{\mathfrak{H}}(f)=0$ imply that the cases of $f=0$ and $f\neq 0$
yield the same inequality, and the two cases therefore coincide for the
remainder of the proof.

$\displaystyle-\frac{1}{2}d^{\star}d|q_{1}|^{2}$ $\displaystyle=$
$\displaystyle|\nabla q_{1}|^{2}+\langle
q_{1},\mathcal{R}q_{1}\rangle+\tfrac{1}{\varepsilon^{2}}|\mathcal{A}q_{1}|^{2}+\tfrac{1}{\varepsilon}\langle
q_{1},\mathfrak{B}q_{1}\rangle$ $\displaystyle\geq$ $\displaystyle|\nabla
q_{1}|^{2}+\langle
q_{1},\mathcal{R}q_{1}\rangle+\tfrac{1}{\varepsilon^{2}}|(A_{\mathfrak{H}}+A_{\varepsilon})q_{1}|^{2}+\tfrac{1}{\varepsilon}\langle
q_{1},(\nabla A_{\mathfrak{H}}+\nabla A_{\varepsilon})q_{1}\rangle$
$\displaystyle\geq$ $\displaystyle|\nabla q_{1}|^{2}+\langle
q_{1},\mathcal{R}q_{1}\rangle+\tfrac{1}{\varepsilon^{2}}|(A_{\mathfrak{H}}+A_{\varepsilon})q_{1}|^{2}+\tfrac{1}{\varepsilon}\langle
q_{1},(\nabla A_{\mathfrak{H}}+\nabla A_{\varepsilon})q_{1}\rangle$
$\displaystyle\geq$ $\displaystyle|\nabla
q_{1}|^{2}+\tfrac{1}{2\varepsilon^{2}}|A_{\mathfrak{H}}q_{1}|^{2}+\tfrac{1}{\varepsilon}\langle
q_{1},(\nabla A_{\varepsilon})q_{1}\rangle$

where in addition to the absorptions done before, we have used that
$\|A_{\varepsilon}\|_{C^{0}(K^{\prime})}\to 0$ which implies that

$|A_{\varepsilon}q_{1}|^{2}\leq\frac{\Lambda_{K}^{2}}{4}|q_{1}|^{2}\leq\frac{1}{4}|A_{\mathfrak{H}}q_{1}|^{2}$
(3.5)

once $\varepsilon$ is sufficiently small. Rearranging shows the differential
inequality now dictates that

$\Delta|q_{1}|^{2}+\frac{|A_{\mathfrak{H}}q_{1}|^{2}}{\varepsilon^{2}}\leq-2|\nabla
q_{1}|^{2}-\tfrac{2}{\varepsilon}\langle q_{1},(\nabla
A_{\varepsilon})q_{1}\rangle$ (3.6)

on $K^{\prime}$.

Proceeding as before (but with an additional factor of $\sqrt{2}$), set
$M:={\frac{\Lambda_{0}}{2e}}$. Taking $u=|q_{1}|^{2}$, in this case (2.11)
becomes

$\displaystyle u(y_{0})$ $\displaystyle=$
$\displaystyle-\int_{B_{0}}G(y,y_{0})(-\Delta_{g}-M^{2})(\beta u)\ dV_{g}$
$\displaystyle\leq$
$\displaystyle\int_{B_{0}}G(y,y_{0})\beta(\Delta_{g}+M^{2})u)\
dV_{g}+C\int_{B_{0}}G(y,y_{0})\left(\frac{1}{R_{0}}|du|+\frac{1}{R_{0}^{2}}|u|\right)\
dV_{g}$

and combining this with 3.6 yields

$\displaystyle|q_{1}|^{2}(y)$ $\displaystyle+$
$\displaystyle\int_{B_{0}}G(y,y_{0})\beta\left(|\nabla
q_{1}|^{2}+\frac{\Lambda_{0}}{8\varepsilon^{2}}|q_{1}|^{2}\right)\ dV_{g}$
(3.7) $\displaystyle\leq$ $\displaystyle
C\int_{B_{0}}G(y,y_{0})\left(\frac{1}{R_{0}}|d|q_{1}|^{2}|+\frac{1}{R_{0}^{2}}|q_{1}|^{2}\right)\
dV_{g}+\frac{1}{\varepsilon}\int_{B_{0}}G(y_{0},y)\beta|\langle q_{1},(\nabla
A_{\varepsilon})q_{1}\rangle|dV_{g}.$ (3.8)

Using Lemma 3.2 below, to absorb the second term on the right in (3.8) into
the two remaining terms of (3.7)–(3.8) leads to

$\displaystyle|q_{1}|^{2}(y)$ $\displaystyle\leq$ $\displaystyle
2C\int_{B_{0}}G(y,y_{0})\left(\frac{1}{R_{0}}|d|q_{1}|^{2}|+\frac{1}{R_{0}^{2}}|q_{1}|^{2}\right)\
dV_{g}$ (3.9)

after which the result follows identically to Theorem 1.1 by repeating the
argument leading to (2.14). ∎

######  Lemma 3.2.

For $\varepsilon$ sufficiently small, the integral

$I=\frac{1}{\varepsilon}\int_{B_{0}}G(y_{0},y)\beta|\langle q_{1},(\nabla
A_{\varepsilon})q_{1}\rangle|dV_{g}$

satisfies

$\displaystyle I$ $\displaystyle\leq$
$\displaystyle\int_{B_{0}}G(y,y_{0})\beta\left(|\nabla
q_{1}|^{2}+\frac{\Lambda_{0}}{8\varepsilon^{2}}|q_{1}|^{2}\right)dV_{g}+C\int_{B_{0}}G(y,y_{0})\left(\frac{1}{R_{0}}|d|q_{1}|^{2}|+\frac{1}{R_{0}^{2}}|q_{1}|^{2}\right)dV_{g}$
(3.10)

###### Proof of Lemma 3.2.

This follows from a weighted interpolation inequality and a dyadic
decomposition. Let

$A_{n}=\big{\\{}|y|\in[r_{n+1},r_{n}]\big{\\}}$

for $n\geq 0$ be a sequence of disjoint annuli covering $B_{0}$ where the
radii $r_{n}$ are defined inductively by

$\displaystyle r_{0}$ $\displaystyle=$ $\displaystyle R_{0}$ $\displaystyle
r_{n+1}$ $\displaystyle=$ $\displaystyle
r_{n}-\min\left\\{{\tfrac{r_{n}}{5},\tfrac{1}{M}}\right\\}.$

Next, subdivide each $A_{n}$ into a disjoint union of sectors $A_{n\ell}$ such
that $\text{diam}(A_{n\ell})\leq|r_{n+1}-r_{n}|$. It is now enough to prove
(3.10) separately for each of the disjoint sectors $A_{n\ell}$.

On these sectors, we have the following weighted interpolation inequality: If
$p$ satisfies

$\frac{1}{p}=\frac{j}{n}+\alpha\left(\frac{1}{r}-\frac{m}{n}\right)+\frac{1-\alpha}{s}.$
(3.11)

then the following inequality holds uniformly over the collection $A_{n\ell}$
and uniformly in $M$ (on which $G=G(y,y_{0})$ depends) .

$\displaystyle\left(\int_{A_{n\ell}}G^{p/2}|\nabla^{j}v|^{p}\ dV\right)^{1/p}$
$\displaystyle\leq$ $\displaystyle
C\left(\int_{A_{n\ell}}G^{r/2}|\nabla^{m}v|^{r}\
dV\right)^{\alpha/r}\left(\int_{A_{n\ell}}G^{s/2}|v|^{s}\
dV\right)^{(1-\alpha)/s}$ (3.12)
$\displaystyle+C\left(\int_{A_{n\ell}}G|v|^{2}\ dV\right)^{1/2}.$

To prove (3.12), notice that without the Green’s function weight (ie setting
$G=1$) this follows from the scale invariance of the standard interpolation
inequalities (the $L^{2}$-term gets comparatively stronger on smaller balls).
The version including $G$ follows from this and invoking the Harnack
inequality ([12], Theorem 8.20) on each $A_{n\ell}$ which shows

$\sup_{A_{n\ell}}|G(y_{0},y)|\leq C\inf_{A_{n\ell}}|G(y_{0},y)|.$

Indeed, by construction, each $A_{n\ell}$ is contained in a ball of radius
$R_{n}$ such that the ball of radius $4R_{n}\subseteq B_{R_{0}}-\\{y_{0}\\}$.
Since $G(y_{0},y)\geq 0$ by the maximum principle and satisfies
$(\Delta_{g}+M^{2})G(y_{0},y)=0$ on $B_{R_{0}}-\bigcup_{k\geq n+3}A_{k}$, the
Harnack inequality ([12], Theorem 8.20) applies. Moreover, by the scaling of
the constant (see [12] Theorem 8.20 and the subsequent comments), the constant
can be taken to be uniform since $r_{n}\leq\tfrac{1}{M}$ by construction (so
that in the notation of 8.20 one has $\nu R=O(1)$).

We conclude the lemma using (3.12). Without disrupting the bounds (2.12) we
may assume that $\beta=\chi^{2}$ is the square of another smooth cut-off
function satisfying the same bounds up to universal constants. In the case of
dimension 4, applying Hölder’s inequality with with $q^{*}=4$ and $p^{*}=4/3$
yields

$\displaystyle\frac{1}{\varepsilon}\int_{A_{n\ell}}G(y_{0},y)\beta|\langle
q_{1},(\nabla A_{\varepsilon})q_{1}\rangle|dV_{g}$ $\displaystyle\leq$
$\displaystyle\frac{1}{\varepsilon}\|A_{\varepsilon}\|_{L^{1,4}(B_{0})}\left(\int_{A_{n\ell}}G^{p/2}|q_{1}\sqrt{\beta}|^{p}\
dV\right)^{2/p}$ (3.13)

where $p=8/3$. Next, we apply the interpolation inequality (3.12) to
$q_{1}\sqrt{\beta}$ with $r=s=2$, and $j=0$ and $m=1$ in which case (3.11)
shows $\alpha=\frac{1}{2}$ hence (3.13) is bounded by

$\displaystyle\leq$
$\displaystyle\frac{C}{\varepsilon}\|A_{\varepsilon}\|_{L^{1,4}(B_{0})}\left[\left(\int_{A_{n\ell}}G|\nabla(\chi
q_{1})|^{2}\ dV\right)^{1/2}\left(\int_{A_{n\ell}}G|\chi q_{1}|^{2}\
dV\right)^{1/2}+C\|G^{1/2}\chi q_{1}\|^{2}_{L^{2}(A_{n\ell})}\right]$
$\displaystyle\leq$
$\displaystyle\frac{C}{\varepsilon}\|A_{\varepsilon}\|_{L^{1,4}(B_{0})}\left(\varepsilon\frac{\|G^{1/2}\nabla(\chi
q_{1})\|^{2}_{L^{2}(A_{n\ell})}}{2}+\frac{\|G^{1/2}\chi
q_{1}\|^{2}_{L^{2}(A_{n\ell})}}{2\varepsilon}+\|G^{1/2}\chi
q_{1}\|^{2}_{L^{2}(A_{n\ell})}\right)$ $\displaystyle\leq$
$\displaystyle{C}\|A_{\varepsilon}\|_{L^{1,4}(B_{0})}\left(\int_{A_{n\ell}}G(y,y_{0})\beta\left(|\nabla
q_{1}|^{2}+\frac{1}{2\varepsilon^{2}}|q_{1}|^{2}\right)\
dV+\int_{A_{n\ell}}G(y_{0},y)\frac{1}{R_{0}^{2}}|q_{1}|^{2}\ dV\right)$

where we have used the bounds (2.12) on $d\chi$ to obtain the last term. The
assumption that $\|A_{\varepsilon}\|_{L^{1,4}}\to 0$ allows us to choose
$\varepsilon$ sufficiently small that
$\left(C+\tfrac{C\Lambda_{0}}{8}\right)\|A_{\varepsilon}\|_{L^{1,4}}\leq 1$
giving (3.10). The case of dimension $n\neq 4$ differs only in the arithmetic
choices of exponents in Hölder’s inequality.

∎

##  4\. Generalized Seiberg-Witten Equations

This section introduces generalized Seiberg-Witten equations; the subsequent
section shows that in the relevant cases these fit into the framework of
Sections 2–3.

###  4.1. Seiberg-Witten Data

Generalized Seiberg-Witten equations are systems of coupled non-linear first-
order PDEs on manifolds of dimension 3 and 4. There is a system of generalized
Seiberg-Witten equations associated to each quaternionic representation of a
compact Lie group $G$. Rather than working in the most general setting, we
will here opt for an abridged exposition which suffices for our purposes. For
a more general introduction, see [46, 45, 1].

We first focus on the three-dimensional case. Let $(Y,g_{0})$ denote an
oriented Riemannian 3-manifold, and fix a spin structure $\mathfrak{s}\to Y$
considered as a principal $\text{Sp}(1)\simeq\text{Spin}(3)$-bundle. We may
write

$T^{\star}Y\simeq\mathfrak{s}\times_{Ad}\text{Im}(\mathbb{H})$ (4.1)

where $Ad:\text{Sp}(1)\to\text{Im}(\mathbb{H})=\mathfrak{sp}(1)$ is the
adjoint representation of $\text{Spin}(3)$. Next, let $G$ be a compact Lie
group, and $V$ a quaternionic vector space with a real inner-product denoted
$\langle-,-\rangle$ carrying a quaternionic representation

$\rho:G\to\text{GL}_{\mathbb{H}}(V)$

respecting the inner product.

Since $\rho$ is quaternionic, it extends to a map (denoted by the same symbol)
$\rho:\text{Sp}(1)\times G\to\text{GL}_{\mathbb{H}}(V)$ given by $(q,g)\mapsto
q\cdot\rho(g)$. Let

$\gamma:\text{Im}(\mathbb{H})\otimes_{\mathbb{R}}\mathfrak{g}\to\text{End}_{\mathbb{H}}(V).$
(4.2)

be the induced linearized action. Additionally, for $\Psi\in V$ define
$\mu:V\to\text{Im}(\mathbb{H})\otimes_{\mathbb{R}}\mathfrak{g}$ by

$\frac{1}{2}\mu(\Psi,\Psi)=\frac{1}{2}\sum_{\alpha,j}\langle\gamma(I_{j}\otimes\mathfrak{t}_{\alpha})\Psi,\Psi\rangle\
I_{j}\otimes t_{\alpha}$ (4.3)

where $I_{j}$ for $j=1,2,3$ is a basis of $\text{Im}(\mathbb{H})$, and
$\\{\mathfrak{t}_{\alpha}\\}$ are a basis of $\mathfrak{g}$. It is
straightforward to check that $\frac{1}{2}\mu$ is the hyperkähler moment map
for the action of $G$ on $V$ by $\rho$.

Working globally on $Y$ now, let $P\to Y$ be a principal $G$-bundle.

######  Definition 4.1.

The Spinor Bundle associated to $\rho$ is the vector bundle

$S=(\mathfrak{s}\times_{Y}P)\times_{\text{Sp}(1)\times G}V.$

It is endowed with a Clifford multiplication and a moment map denoted
(respectively) by

$\gamma:\Omega^{1}(\mathfrak{g}_{P})\to\text{End}(S)\hskip
56.9055pt\mu:S\to\Omega^{1}(\mathfrak{g}_{P})$

given fiberwise by the maps (4.2) and (4.3). Here,
$\Omega^{1}(\mathfrak{g}_{P})$ is the space of 1-forms valued in the adjoint
bundle of $P$, viewed as the associated bundle via (4.1) and the adjoint
representation of $G$.

Each set of data $(G,V,\rho)$ gives rise to a system of generalized Seiberg-
Witten equations on $(Y,g_{0})$. These are the following PDEs for a pair
$(\Psi,A)\in\Gamma(S)\times\mathscr{A}(P)$ consisting of a spinor $\Psi$ and a
connection $A$ on the principal bundle $P$ respectively. Such a pair is called
a configuration.

######  Definition 4.2.

The generalized Seiberg-Witten equations determined by $(P,S,\gamma,\mu)$ are

$\displaystyle\not{D}_{A}\Psi$ $\displaystyle=$ $\displaystyle 0$ (4.4)
$\displaystyle\star F_{A}+\tfrac{1}{2}\mu(\Psi,\Psi)$ $\displaystyle=$
$\displaystyle 0$ (4.5)

where $\not{D}_{A}$ denotes the Dirac operator on $S$ determined by the spin
connection and $A$ using the Clifford multiplication $\gamma$, and $F_{A}$ is
the curvature of $A$. These equations are invariant under the Gauge group

$\mathcal{G}=\Gamma(P\times_{Ad}G).$

The discussion above carries over to the case of an oriented Riemannian
4-manifold $(X,g_{0})$ with only two minor modifications.

  1. (1)

As not all 4-manifolds are spin, we impose the additional requirement that
there exists a central element $-1\in Z(G)$ so that $\rho(-1)=-\text{Id}$, and
consider principal bundles $P,Q$ with structure groups $G$ and
$\text{Spin}^{G}(4)=(\text{Spin}(4)\times G)/\mathbb{Z}_{2}$ respectively.
Here, $\mathbb{Z}_{2}$ acts by $(1,1)\mapsto(-1,-1)$.

  2. (2)

In (4.1), $\Lambda^{1}(Y)$ is replaced by
$\Lambda^{2}_{+}(X)=Q\times_{\sigma}\text{Im}(\mathbb{H})$ where
$\sigma:\text{Spin}^{G}(4)\to SO(\text{Im}(\mathbb{H}))$ is the composition of
projection to $SO(4)$ via $\text{Spin}(4)$ and the standard 3-dimensional
representation of $SO(4)$. Similarly in (4.4–4.5), $\star F_{A}$ is replaced
by $F_{A}^{+}$ and $\not{D}_{A}$ by $\not{D}_{A}^{+}$.

See [45] for a more complete discussion of the 4-dimensional case.

Most equation of interest in mathematical gauge theory arise as generalized
Seiberg-Witten equations for particular choices of the data $(G,V,\rho)$.

######  Example 4.3.

The Anti-Self-Dual (ASD) Yang-Mills equations and standard Seiberg-Witten
equations (and their dimensional reductions) are generalized Seiberg-Witten
equations obtained from the following data.

  1. $\bullet$

The data $G=SU(2)$ and $V=\\{0\\}$ with $\rho$ being the trivial
representation gives the ASD Yang-Mills equations.

  2. $\bullet$

The data $G=U(1)$ and $V=\mathbb{H}$ with $\rho$ being multiplication by
$e^{i\theta}\in U(1)$ on the right reproduces the standard Seiberg-Witten
equations.

We now distinguish four cases, (I)–(IV) as in the statement of Theorem 1.3.
These cases will be referred to repeatedly throughout the proof of Theorem
1.3.

######  Example 4.4 (Case (I)).

The data $G=U(1)$ and $V=\mathbb{H}\otimes_{\mathbb{C}}\mathbb{C}^{r}$ with
$\rho$ being multiplication by $e^{i\theta}\in U(1)$ on the right on the first
factor leads to the Seiberg-Witten equations with $r$ spinors.
More concretely, in this case $S=W\otimes E$ where $W\to Y$ is the spinor
bundle of a $\text{spin}^{c}$ structure and $E\to Y$ an auxiliary bundle of
(complex) rank $r$ with trivial determinant. Then the equations are

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

where $\not{D}_{A}$ is the Dirac operator on $S_{V}$ and the moment map $\mu$
can be described as follows. Let $e^{j}$ denote a local frame of $T^{\star}Y$;
the local frame of $W$ arising as the $\pm i$ eigenspaces of Clifford
multiplication by $e^{1}$, a spinor can be written $\Psi=(\alpha,\beta)$ where
$\alpha,\beta\in\Gamma(V)$. Then, the moment map is,

$\frac{1}{2}\mu(\Psi,\Psi)=\frac{1}{2}\sum_{j=1}^{3}\langle
ie^{j}.\Psi,\Psi\rangle
ie^{j}=\frac{i}{2}\Big{(}(|\beta|^{2}-|\alpha|^{2})e^{1}\ ,\
\text{Re}(-\overline{\alpha}\beta)e^{2}\ ,\
\text{Im}(-\overline{\alpha}\beta)e^{3}\Big{)}.$ (4.6)

In particular $\mu$ is the sum of the standard Seiberg-Witten moment map over
the $r$-spinors. See [25, 4] for further details.

######  Example 4.5 (Case (III)).

On a 4-manifold $X$ the equations and moment map are analogously related to
the standard 4-dimensional Seiberg-Witten equations. To be precise, they take
the form

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

where $\mu$ is defined analogously to (4.6) using an orthonormal basis
$\omega^{j}$ of $\Lambda^{2}_{+}(i\mathbb{R})$ in place of the basis $e^{j}$.

######  Example 4.6.

(Case (II)). The data $G=SU(2)$ and
$V=\mathfrak{su}(2)\otimes_{\mathbb{R}}\mathbb{H}$ with $\rho$ being the
quaternionificaition of the adjoint representation gives rise to the equations
for a flat $\text{SL}(2,\mathbb{C})$ connection in dimension $3$. In this
case,

$S=(\Lambda^{0}\oplus\Lambda^{1})(\mathfrak{g}_{P})\hskip
56.9055pt\not{D}_{A}=\mathbb{d}_{A}=\begin{pmatrix}0&-d_{A}^{\star}\\\
-d_{A}&\star d_{A}\end{pmatrix}\begin{pmatrix}\Psi_{0}\\\
\Psi_{1}\end{pmatrix}$

and the moment map is
$\frac{1}{2}\mu(\Psi,\Psi)=-\frac{1}{2}\star[\Psi\wedge\Psi].$ The equations
therefore become

$\displaystyle\mathbb{d}_{A}\Psi$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle\star F_{A}-\tfrac{1}{2}\star[\Psi\wedge\Psi]$ $\displaystyle=$
$\displaystyle 0$

######  Example 4.7.

(Case (IV)). In dimension 4, the data of Example 4.6 gives rise to the complex
ASD equations. Analogously to the 3-dimensional case, in dimension $4$ one has

$S^{+}=\Lambda^{1}(\mathfrak{g}_{P})\hskip
28.45274ptS^{-}=(\Lambda^{0}\oplus\Lambda^{2}_{-})(\mathfrak{g}_{P})\hskip
56.9055pt\mathbb{d}_{A}^{+}=(d_{A}^{\star},d_{A}^{-})$

and $\frac{1}{2}\mu(\Psi,\Psi)=-\frac{1}{2}[\Psi\wedge\Psi]^{+}.$ The
corresponding equations are

$\displaystyle\mathbb{d}^{+}_{A}\Psi$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle F_{A}^{+}-\tfrac{1}{2}[\Psi\wedge\Psi]^{+}$ $\displaystyle=$
$\displaystyle 0.$

######  Example 4.8.

(Vafa-Witten Equations) The same data as in the previous example can also give
rise to the Vafa-Witten equations on a four-manifold $X^{4}$ (depending on a
choice of auxiliary data omitted from the discussion here—see [45], Examples
2.31 and 2.36). In this case, one has

$S^{+}=(\Lambda^{0}\oplus\Lambda^{2}_{+})(\mathfrak{g}_{P})\hskip
28.45274ptS^{-}=\Lambda^{1}(\mathfrak{g}_{P})\hskip
56.9055pt\not{D}_{A}(C,B)=d_{A}C+d_{A}^{\star}B$

and if $\Psi=(C,B)\in(\Omega^{0}\oplus\Omega^{2}_{+})(\mathfrak{g}_{P})$ then
the moment map is

$\frac{1}{2}\mu(\Psi,\Psi)=[C,B]+\frac{1}{2}[B\times B]$

where $[\\_\times\\_]$ is the product induced by viewing $\Lambda^{2}_{+}(TX)$
as the bundle of Lie algebras arising as the associated bundle of
$SO(4)$-frames on $X$ via the positive irreducible component of the adjoint
representation on $\mathfrak{so}(4)$. Explicitly, in an orthonormal frame
$\omega_{i}$ of $\Lambda^{2}_{+}$ it is given on a self-dual 2-form
$B=\omega_{i}\otimes B_{i}$ by $[B\times
B]=\epsilon_{ijk}[B_{i},B_{j}]\omega_{k}$ where summation over repeated
indices is implicit.

######  Example 4.9.

($\text{ADHM}_{r,k}$ Seiberg-Witten Equations) For $G=U(k)$ and

$V_{r,k}=\text{Hom}_{\mathbb{C}}(\mathbb{C}^{r},\mathbb{H}\otimes_{\mathbb{C}}\mathbb{C}^{k})\oplus(\mathbb{H}^{\vee}\otimes_{\mathbb{R}}\mathfrak{u}(k))$

where $\rho$ acts on the $\mathbb{C}^{k}$ factor via the standard
representation and the $\mathfrak{u}(k)$ factor via the adjoint gives rise to
the $\text{ADHM}_{r,k}$ Seiberg-Witten Equations. Here $\mathbb{H}^{\vee}$
denotes the dual space. The zero-locus $\mu^{-1}(0)$ of the moment map in this
situation coincides with the ADHM construction of the moduli space of ASD
$SU(r)$-instantons of charge $k$ on $\mathbb{R}^{4}$. These equations are
conjectured to connect Yang-Mills theory on manifolds with special holonomy to
Seiberg-Witten theory on calibrated submanifolds (see [5, 15, 2]).

When $k=1$ these equations coincide with Example 4.5. For $(r,k)=(1,2)$ the
spinor bundle is (effectively) the sum of those in Case (I) and Case (II),
with the moment map being the sum of the ($U(2)$-analogue) of the moment maps
for those. This particular case is studied in detail in [46]

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

This subsection summarizes known compactness results for the moduli space of
solutions to generalized Seiberg-Witten equations; these were established in
[38, 37, 16, 40, 46, 41]. See also [25, 3, 16, 46] for additional details and
exposition. It is well-known that the moduli space of solutions to the
standard Seiberg-Witten equations is compact [24, 17]. This is a consequence
of the inequality

$\langle\gamma(\mu(\Psi,\Psi))\Psi,\Psi\rangle\geq\frac{1}{4}|\Psi|^{4}$

which leads to an a priori bound on the $L^{2}$-norm of the spinor for a
solution of (4.4–4.5). For general Seiberg-Witten data, however, no such
inequality can hold because $\mu^{-1}(0)\neq\emptyset$ in general.
Consequently, there may be sequences of solutions $(\Psi_{i},A_{i})$ with
$\|\Psi_{i}\|_{L^{2}}\to\infty$, which have no convergent subsequences and
therefore lead to a loss of compactness of the moduli space.

In this situation, one may attempt to compactify the moduli space by “blowing
up” the configuration space, i.e. extending it to include a boundary stratum
at infinity. More specifically, we re-parameterize the subset of the
configuration space with $\|\Psi\|_{L^{2}}>0$ by replacing $\Psi$ by a pair
$(\Phi,\varepsilon)$ where $\|\Psi\|_{L^{2}}=\frac{1}{\varepsilon}$ and
$\Phi=\varepsilon\Psi$ is a spinor with unit $L^{2}$-norm. After including the
boundary stratum consisting of configurations with $\varepsilon=0$, the blown-
up configuration space consists of triples
$(\Phi,A,\varepsilon)\in\mathbb{S}(\Gamma(S))\times\mathscr{A}(P)\times[0,\infty]$,
where $\mathbb{S}(\Gamma(S))$ denotes the sphere of spinors with unit
$L^{2}$-norm. The corresponding blown-up Seiberg-Witten equation is

$\displaystyle\not{D}_{A}\Phi$ $\displaystyle=$ $\displaystyle 0$ (4.7)
$\displaystyle\star\varepsilon^{2}F_{A}+\tfrac{1}{2}\mu(\Phi,\Phi)$
$\displaystyle=$ $\displaystyle 0$ (4.8) $\displaystyle\|\Phi\|_{L^{2}}$
$\displaystyle=$ $\displaystyle 1.$ (4.9)

Intuitively, one expects that a sequence of solutions to the original
equations with diverging $L^{2}$-norm should converge to a solution of the
$\varepsilon=0$ verison of the blown-up equations, and that including these
solutions as boundary strata would result in a compact moduli space. The
upcoming Theorem 4.10 establishes a version of this statement, but there are
several important caveats:

  1. 1)

the $\varepsilon=0$ version of (4.8) demands that $\Phi$ lies in the set
$\mu^{-1}(0)$ in each fiber; the latter is a closed subset of each fiber of
$S$, but is not a manifold because $0$ is a cone point in each fiber.

  2. 2)

The energy density $|F_{A}|^{2}$ of the curvature may concentrate along
subsets of $Y$ (resp. $X$) in the limit $\varepsilon\to 0$.

Because of these complications, the limit of a sequence of solutions only
satisfies the $\varepsilon=0$ version of (4.8) away from a closed subset
denoted $\mathcal{Z}$ called the singular set. In the case of a non-abelian
gauge group on a 4-manifold, the bubbling locus arising from Uhlenbeck
compactness is also included in $\mathcal{Z}$.

The following theorem combines compactness results for several generalized
Seiberg-Witten equations which were proved independently by multiple authors.
It unifies results on Case (I) the Seiberg-Witten equations with $r=2$ spinors
in 3 dimensions [16], Case (II) the equations for a flat
$\text{SL}(2,\mathbb{C})$-connection on a 3-manifold [38, 46], Case (III) the
Seiberg-Witten equations with $r=2$ spinors in 4 dimensions [40], and Case
(IV) the complex ASD equation in 4 dimensions [37]. In addition, it includes
regularity statements for the singular set proved in [39, 50]. In each case,
if a sequence of solutions has a subsequence on which the $L^{2}$-norm remains
bounded (i.e. if $\limsup\varepsilon>0$) then standard compactness arguments
apply to show a subsequence converges; thus we state the theorem only in the
case where $\varepsilon\to 0$.

######  Theorem 4.10.

([16, 38, 40, 37, 39, 50]) Suppose that $Y$ is a closed, oriented 3-manifold
(respectively, $X$ a 4-manifold) and $(P,G,\rho,\mu)$ generalized Seiberg-
Witten data corresponding to Cases (I), (II) from the statement of Theorem 1.3
(resp. Cases (III), (IV)). Given a sequence $(\Phi_{i},A_{i},\varepsilon_{i})$
of blown-up configurations satisfying (4.7–4.9), i.e.

$\not{D}_{A_{i}}\Phi_{i}=0\hskip
42.67912pt\star\varepsilon_{i}^{2}F_{A_{i}}+\frac{1}{2}\mu(\Phi_{i},\Phi_{i})=0\hskip
42.67912pt\|\Phi_{i}\|_{L^{2}}=1$

(resp. $\not{D}_{A_{i}}^{+}$ and $F_{A_{i}}^{+}$) with respect to a sequence
of metrics $g_{i}\to g_{0}$ on $Y$ (resp. $X$), such that $\varepsilon_{i}\to
0$.

Then, there exists a triple $(\mathcal{Z}_{0},\Phi_{0},A_{0})$ where

  * •

$\mathcal{Z}_{0}\subseteq Y$ (resp. $X$) is a closed rectifiable subset of
Haudorff codimension at least 2.

  * •

$\Phi_{0}$ is a spinor on $Y-\mathcal{Z}_{0}$ such that $|\Phi_{0}|$ extends
as a continuous function to $Y$ (resp. $X$) with
$\mathcal{Z}_{0}=|\Phi_{0}|^{-1}(0)$.

  * •

$A_{0}$ is a connection on $P|_{Y-\mathcal{Z}_{0}}$ (resp.
$P|_{X-\mathcal{Z}_{0}})$,

such that $(\Phi_{0},A_{0})$ satisfies the $\varepsilon=0$ version of
(4.7–4.9) on $Y-\mathcal{Z}_{0}$ (resp. $X-\mathcal{Z}_{0}$) with respect to
the metric $g_{0}$. Furthermore, there is an $\alpha>0$ such that and after
passing to a subsequence and up to gauge transformations defined on
$Y-\mathcal{Z}_{0}$ (resp. $X-\mathcal{Z}_{0}$),

$\Phi_{i}\overset{L^{2,2}_{loc}}{\rightharpoonup}\Phi_{0}\hskip
42.67912ptA_{i}\overset{L^{1,2}_{loc}}{\rightharpoonup}A_{0}\hskip
42.67912pt|\Phi_{i}|\overset{C^{0,\alpha}}{\to}|\Phi_{0}|.$ (4.10)

Here, local convergence means on compact subsets of $Y-\mathcal{Z}_{0}$ (resp.
$X$), and the half-arrows in the first two statements denote convergence in
the weak topology. In Case (III) the convergence $\Phi_{i}\to\Phi_{0}$ is
(strongly) $L^{2,2}_{loc}$ and in Case (IV) it is only (strongly)
$L^{1,2}_{loc}$.

Finally, in Case (II) and Case (IV) the limiting connection $A_{0}$ has
harmonic curvature, i.e. it satisfies
$d_{A_{0}}F_{A_{0}}=d_{A_{0}}^{\star}F_{A_{0}}=0$ where $d_{A_{0}}$ is the
exterior covariant derivative. ∎

The limiting configuration $(\mathcal{Z}_{0},\Phi_{0},A_{0})$ satisfies the
$\varepsilon=0$ version of (4.7–4.9), which reads

$\not{D}_{A_{0}}\Phi_{0}=0\hskip 42.67912pt\mu(\Phi_{0},\Phi_{0})=0\hskip
42.67912pt\|\Phi_{0}\|_{L^{2}}=1,$ (4.11)

and is considered up to the action of gauge transformations on
$Y-\mathcal{Z}_{0}$ (resp. $X-\mathcal{Z}_{0}$). This equation is not
elliptic, even modulo gauge, as the symbol of ${\mathbb{d}_{A}}$ degenerates
in the limit $\varepsilon\to 0$. The Haydys Correspondence (see [25] Section
2, [3]), however, shows that by exploiting gauge invariance in a different way
than gauge-fixing, (4.11) can be recast as an elliptic equation whose symbol
degenerates only along $\mathcal{Z}_{0}$.

The Haydys Correspondence may be paraphrased as follows (see [3] Section 4 and
[14] for details). Let $\pi:\mu^{-1}(0)\to\mathfrak{X}=\mu^{-1}(0)/G$ be the
projection map to the fiberwise quotient by the action of $G$; $\mathfrak{X}$
is a bundle whose fibers are hyperkähler orbifolds isometric to the
hyperkähler quotient $V///G$.

Given a solution $(\mathcal{Z}_{0},\Phi_{0},A_{0})$ of (4.11), the projection
$s=\pi(\Phi_{0})\in\Gamma(\mathfrak{X})$ is a solution of a different PDE, the
Fueter equation $\mathfrak{F}(s)=0$. The content of the Haydys correspondence
is that one can recover the triple $(\mathcal{Z}_{0},\Phi_{0},A_{0})$ from $s$
even though $s$ retains no information about the connection $A_{0}$ or the
action of $G$. This is done by choosing a local lift $\Phi_{0}$ of $s$ to
$\mu^{-1}(0)\subset\Gamma(S)$ on $Y-\mathcal{Z}_{0}$ (resp.
$X-\mathcal{Z}_{0}$). Such a lift determines a splitting of $S$ along the
image of $\Phi$ as follows. First, decompose $S=T(\mu^{-1}(0))\oplus
T(\mu^{-1}(0))^{\perp}$; the first factor further decomposes as
$T(\mu^{-1}(0))=S^{\text{Re}}_{\Phi_{0}}\oplus\mathfrak{g}\Phi_{0}$ 222The
notation of the superscripts is chosen to agree with the case of the Seiberg-
Witten equations with $r=2$ spinors where the spinor bundle admits a real
structure $\tau$ with $\tau^{2}=Id$, see Sections 2–3 of [25]. Other authors
[3] denote these $\mathfrak{H}_{\Phi_{0}},\mathfrak{N}_{\Phi_{0}}$, while we
reserve the latter notation for the splitting in (2.4)  , where
$\mathfrak{g}\Phi_{0}=\\{v\Phi_{0}\ |\ v\in\mathfrak{g}_{P}\\}$. The
decomposition determined by $\Phi_{0}$ is then

$S|_{Y-\mathcal{Z}_{0}}=S^{\text{Re}}_{\Phi_{0}}\oplus
S^{\text{Im}}_{\Phi_{0}}$ (4.12)

where $S^{\text{Im}}_{\Phi_{0}}=\mathfrak{g}\Phi_{0}\oplus
T(\mu^{-1}(0))^{\perp}$. It can then be shown that the condition that
$\nabla_{A_{0}}\Phi_{0}\in\Gamma(S^{\text{Re}}_{\Phi_{0}})$ determines $A_{0}$
uniquely, in which case $\mathfrak{F}(s)=0$ is equivalent to (4.11). Notice
that one either side of the Haydys correspondence the singular behavior along
$\mathcal{Z}_{0}$ cannot be eliminated.

In Cases (I)–(IV) of Theorem 1.3 (and Examples 4.8, 4.9), the Haydys
Correspondence and the Fueter equation admit a simplification due to the
following additional structure: there exists a linear subspace
$E\subset\mu^{-1}(0)\subset V$ such that every $G$-orbit intersects $E$ in
exactly 2 points. In this situation, for a local lift valued in $E$, one has
$S^{\text{Re}}_{\Phi_{0}}=E$ and the Fueter equation is simply the Dirac
equation on spinors considered up to sign. The data of a solution of (4.11) is
then equivalent to the following data.

######  Definition 4.11.

A $\mathbb{Z}_{2}$-Harmonic Spinor valued in a (real) Clifford module $E\to Y$
(resp. $X$) is a triple $(\mathcal{Z}_{0},\ell_{0},\Phi_{0}),$ where

  1. (1)

$\mathcal{Z}_{0}\subset Y$ (resp. $X$) is a closed, rectifiable subset of
Hausdorff codimension 2,

  2. (2)

$\ell_{0}\to Y-\mathcal{Z}_{0}$ (resp. $X-\mathcal{Z}_{0}$) is real line
bundle, and

  3. (3)

$\Phi_{0}\in\Gamma(E\otimes_{\mathbb{R}}\ell_{0})$ is a spinor with
$\nabla\Phi_{0}\in L^{2}$ and whose norm $|\Phi_{0}|$ extends to $Y$ (resp.
$X$) as a Hölder continuous function,

such that

$\|\Phi_{0}\|_{L^{2}}=1\hskip
42.67912pt\not{D}_{\mathcal{Z}_{0}}\Phi_{0}=0\hskip
42.67912pt|\Phi_{0}|^{-1}(0)=\mathcal{Z}_{0}.$ (4.13)

$\mathbb{Z}_{2}$-harmonic spinors are always considered up to the equivalence
$\Phi_{0}\mapsto-\Phi_{0}$.

In (4.13), the Dirac operator is formed using the connection arising from the
connection on $E$ and the unique flat connection with holonomy in
$\mathbb{Z}_{2}$ on the line bundle $\ell_{0}$. Under the Haydys
correspondence, this unique flat connection of $\ell_{0}$ is equivalent to the
connection arising from $A_{0}$ in the conclusion of Theorem 4.10 (see [25]
Section 3) and the spin connection. For all the cases of Theorem 1.3, the
$\mathbb{Z}_{2}$-harmonic spinors that arise are sections of Clifford modules
of real rank 4. In Case (II), and Case (IV) the $\mathbb{Z}_{2}$-harmonic
spinors arising from Theorem 4.10 are also called $\mathbb{Z}_{2}$-harmonic
1-forms, i.e. spinors for the Clifford modules
$(\Omega^{0}\oplus\Omega^{1})(\mathbb{R})$ or $\Omega^{1}(\mathbb{R})$ in 3
and 4 dimensions respectively.

##  5\. Concentration Properties of Generalized Seiberg-Witten Equations

As explained in the introduction, it is desirable to improve the convergence
statements in Theorem 4.10 to $C^{\infty}_{loc}$. Although the equations are
elliptic for $\varepsilon\neq 0$, naive attempts to bootstrap convergence are
foiled by increasingly large powers of $\varepsilon$ entering the elliptic
estimates (see Section 6). The abstract framework introduced in Section 2–3
can be applied to overcome this problem. Let
$(\Phi_{i},A_{i},\varepsilon_{i})$ be a sequence of solutions to (4.7–4.9)
converging to a $\mathbb{Z}_{2}$-harmonic spinor
$(\mathcal{Z}_{0},A_{0},\Phi_{0})$ in the sense of Theorem 4.10. Then the un-
renormalized configuration may be written as a perturbation of the limit
$(\frac{\Phi_{i}}{\varepsilon_{i}},A_{i})=(\frac{\Phi_{0}}{\varepsilon_{i}},A_{0})+(\varphi_{i},a_{i})$,
where $(\varphi_{i},a_{i})$ solve the equation

$\mathcal{L}_{(\frac{\Phi_{0}}{\varepsilon_{i}},A_{0})}(\varphi_{i},a_{i})+Q(\varphi_{i},a_{i})=-E_{0}$
(5.1)

Here, $\mathcal{L}_{(\Psi,A)}$ denotes the linearized Seiberg-Witten equations
linearized at the configuration $(\Psi,A)$, $Q(-,-)$ is a quadratic term, and
$E_{0}=SW(\Phi_{0},A_{0})$ is the error by which the limiting configuration
fails to solve the Seiberg-Witten equations. The next two subsections show
that (5.1) behaves as a concentrating Dirac operator with fixed degeneracy as
$\varepsilon\to 0$, with the singular set $\mathcal{Z}_{0}$ occupying the role
of the set denoted by the same symbol in Sections 2–3.

###  5.1. The Concentration Property

In this subsection, it is shown that the linearization of the generalized
Seiberg-Witten equations associated to any data satisfy the concentration
property of Definition 2.3. In Subsection 5.2, it is shown that the non-
linearity of the equations satisfies the similar criteria necessary for
Corollary 1.2 to apply.

On a 3-manifold, the linearization of the generalized Seiberg-Witten equations
(4.4–4.5) is as follows. Let $(\frac{\Phi}{\varepsilon},A)$ denote a
configuration, where $\|\Phi\|_{L^{2}}=1$ and $\varepsilon>0$. The
linearization at $(\frac{\Phi}{\varepsilon},A)$ acting on a variation
$(\varphi,a)$ is

$\frac{d}{ds}\Big{|}_{s=0}SW\left(\tfrac{\Phi}{\varepsilon}+s\varphi,A+sA\right)=\begin{pmatrix}\not{D}_{A}\varphi+\gamma(a)\tfrac{\Phi}{\varepsilon}\vspace{.15cm}\\\
\tfrac{\mu(\varphi,\Phi)}{\varepsilon}+\star d_{A}a\end{pmatrix}.$

where $\mu(\varphi,\psi)$ denotes the polarization of the moment map. To make
this into an elliptic system, we supplement the configuration
$(\frac{\Phi}{\varepsilon},A)$ with an auxiliary $0$-form
$a_{0}\in\Omega^{0}(\mathfrak{g}_{P})$ and impose the gauge-fixing condition

$-d_{A}^{\star}a+\frac{\mu_{0}(\varphi,\Phi)}{\varepsilon}=0$ (5.2)

where we have updated our notation so that
$a=(a_{0},a_{1})\in(\Omega^{0}\oplus\Omega^{1})(\mathfrak{g}_{P})$ and
$\mu=(\mu_{0},\mu_{1})\in(\Omega^{0}\oplus\Omega^{1})(\mathfrak{g}_{P})$.
Here, $\mu_{0}:S\otimes S\to\mathfrak{g}_{P}$ is defined in a local
trivialization $t_{\alpha}$ of $\mathfrak{g}_{P}$ by
$\mu_{0}(\varphi,\psi):=\sum_{\alpha}\langle\mathfrak{t}_{\alpha}\varphi,\psi\rangle\mathfrak{t}_{\alpha}$.
The equations may then be written in the suggestive form:

$\mathcal{L}_{(\Phi,A,\varepsilon)}\begin{pmatrix}\varphi\\\
a\end{pmatrix}=\left(D+\frac{1}{\varepsilon}\mathcal{A}\right)\begin{pmatrix}\varphi\\\
a\end{pmatrix}$ (5.3)

where

$D=\begin{pmatrix}\not{D}_{A}&0\vspace{.15cm}\\\
0&\mathbb{d}_{A}\end{pmatrix}\ \ \hskip 85.35826pt\ \
\mathcal{A}=\begin{pmatrix}0&\gamma(\\_)\Phi\vspace{.15cm}\\\
\mu(\\_,\Phi)&0\end{pmatrix},$ (5.4)

and $\mathbb{d}_{A}=\begin{pmatrix}0&-d_{A}^{\star}\\\ -d_{A}&\star
d_{A}\end{pmatrix}\begin{pmatrix}a_{0}\\\ a_{1}\end{pmatrix}.$ On a 4-manifold
$X$, the auxiliary form $a_{0}$ is not necessary for ellipticity, and the
formula (5.3) for $\mathcal{L}_{(\Phi,A,\varepsilon)}$ is the same after
replacing $\not{D}_{A}$ by $\not{D}_{A}^{+}$ and $\mathbb{d}_{A}$ by
$\mathbb{d}_{A}^{+}=(-d_{A}^{\star},d_{A}^{+})$.

The symbol of $D$ in 5.4 is

$\sigma_{D}(\xi)=\begin{pmatrix}\rho(\xi)&0\vspace{.15cm}\\\
0&\mathbb{cl}(\xi)\end{pmatrix}\hskip 28.45274pt\text{for }\hskip
28.45274pt\xi\in\Omega^{1}(T^{\star}Y).$

where $\rho,{\bf cl}$ are the symbols of $\not{D}_{A},\mathbb{d}_{A}$
respectively (or $\not{D}_{A}^{+},\mathbb{d}_{A}^{+}$ in dimension 4). The
next lemma is a particular instance of a more general result concerning
commuting Clifford pairs discussed in [18, 19].

######  Lemma 5.1.

The linearization of the generalized Seiberg-Witten equations written in the
form (5.3)–(5.4) obeys the Concentration Property of Definition 2.3, i.e.
$\sigma_{D}$ and $\mathcal{A}$ satisfy

$\mathcal{A}^{\star}\sigma_{D}(\xi)=\sigma_{D}(\xi)^{\star}\mathcal{A}$ (5.5)

for all $\xi\in T^{\star}Y$.

###### Proof.

For each $\xi\in\Omega^{1}(\mathbb{R})$, (5.5) is equivalent to the following
equalities for all $a\in(\Omega^{0}\oplus\Omega^{1})(\mathfrak{g}_{P})$, and
$\varphi,\psi\in\Gamma(S)$:

$\displaystyle\rho(\xi)\gamma(a)$ $\displaystyle=$ $\displaystyle-\gamma({\bf
cl}(\xi)a)$ (5.6) $\displaystyle{\bf cl}(\xi)\mu(\varphi,\psi)$
$\displaystyle=$ $\displaystyle-\mu(\rho(\xi)\varphi,\psi).$ (5.7)

In (5.6), ${\bf cl}$ is extended to act on $\mathfrak{g}_{P}$-valued forms via
the form components. The expressions (5.6–5.7) are easily verified in an
oriented orthonormal frame of $T^{\star}Y$ (resp. $T^{\star}X$) and
$\mathfrak{g}_{P}$.

∎

###  5.2. Fixed Degeneracies

This subsection shows that the fixed degeneracy assumption of Definition 2.3
is satisfied for the linearized Seiberg-Witten equations, and that the non-
linear terms have the form necessary for Corollary 1.2 to apply.

To begin, there is the following alternative description of the subspaces
$S^{\text{Re}},S^{\text{Im}}$ from (4.12). Since $\frac{1}{2}\mu(\Phi,\Phi)$
is bilinear, its linearization at a spinor $\Phi_{0}$ is given by
$\mu(-,\Phi_{0})$, where $\mu$ is now interpretted as the bilinear form
arising as the polarization of the original quadratic map. On
$Y-\mathcal{Z}_{0}$ where $\Phi_{0}$ is non-vanishing, one then has ,

$S^{\text{Re}}=\ker(\mu(-,\Phi_{0}))\hskip 36.98866pt\text{}\hskip
36.98866ptS^{\text{Im}}=(S^{\text{Re}})^{\perp}$ (5.8)

(see [1], Proposition 2.1.5 and [25], Section 2). The next proposition is
proved on a case by case basis for the different equations to which Theorem
1.3 applies. The result could be shown using a more abstract framework, but we
find it instructive to give explicit descriptions as the splitting described
by the proposition provides a novel way of writing many of these equations
which may be useful elsewhere.

######  Proposition 5.2.

The linearization of the generalized Seiberg-Witten equations written in the
form (5.3)–(5.4) obeys the fixed degeneracy assumption of Definition 2.3, i.e.
in dimension $n=3$ there is a splitting of vector bundles

$S\oplus(\Omega^{0}\oplus\Omega^{1})(\mathfrak{g}_{P})=\mathfrak{N}\oplus\mathfrak{H}$

which respects Clifford multiplication $\gamma$ and is parallel with respect
to $\nabla_{A_{0}}$ such that the map $\mathcal{A}$ of (5.4) has the block
diagonal form (2.4).

In dimension $n=4$ there are splittings

$S^{+}\oplus\Omega^{1}(\mathfrak{g}_{P})=\mathfrak{N}^{+}\oplus\mathfrak{H}^{+}\hskip
28.45274pt\text{ and }\hskip
28.45274ptS^{-}\oplus(\Omega^{0}\oplus\Omega^{2}_{+})(\mathfrak{g}_{P})=\mathfrak{N}^{-}\oplus\mathfrak{H}^{-}$

for which the same conclusions hold.

###### Proof.

The proposition is proved separately for cases (I)–(VI) of Theorem 1.3.

Case (I): (Two spinor Seiberg-Witten on $Y^{3}$). In this case, the
$\mathbb{Z}_{2}$-harmonic spinors are as in Definition 4.11 with $E$ the
spinor bundle of a spin structure on $Y$. A limiting configuration
$(\mathcal{Z}_{0},A_{0},\Phi_{0})$ satisfying (4.11) gives rise to such a
$\mathbb{Z}_{2}$-harmonic spinor as follows: the Haydys Correspondence gives
an isomorphism $S^{\text{Re}}_{\Phi_{0}}\simeq E\otimes\ell_{0}$, and under
this isomorphism the connection induced on $S^{\text{Re}}$ by $A_{0}$ is
intertwined with the connection formed from the spin connection on $E$ and the
unique flat connection on $\ell_{0}$ with holonomy in $\mathbb{Z}_{2}$. Thus
in this case, the limiting connection $A_{0}$ on $Y-\mathcal{Z}_{0}$ is itself
flat with holonomy in $\mathbb{Z}_{2}$ (see also [16] Appendix I and Sections
2–3 of [25]). We tacitly also refer to the triple
$(\mathcal{Z}_{0},A_{0},\Phi_{0})$ as a $\mathbb{Z}_{2}$-harmonic spinor.

Let $S^{\text{Re}}\to Y-\mathcal{Z}_{0}$ be the bundle defined in (5.8). Then
define

$\mathfrak{N}:=S^{\text{Re}}\hskip
85.35826pt\mathfrak{H}:=S^{\text{Im}}\oplus(\Omega^{0}\oplus\Omega^{1})(i\mathbb{R}).$
(5.9)

The splitting 5.9 is respected by Clifford multiplication by forms in
$\alpha\in\Omega^{0}(\mathbb{R})\oplus\Omega^{1}(\mathbb{R})$. Indeed, the
linearized moment map (cf. 4.6) now takes the form

$\mu(\varphi,\Phi_{0})=\sum_{j=0}^{3}\langle ie^{j}\varphi,\Phi_{0}\rangle
e^{j}\otimes i$ (5.10)

where $\\{e^{0}=1,e^{1},e^{2},e^{3}\\}$ is an orthonormal frame of
$(\Omega^{0}\oplus\Omega^{1})(\mathbb{R})$ and $i=\sqrt{-1}$ is the basis
element of the Lie algebra of $U(1)$. To show Clifford multiplication respects
the splitting, it suffices to show that $\varphi\in S^{\text{Re}}$, i.e.
$\mu(\varphi,\Phi_{0})=0$ then $\mu(e^{k}.\varphi,\Phi_{0})=0$ as well. This
following from the observation that replacing $e^{j}$ by $e^{j}.e^{k}$ in
(5.10) simply results in a permutation of the frame
$\\{e^{0},e^{1},e^{2},e^{3}\\}$. Since Clifford multiplication respects
$S^{\text{Re}}$, it also respects the orthogonal complement $S^{\text{Im}}$.

Next, we show that the connection $\nabla_{A_{0}}$ respects the splitting as
well, i.e. $\nabla_{A_{0}}\varphi^{\text{Re}}\in S^{\text{Re}}$ and likewise
for $S^{\text{Im}}$. Indeed, the preceding paragraph implies that
$S^{\text{Re}}=\\{b.\Phi_{0}\ |\
b\in\Omega^{0}(\mathbb{R})\oplus\Omega^{1}(\mathbb{R})\\}$. Similarly,
$S^{\text{Im}}=\\{(ia).\Phi_{0}\ |\
ia\in\Omega^{0}(\mathbb{R})\oplus\Omega^{1}(i\mathbb{R})\\}$. Using this
description, let $\varphi^{\text{Re}}=b.\Phi_{0}\in\Gamma(S^{\text{Re}})$ be a
spinor in $S^{\text{Re}}$. Then

$\nabla_{A_{0}}\varphi^{\text{Re}}=\nabla_{A_{0}}(b.\Phi_{0})=db.\Phi_{0}+b.\nabla_{A_{0}}\Phi_{0}\in\Gamma(S^{\text{Re}})$

since $\nabla_{A_{0}}\Phi_{0}\in S^{\text{Re}}$ by the Haydys Correspondence,
and $db.\Phi_{0}=-(\star db).\Phi_{0}\in S^{\text{Re}}$ by the preceding
paragraph. An identical argument applies to show that $S^{\text{Im}}$ is
preserved as well. The decomposition 5.9 therefore satisfies both hypotheses
of Definition 2.1.

Writing a configuration $(\varphi^{\text{Re}},\varphi^{\text{Im}},a)$ in this
decomposition, the linearized Seiberg-Witten equations at
$(\frac{\Phi_{0}}{\varepsilon},A_{0})$ take the form (5.3) where

$D=\begin{pmatrix}\not{D}_{A_{0}}^{\text{Re}}&0&0\\\
0&\not{D}_{A_{0}}^{\text{Im}}&0\\\
0&0&\mathbb{d}\end{pmatrix}\begin{pmatrix}\varphi^{\text{Re}}\\\
\varphi^{\text{Im}}\\\ a\end{pmatrix}\hskip
56.9055pt\mathcal{A}=\begin{pmatrix}0&0&0\\\ 0&0&\gamma(\\_){\Phi_{0}}\\\
0&{\mu(\\_,\Phi_{0})}&0\end{pmatrix}\begin{pmatrix}\varphi^{\text{Re}}\\\
\varphi^{\text{Im}}\\\ a\end{pmatrix}.$ (5.11)

Case (II): (Flat $\text{SL}(2,\mathbb{C})$ Connections on $Y^{3}$). In this
case, the $\mathbb{Z}_{2}$-harmonic spinors are $\mathbb{Z}_{2}$-harmonic
1-forms, i.e. ones with $E=(\Omega^{0}\oplus\Omega^{1})(\mathbb{R})$. As in
the previous case, the Haydys correspondence gives rise to an isomorphism
$S^{\text{Re}}\simeq E\otimes\ell_{0}$, which intertwines the connection
induced by $A_{0}$ and the connection on $E\otimes\ell_{0}$ formed from the
Levi-Civitas connection and the unique flat connection with $\mathbb{Z}_{2}$
holonomy on $\ell_{0}$. Again, we tacitly refer to the triple
$(\mathcal{Z}_{0},A_{0},\Phi_{0})$ as a $\mathbb{Z}_{2}$-harmonic 1-form.

Let $S^{\text{Re}},S^{\text{Im}}$ be defined by (5.8). This case differs
slightly from the previous one because
$\gamma(-)\Phi_{0}:(\Omega^{0}\oplus\Omega^{1})(\mathfrak{g}_{P})\to
S^{\text{Im}}$ now has kernel. Explicitly,
$\gamma(a)\Phi_{0}=[a\wedge\Phi_{0}]$ and the kernel consists of
$\mathfrak{g}_{P}$-valued forms whose $\mathfrak{g}_{P}$-component is parallel
to $\Phi_{0}$. Define

$\Omega^{\text{Re}}=\ker(\gamma(-)\Phi_{0})\hskip
85.35826pt\Omega^{\text{Im}}=\ker(\gamma(-)\Phi_{0})^{\perp}$

then the relevant splitting is

$\mathfrak{N}=S^{\text{Re}}\oplus\Omega^{\text{Re}}\hskip
85.35826pt\mathfrak{H}=S^{\text{Im}}\oplus\Omega^{\text{Im}}.$ (5.12)

As before, we have $S^{\text{Re}}=\\{b.\Phi_{0}\ |\
b\in(\Omega^{0}\oplus\Omega^{1})(\mathbb{R})\\}$, and
$S^{\text{Im}}=\\{\gamma(a)\Phi_{0}\ |\ a\in\Omega^{\text{Im}}\\}$. The proof
that $\nabla_{A_{0}}$ respects $S^{\text{Re}}$ follows by the same argument as
in Case (i), and differentiating the orthogonality relation
$0=\langle\varphi^{\text{Re}},\varphi^{\text{Im}}\rangle$ shows
$\nabla_{A_{0}}$ preserves $S^{\text{Im}}$ as well. In this case, since
$S=(\Omega^{0}\oplus\Omega^{1})(\mathfrak{g}_{P})$, and
$\mu(-,\Phi_{0})=[-\wedge\Phi_{0}]=\gamma(-)\Phi_{0}$ one secretly has
$\Omega^{\text{Re}}=S^{\text{Re}}$ and likewise for the imaginary components;
thus the splitting of the form components, being the same splitting, is also
parallel.

By the description of $S^{\text{Im}}$ following (5.12), the image of
$\gamma(-)\Phi_{0}$ lies in $S^{\text{Im}}$. Similarly, $\mu(-,\Phi_{0})$ has
image only in $\Omega^{\text{Im}}$ since $\langle
a^{\text{Re}},\mu(\varphi,\Phi_{0})\rangle=\langle\gamma(a^{\text{Re}})\Phi_{0},\varphi\rangle=0$
by definition of $\Omega^{\text{Re}}$. Writing a configuration
$(\varphi^{\text{Re}},a^{\text{Re}},\varphi^{\text{Im}},a^{\text{Im}})$ in the
decomposition (5.12), the linearized Seiberg-Witten equations at
$(\frac{\Phi_{0}}{\varepsilon},A_{0})$ take the form (5.3) where

$D=\begin{pmatrix}\not{D}_{A_{0}}^{\text{Re}}&0&0&0\\\
0&\mathbb{d}_{A_{0}}^{\text{Re}}&0&0\\\ 0&0&\not{D}_{A_{0}}^{\text{Im}}&0\\\
0&0&0&\mathbb{d}_{A_{0}}^{\text{Im}}\end{pmatrix}\begin{pmatrix}\varphi^{\text{Re}}\\\
a^{\text{Re}}\\\ \varphi^{\text{Im}}\\\ a^{\text{Im}}\end{pmatrix}\hskip
56.9055pt\mathcal{A}=\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\
0&0&0&\gamma(\\_){\Phi_{0}}\\\
0&0&{\mu(\\_,\Phi_{0})}&0\end{pmatrix}\begin{pmatrix}\varphi^{\text{Re}}\\\
a^{\text{Re}}\\\ \varphi^{\text{Im}}\\\ a^{\text{Im}}\end{pmatrix}$ (5.13)

Case (III): (Two spinor Seiberg-Witten on $X^{4}$). The four-dimensional case
for the two-spinor Seiberg-Witten equations is virtually identical to the
three-dimensional case: let $(S^{+})^{\text{Re}}=\ker(\mu(-,\Phi_{0}))$ as in
(5.8). Set

$\mathfrak{N}^{+}=(S^{+})^{\text{Re}}\hskip
85.35826pt\mathfrak{H}^{+}:=(S^{+})^{\text{Im}}\oplus\Omega^{1}(i\mathbb{R}).$
(5.14)

Then, let $(S^{-})^{\text{Re}}=\\{\alpha.\varphi^{\text{Re}}\ |\
\alpha\in\Omega^{1}(\mathbb{R})\ ,\ \varphi\in(S^{+})^{\text{Re}}\\}$ and
likewise for $(S^{-})^{\text{Im}}$, and define

$\mathfrak{N}^{-}=(S^{-})^{\text{Re}}\hskip
85.35826pt\mathfrak{H}^{-}:=(S^{+})^{\text{Im}}\oplus(\Omega^{0}\oplus\Omega^{2}_{+})(i\mathbb{R}).$
(5.15)

The proof that the fixed degeneracy hypothesis is satisfied carries over
mutatis mutandis from the 3-dimensional case, and the expressions (5.11) are
identical regarded as matrices for the splittings (5.14) and (5.15).

Case (IV): (Complex $ASD$ Equations on $X^{4}$). This case is the four-
dimensional version of Case (II) in the same way that Case (IV) is the four-
dimensional version of Case (I).

∎

######  Remark 5.3.

To elaborate on Remark 1.6, the requirement that the splitting be parallel is
the barrier to extending Theorem 1.3 to other generalized Seiberg-Witten
equations (e.g. the Seiberg-Witten equations with $r>2$ spinors). In such
cases, $\mu^{-1}(0)$ is a not simply the quotient of a vector space by a
finite group, and the proof that the splitting is parallel breaks down. (In
such cases, and the limiting Fueter equation has no interpretation as a linear
Dirac equation). Analytically, the failure of the splitting to be parallel
causes the assertion (2.7), as it would in general involve terms including the
covariant derivative of $q_{0}$ (cf [3] Section 6.2).

The next lemma verifies the first two hypotheses of Corollary 1.2 apply in the
case of generalized Seiberg-Witten equations. The third hypothesis, i.e. the
estimates (1.7), are the subject of the upcoming Lemmas 6.2 and 6.3 in the
next section.

######  Lemma 5.4.

The generalized Seiberg-Witten equations in Cases (I) – (VI) can be written in
the form

$(D+\tfrac{1}{\varepsilon}\mathcal{A})\mathfrak{q}+Q(\mathfrak{q})=f$ (5.16)

where $\mathfrak{q}=(\varphi,a)$ and $f$ and $Q$ satisfy the following:

  * (i)

$f\in\Gamma(\mathfrak{N})$

  * (ii)

$Q(\mathfrak{q})=Q_{1}(\mathfrak{q})\pi_{\mathfrak{H}}(\mathfrak{q})$ for a
linear operator $Q_{1}$

  * (iii)

$Q_{1}^{\star}\sigma_{D}(\xi)=\sigma_{D}(\xi)^{\star}Q_{1}$ for all $\xi\in
T^{\star}Y$ (resp. $T^{\star}X$).

###### Proof.

The (un-renormalized) $\mathbb{Z}_{2}$-harmonic spinor satisfies the equations

$\not{D}_{A_{0}}\left(\tfrac{\Phi_{0}}{\varepsilon}\right)=0\hskip
56.9055pt\tfrac{1}{2}\tfrac{\mu(\Phi_{0},\Phi_{0})}{\varepsilon^{2}}=0$

on $Y-\mathcal{Z}_{0}$ (resp. $F_{A_{0}}^{+},\not{D}_{A_{0}}^{+}$ on
$X-\mathcal{Z}_{0}$). In particular, $(\tfrac{\Phi_{0}}{\varepsilon},A_{0})$
fails to solve the Seiberg-Witten equations on $Y-\mathcal{Z}_{0}$ (resp.
$X-\mathcal{Z}_{0}$) by $\star F_{A_{0}}$ (resp. $F_{A_{0}}^{+}$). Thus
$(\tfrac{\Phi_{0}}{\varepsilon},A_{0})+(\varphi,a)$ satisfies the Seiberg-
Witten equations if and only if $(\varphi,a)$ satisfies the deformation
equation (cf. (5.1)

$\Big{(}\mathcal{L}_{\left(\tfrac{\Phi_{0}}{\varepsilon},A_{0}\right)}+Q\Big{)}(\varphi,a)=-E_{0}.$
(5.17)

where $Q(\varphi,a)=(\gamma(a)\varphi\ ,\ \mu(\varphi,\varphi))$, and
$E_{0}=-\star F_{A_{0}}$ (resp. $-F_{A_{0}}^{+}$). The Haydys Correspondence
implies that $F_{A_{0}}\in\Omega^{\text{Re}}$ (see Appendix C of [2] ), hence
$E_{0}\in\Gamma(\mathfrak{N})$ and statement (i) is satisfied. The proof now
proceeds in each of the cases individually.

Case (I): (Two Spinor Seiberg-Witten on $Y^{3}$). As in Case (I) in the proof
of Lemma 5.2, Clifford multiplication by $\mathbb{R}$-valued forms preserves
the splitting $S=S^{\text{Re}}\oplus S^{\text{Im}}$ while Clifford
multiplication by $\mathfrak{g}_{P}=i\mathbb{R}$-valued forms reverses it.
Moreover, in this case, the Haydys Correspondence implies that $F_{A_{0}}=0$.
Thus in the splitting (5.9) of Lemma 5.2 the non-linear deformation equation
(5.17) on triples $(\varphi^{\text{Re}},\varphi^{\text{Im}},a)$ takes the form

$\displaystyle\underbrace{\begin{pmatrix}\not{D}^{\text{Re}}_{A_{0}}\varphi^{\text{Re}}\\\
\not{D}^{\text{Im}}_{A_{0}}\varphi^{\text{Im}}+\gamma(a)\frac{\Phi_{0}}{\varepsilon}\\\
\mathbb{d}a\ +\
\tfrac{\mu(\varphi^{\text{Im}},\Phi_{0})}{\varepsilon}\end{pmatrix}}_{\mathcal{L}_{(\Phi_{0},A_{0}})}\
\ +\ \ \underbrace{\begin{pmatrix}\gamma(a)\varphi^{\text{Im}}\\\
\gamma(a)\varphi^{\text{Re}}\\\
2\mu(\varphi^{\text{Im}},\varphi^{\text{Re}}\end{pmatrix}}_{Q(\varphi,a)}\ \
=\ \ \begin{pmatrix}0\\\ 0\\\ 0\end{pmatrix}.$ (5.18)

Note that each term of $Q(\varphi,a)$ contains at least a linear factor in
$(\varphi^{\text{Im}},a)$ which is the assertion of statement (ii) of the
lemma. The equations can therefore be written as:

$\displaystyle\left(\begin{pmatrix}\not{D}_{A_{0}}^{\text{Re}}&0&0\\\
0&\not{D}_{A_{0}}^{\text{Im}}&0\\\
0&0&\mathbb{d}\end{pmatrix}+\frac{1}{\varepsilon}\begin{pmatrix}0&0&\gamma(\\_)\varepsilon\varphi^{\text{Im}}\\\
0&0&\gamma(\\_)(\Phi_{0}+\varepsilon\varphi^{\text{Re}})\\\
0&\mu(\\_,\Phi_{0}+\varepsilon\varphi^{\text{Re}})&0\end{pmatrix}\right)\begin{pmatrix}\varphi^{\text{Re}}\\\
\varphi^{\text{Im}}\\\ a\end{pmatrix}=0.$ (5.19)

which has the form (5.16). Equivalently, the equation has been recast as a
concentrating Dirac operator with $\mathcal{A}$ in the form (3.4) with

$\begin{pmatrix}0&0\\\ 0&A_{\mathfrak{H}}\end{pmatrix}=\begin{pmatrix}0&0&0\\\
0&0&\gamma(\\_)\Phi_{0}\\\ 0&\mu(\\_,\Phi_{0})&0\end{pmatrix}\hskip
56.9055ptA_{\varepsilon}=\begin{pmatrix}0&0&\gamma(\\_)\varepsilon\varphi^{\text{Im}}\\\
0&0&\gamma(\\_)\varepsilon\varphi^{\text{Re}}\\\
\mu(\\_,\varepsilon\varphi^{\text{Im}})&\mu(\\_,\varepsilon\varphi^{\text{Im}})&0\end{pmatrix}.$
(5.20)

where $A_{\mathfrak{H}}$ is the lower $2\times 2$ block. In this form, it is
now obvious that item (iii) holds by Lemma 5.1, (as the proof of the
commutation relation applied for any spinor).

Case (II): (Flat $\text{SL}(2,\mathbb{C})$ Connections on $Y^{3}$) The only
salient difference between this case and the previous one is that there is an
additional non-linear term $a\wedge a$ arising from the non-abelianness of the
connection. In this case, item (ii) of the lemma follows from the following
observation: since $a\in\Omega^{\text{Re}},\varphi\in S^{\text{Re}}$ means
that that $a,\varphi$ are $\mathfrak{g}_{P}$-valued 1-forms whose Lie algebra
component is parallel to $\Phi_{0}$, the commutator
$\gamma(a^{\text{Re}})\varphi^{\text{Re}}=[a^{\text{Re}}\wedge\varphi^{\text{Re}}]=0$.
Likewise,
$\mu(\varphi^{\text{Re}},\varphi^{\text{Re}})=[\varphi^{\text{Re}}\wedge\varphi^{\text{Re}}]=0$
and $a^{\text{Re}}\wedge a^{\text{Re}}=0$. Consequently, (5.17) in this case
takes the form

$\displaystyle\not{D}_{A_{0}}^{\text{Re}}\varphi^{\text{Re}}\hskip 28.45274pt\
+\ \hskip 49.79231pt\
\Pi^{\text{Re}}(\gamma(a^{\text{Re}})\varphi^{\text{Im}}\ +\
\gamma(a^{\text{Im}})\varphi^{\text{Re}}\ +\
\gamma(a^{\text{Im}})\varphi^{\text{Im}})$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle\mathbb{d}_{A_{0}}^{\text{Re}}a^{\text{Re}}\hskip 28.45274pt\ +\
\Pi^{\text{Re}}(2\mu(\varphi^{\text{Re}},\varphi^{\text{Im}})\ +\
\mu(\varphi^{\text{Im}},\varphi^{\text{Im}})\ +\
(a^{\text{Re}}+a^{\text{Im}})\wedge(a^{\text{Im}})$ $\displaystyle=$
$\displaystyle-\star F_{A_{0}}$
$\displaystyle\not{D}_{A_{0}}^{\text{Im}}\varphi^{\text{Im}}+\gamma(a)\tfrac{\Phi_{0}}{\varepsilon}\
\ \ +\ \hskip 49.79231pt\
\Pi^{\text{Im}}(\gamma(a^{\text{Re}})\varphi^{\text{Im}}\ +\
\gamma(a^{\text{Im}})\varphi^{\text{Re}}\ +\
\gamma(a^{\text{Im}})\varphi^{\text{Im}})$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle\underbrace{\mathbb{d}_{A_{0}}^{\text{Im}}a^{\text{Im}}+\tfrac{\mu(\varphi^{\text{Im}},\Phi_{0})}{\varepsilon}}_{\mathcal{L}_{(\Phi_{0},A_{0})}}\
+\ \underbrace{\Pi^{\text{Im}}(2\mu(\varphi^{\text{Re}},\varphi^{\text{Im}})\
+\ \mu(\varphi^{\text{Im}},\varphi^{\text{Im}})\ +\
(a^{\text{Re}}+a^{\text{Im}})\wedge(a^{\text{Im}})}_{Q(\varphi,a)}$
$\displaystyle=$ $\displaystyle 0.$ (5.21)

Thus, with $A_{\mathfrak{H}}$ being the lower block of (5.13) and

$A_{\varepsilon}=\begin{pmatrix}0&\Pi^{\text{Re}}(\gamma(\\_)\varepsilon\varphi^{\text{Im}})&0&\Pi^{\text{Re}}(\gamma(\\_)\varepsilon\varphi)\\\
\Pi^{\text{Re}}(\mu(\\_,\varepsilon\varphi^{\text{Im}}))&\Pi^{\text{Re}}(\\_\wedge\varepsilon
a)&\Pi^{\text{Re}}(\mu(\\_,\varepsilon\varphi))&0\\\
0&\Pi^{\text{Im}}(\gamma(\\_)\varepsilon\varphi^{\text{Im}})&0&\Pi^{\text{Im}}(\gamma(\\_)\varepsilon\varphi)\\\
\Pi^{\text{Im}}(\mu(\\_,\varepsilon\varphi^{\text{Im}}))&0&\Pi^{\text{Im}}(\mu(\\_,\varepsilon\varphi))&\Pi^{\text{Im}}(\\_\wedge\varepsilon
a)\end{pmatrix}\begin{pmatrix}\varphi^{\text{Re}}\\\ a^{\text{Re}}\\\
\varphi^{\text{Im}}\\\ a^{\text{Im}}\end{pmatrix}$ (5.22)

the operator has the desired form. Item (iii) of the lemma follows identically
to the previous case.

Cases (III) – (IV): These cases are analogous to Cases (I) and (II) in the
same way as Lemma 5.2.

∎

##  6\. Bootstrapping

The convergence $\Phi_{i}\overset{L^{2,2}}{\rightharpoonup}\Phi_{0}$ and
$A_{i}\overset{L^{1,2}}{\rightharpoonup}A_{0}$ obtained in Theorem 4.10 cannot
be naively bootstrapped to higher Sobolev spaces using elliptic estimates in
the standard way. Indeed, if it is known that $A_{i}\to
A_{0},\Phi_{i}\to\Phi_{0}$ in $L^{k,2}$ on a compact subset $K^{\prime}\Subset
Y-\mathcal{Z}_{0}$, or equivalently using the notation of (5.1), $a_{i}\ ,\
\varepsilon_{i}\varphi_{i}\to 0$ in $L^{k,2}$. Recalling that Theorem 4.10
asserts convergence for the renormalized spinor
$\Phi_{i}=\Phi_{0}+\varepsilon_{i}\varphi_{i}$, then the elliptic estimate on
a compact subsets $K\Subset K^{\prime}\Subset Y-\mathcal{Z}_{0}$ reads:

$\|a_{i}\|_{L^{k+1,2}(K)}\leq
C_{k}\left(\frac{1}{\varepsilon_{i}^{2}}\|\mu(\Phi_{0},\varepsilon_{i}\varphi_{i})+\mu(\varepsilon_{i}\varphi_{i},\varepsilon_{i}\varphi_{i})\|_{L^{k,2}(K^{\prime})}+\|a\|_{L^{k,2}(K^{\prime})}\right).$

Because of the factor of $\varepsilon_{i}^{-2}$ on the right hand side,
convergence $a_{i}\to 0$ in $L^{k+1,2}$ does not follow. Concluding
convergence in this way requires knowing that $\varphi_{i}\to 0$ in $L^{k,2}$
at least as fast as $\varepsilon_{i}^{2}$. The exponential convergence
furnished by Corollary 1.2 overcomes this issue and allows the bootstrapping
to proceed.

Lemmas 5.1 and 5.2 show that the assumptions for Theorem 1.1 are satisfied for
the linearized Seiberg-Witten equations in all case (I)–(IV). The next two
lemmas show that the non-linear terms satisfy the final assumption 1.7 the
necessary to apply Corollary 1.2 in the 3-dimensional and 4-dimensional cases
respectively. First, we note the following fact:

######  Lemma 6.1.

Suppose $(\Phi_{i},A_{i})\to(\mathcal{Z}_{0},\Phi_{0},A_{0})$ is a sequence of
generalized Seiberg-Witten solutions converging to a $\mathbb{Z}_{2}$-harmonic
spinor in the sense of Theorem 4.10. The limiting connection $A_{0}$ in
dimensions $n=3$ and $n=4$ satisfies

$\|F_{A_{0}}\|_{C^{2}(K^{\prime})}<\infty.$

###### Proof.

In Cases (I) and (III), the Haydys Correspondence implies that $F_{A_{0}}=0$,
and the result is immediate.

In Cases (II) and (IV), the limiting curvature $F_{A_{0}}$ in Theorem 4.10 is
harmonic, i.e.

$(d_{A_{0}}+d_{A_{0}}^{\star})F_{A_{0}}=0$ (6.1)

(in fact, Remark 1.37 of [46] shows $F_{A_{0}}$ vanishes in Case (II)).
Because $A_{0}$ is only known to be $L^{1,2}$, the coefficients of (6.1) are
too rough to initiate a standard bootstrapping argument. However, there is
more information in this case: the Haydys Correspondence with stabilizers (see
Appendix C of [2]) shows that $F_{A_{0}}\in\Omega^{\text{Re}}$. Since $A_{0}$
respects the splitting $\Omega^{\text{Re}}\oplus\Omega^{\text{Im}}$ by
Proposition 5.2, (6.1) restricts to an equation on the smooth bundle
$\Omega^{\text{Re}}\to Y-\mathcal{Z}_{0}$ (resp. $X-\mathcal{Z}_{0}$).

As in the proof of Proposition 5.2, one has $\Omega^{\text{Re}}\simeq
S^{\text{Re}}\simeq E\otimes_{\mathbb{R}}\ell_{0}$, where
$E=\Lambda^{1}(\mathbb{R})$ is a rank 4 real Clifford module on $Y$ (resp
$X$), and $\ell_{0}\to Y-\mathcal{Z}_{0}$ (resp. $X-\mathcal{Z}_{0}$) is real
line bundle. Moreover, (cf. the discussion after Definition 4.11), the
restriction of $A_{0}$ viewed as a connection on
$E\otimes_{\mathbb{R}}\ell_{0}$ via this isomorphism is the connection arising
from a smooth spin connection on $E$ and the unique flat connection with
holonomy in $\mathbb{Z}_{2}$ on $\ell_{0}$. Thus (6.1) becomes

$(d_{\Gamma}+d_{\Gamma}^{\star})F_{A_{0}}^{\text{Re}}=0$

where $d_{\Gamma}$ is (up to isomorphism), the exterior covariant derivative
on $\Lambda^{\bullet}(\mathbb{R})\otimes\ell_{0}$. By elliptic regularity, and
since $F_{A_{0}}=F_{A_{0}}^{\text{Re}}$, one has $\|F_{A_{0}}\|_{L^{k,2}}\leq
C_{k}\|F_{A_{0}}\|_{L^{2}}<\infty$ for every $k$, and the conclusion follows
from the Sobolev embedding theorem.

∎

######  Lemma 6.2.

Let $Y$ be a closed, oriented 3-manifold and suppose that
$(\Phi_{i},A_{i},\varepsilon_{i})\to(\mathcal{Z}_{0},A_{0},\Phi_{0})$ is a
sequence of solution to (4.7–4.9) converging to a $\mathbb{Z}_{2}$-harmonic
spinor in the sense of Theorem 4.10 in Case (I) or Case (II). Let
$\Phi_{i}=\Phi_{0}+\varepsilon_{i}\varphi_{i}$ and $A_{i}=A_{0}+a_{i}$. Then

$\|A_{\varepsilon}\|_{L^{1,3}(K)}\to 0\hskip 28.45274pt\text{and }\hskip
28.45274pt\|A_{\varepsilon}\|_{C^{0}(K)}\to 0$ (6.2)

on compact subsets $K\Subset Y-\mathcal{Z}_{0}$, where $A_{\varepsilon}$ is
the matrix (5.20, 5.22) from the proof of Lemma 5.4.

###### Proof.

In Case (I), the matrix $A_{\varepsilon}$ of (5.20) contains only terms
involving $\varepsilon_{i}\varphi_{i}$. In this case, the conclusion is
immediate from Theorem 4.10 and the Sobolev embedding; indeed, the conclusion
of Theorem 4.10 shows that $\varepsilon_{i}\varphi_{i}=\Phi_{i}-\Phi_{0}\to 0$
in $L^{2,2}$. In particular, $\varepsilon_{i}\varphi_{i}\to 0$ in $L^{1,6}$,
thus a fortiori in $L^{1,3}$ and in $C^{0}$ by the embedding
$C^{0,\alpha}\hookrightarrow L^{1,6}$. In Case (II), the matrix
$A_{\varepsilon}$ from (5.22) includes entries involving
$\varepsilon_{i}\varphi_{i}$, which converge as above, and also entries of the
form $\varepsilon_{i}a_{i}$. To prove the lemma, it therefore suffices to show
that $\|\varepsilon_{i}a_{i}\|_{L^{1,p}}\to 0$ for any $p>3$.

For the remainder of the proof, the subscript $i$ is kept implicit in the
notation. Letting $\psi=\varepsilon\varphi$ be the re-normalized deformation
of the spinor, the curvature equation for the deformation $a$ reads:

$\star
F_{A_{0}}+\mathbb{d}_{A_{0}}a=\frac{1}{\varepsilon^{2}}\mu(\Phi_{0}+\psi,\Phi_{0}+\psi)-a\wedge
a.$ (6.3)

where $\mathbb{d}_{A_{0}}$ is the Dirac operator from Example 4.6. The
elliptic estimate for $\mathbb{d}_{A_{0}}$ applied to $a$ yields

$\displaystyle\|a\|_{L^{1,p}}$ $\displaystyle\leq$ $\displaystyle
C_{k,p}\left(\frac{1}{\varepsilon^{2}}\|\mu(\Phi_{0},\psi)\|_{L^{p}}+\frac{1}{\varepsilon^{2}}\|\mu(\psi,\psi)\|_{L^{p}}+\|a\wedge
a\|_{L^{p}}+\|F_{A_{0}}\|_{L^{p}}+\|a\|_{L^{2}}\right)$ (6.4)

holds for $p\geq 2$. Differentiating (6.3) and commuting covariant derivatives
yields the estimate

$\displaystyle\|\nabla_{A_{0}}a\|_{L^{1,p}}$ $\displaystyle\leq$
$\displaystyle
C_{k,p}\Big{(}\frac{1}{\varepsilon^{2}}\|\nabla_{A_{0}}\mu(\Phi_{0},\psi)\|_{L^{p}}+\frac{1}{\varepsilon^{2}}\|\nabla_{A_{0}}\mu(\psi,\psi)\|_{L^{p}}+\|\nabla_{A_{0}}a\wedge
a\|_{L^{p}}$ (6.5) $\displaystyle\ \ \ \ +\
\|\nabla_{A_{0}}F_{A_{0}}\|_{L^{p}}\ +\ \|\ [F_{A_{0}}\wedge a]\
\|_{L^{p}}+\|a\|_{L^{p}}\Big{)}.$

for the covariant derivative (commuting covariant derivatives gives rise to
the curvature term $[F_{A_{0}}\wedge a]$ along with a bounded Riemannian
curvature term which has been absorbed into the final term of (6.5)).

Now we bootstrap. To begin, we know that $\Phi_{0}\in L^{2,2}$,
$\psi=\varepsilon\varphi\to 0$ in $L^{2,2}$, and $a\to 0$ in $L^{1,2}$.

Step 0: By the Sobolev embeddings $L^{1,2}\hookrightarrow L^{6}$, and
$L^{2,2}\hookrightarrow L^{2,2}$, the following quantities are bounded

uniformly in $\varepsilon$: $\|a\|_{L^{6}}$ , $\|\mu(\Phi_{0},\psi)\|_{L^{3}}$
, $\|\mu(\psi,\psi)\|_{L^{3}}$. Additionally, $\|F_{A_{0}}\|_{L^{3}}$ is
bounded by

Lemma 6.1.

Step 1: Apply the elliptic estimate (6.4) with $p=3$ to conclude that
$\|\varepsilon^{2}a\|_{L^{1,3}}$ is bounded in $L^{1,3}$.

Step 2: By Hölder’s inequality with $p=3/2$ and $q=3$,

$\|\nabla_{A_{0}}a\wedge a\|_{L^{2}}\leq\|a\|_{L^{1,3}}\|a\|_{L^{6}}$

is bounded. Likewise for
$\|\mu(\nabla\Phi_{0},\psi)\|_{L^{2}},\|\mu(\Phi_{0},\nabla\psi)\|_{L^{2}},\|\mu(\nabla\psi,\psi)\|_{L^{2}}$
using the bounds on

$\Phi_{0},\psi$ from Theorem 4.10.

Step 3: Using Lemma 6.1 to bound the terms involving $F_{A_{0}}$, apply the
elliptic estimate (6.5) with $p=2$ to conclude that
$\|\varepsilon^{2}a\|_{L^{2,2}}$ is bounded.

Step 4: By Cauchy-Schwartz,

$\|\nabla_{A_{0}}a\wedge a\|_{L^{5/2}}\leq\|a\|_{L^{1,5}}\|a\|_{L^{5}}$

and likewise for
$\|\mu(\nabla\Phi_{0},\psi)\|_{L^{5/2}},\|\mu(\Phi_{0},\nabla\psi)\|_{L^{5/2}},\|\mu(\nabla\psi,\psi)\|_{L^{5/2}}$
using the bounds on

$\Phi_{0},\psi$ from Theorem 4.10.

Step 5: Using Lemma 6.1 to bound the terms involving $F_{A_{0}}$, apply the
elliptic estimate (6.5) with $p=5/2$ to conclude that
$\|\varepsilon^{2}a\|_{L^{2,5/2}}$ is bounded.

Step 6: Applying the interpolation inequality with $\delta<<1$,

$\|a\|_{L^{1,p}}\leq
C_{K}\left(\|a\|_{L^{2,5/2}}^{1/2}\|a\|^{1/2}_{L^{6-\delta}}+\|a\|_{L^{2}}\right)$

where
$\frac{1}{p}=\frac{1}{3}+\frac{1}{2}\left(\frac{1}{5/2}-\frac{2}{3}\right)+\tfrac{1-1/2}{6-\delta}$
to conclude $\|\varepsilon a\|_{L^{1,p}}\to 0$ for
$p=\tfrac{60}{17}-\delta^{\prime}>3$.

It follows that $\varepsilon_{i}a_{i}\to 0$ in $L^{1,3}$ and $C^{0}$ by the
Sobolev embedding since $K$ is compact. ∎

######  Lemma 6.3.

Let $X$ be a closed, oriented 4-manifold and suppose that
$(\Phi_{i},A_{i},\varepsilon_{i})\to(\mathcal{Z}_{0},A_{0},\Phi_{0})$ is a
sequence of solution to (4.7–4.9) converging to a $\mathbb{Z}_{2}$-harmonic
spinor in the sense of Theorem 4.10 in Case (III) or Case (IV). Let
$(\varphi_{i},a_{i})$ denote the difference from $(\Phi_{0},A_{0})$ as in
(5.1). Then

$\|A_{\varepsilon}\|_{L^{1,4}(K)}\to 0\hskip 28.45274pt\text{and }\hskip
28.45274pt\|A_{\varepsilon}\|_{C^{0}(K)}\to 0$ (6.6)

on compact subsets $K\Subset Y-\mathcal{Z}_{0}$, where $A_{\varepsilon}$ is
the four-dimensional analogue of the matrices (5.20, 5.22) from the proof of
Lemma 5.4 in Cases (III) and (IV) respectively. .

###### Proof.

In Case (III) the conclusion again follows directly from Theorem 4.10 and the
Sobolev embedding. Case (IV) follows from a similar bootstrapping procedure as
in the previous lemma, but repeatedly applying Steps 1–5 until convergence in
$L^{2,4+\gamma}$ of $\varepsilon^{2}a_{i}$ is obtained for $0<\gamma<<1$.
Then, interpolating as in Step 6 of Lemma 6.2 establishes that $\varepsilon
a_{i}\to 0$ in $L^{1,4+\gamma}$. The second bound of (6.6) follows from the
Sobolev embedding $L^{1,4+\gamma}\hookrightarrow C^{0,\alpha}$. ∎

The convergence to a $\mathbb{Z}_{2}$-harmonic spinor now, by design, fits
into the abstract framework of Sections 2–3. The following proposition retains
the setting of the previous lemmas, and additionally uses the notation of
$S^{\text{Im}},\Omega^{\text{Im}}$ from Section 5.2.

######  Proposition 6.4.

Suppose that
$(\Phi_{i},A_{i},\varepsilon_{i})\to(\mathcal{Z}_{0},A_{0},\Phi_{0})$ is a
sequence of solutions to (4.7–4.9) converging in the sense of Theorem 4.10 in
Cases (I)–(III); in Case (IV) further assume that $A_{i}\to A_{0}$ in
$L^{1,p}_{loc}$ and $\Phi_{i}\to\Phi_{0}$ in $L^{2,p}_{loc}$ for $p>2$. Let
$(\varphi_{i},a_{i})=(\varphi^{\text{Re}},\varphi^{\text{Im}},a^{\text{Re}},a^{\text{Im}})$
333In Cases (I) and (III), $\Omega^{\text{Re}}$ is empty hence
$a=a^{\text{Im}}$. denote the perturbations from the limiting data which
satisfy (5.1). Then there exist constants $C,c$ depending only on $\Phi_{0}$
and background data such that

$\|(\varphi_{i}^{\text{Im}},a_{i}^{\text{Im}})\|_{C^{0}(K)}\leq\frac{C}{R_{K}^{n/2}}\text{Exp}\left(-\frac{cR_{K}}{\varepsilon_{i}}\right)\|(\varphi_{i},a_{i})\|_{L^{1,2}(K^{\prime})}$
(6.7)

on compact subsets $K\Subset K^{\prime}\Subset Y-\mathcal{Z}_{0}$ (resp.
$X-\mathcal{Z}_{0}$), where $n=3,4$ is the dimension in the respective cases,
and $R_{K}=\text{dist}(K,\mathcal{Z}_{0})$.

###### Proof.

The regularity on $A_{\varepsilon}$ established in Lemmas 6.2–6.3 is
sufficient to apply Proposition 6.1 of [3] to show that there is a gauge
transformation putting the deformation $(\varphi_{i},a_{i})$ in the gauge
(5.2) (note that [3], Proposition 6.1) applies identically in 4-dimensions for
the equivalent Sobolev range). As in Claim 6.1, the limiting connection
$F_{A_{0}}\in\Omega^{\text{Re}}$, so the limiting configuration solves the
Seiberg-Witten equations in the $S^{\text{Im}},\Omega^{Im}$ components (i.e.
$SW(\Phi_{0},A_{0})=-F_{A_{0}}\in\Omega^{\text{Re}}$ as in 5.1). Thus In this
gauge, Proposition 5.1, Lemma 5.2, and Lemmas 6.2–6.3 show that the assumption
of Corollary 1.2 hold in the respective cases. The conclusion then follows
directly. ∎

Using the exponential convergence of Proposition 6.4, we may now conclude the
proof of Theorem 1.3:

###### Proof of Theorem 1.3.

Let
$(\Phi_{0}+\psi_{i},a_{i})=\left(\varepsilon_{i}\left(\frac{\Phi_{0}}{\varepsilon_{i}}+\varphi_{i}\right),a_{i}\right)$
denote the re-normalized sequence. For the remainder of the proof, the
subscript is kept implicit. Let $K$ denote a compact subset, and let
$K_{0}\Supset K_{1}\Supset K_{2}\ldots$ be a nested sequence of compact
subsets of $Y-\mathcal{Z}_{0}$ so that $K\subseteq{\bigcap K_{j}}$. By the
assumption that the sequence converges in the sense of Theorem 4.10,
Proposition 6.4 applies to show

$\|(\varphi^{\text{Im}},a)\|_{C^{0}(K_{0})}\leq
C_{K}\text{Exp}\left(-\frac{c}{\varepsilon}\right)\|(\varphi^{\text{Re}},\varphi^{\text{Im}},a)\|_{L^{1,2}(Y-\mathcal{Z}_{0})}\leq\frac{C}{\varepsilon}\text{Exp}\left(-\frac{c}{\varepsilon}\right).$

It now follows from the curvature equation

$F_{A_{i}}+\frac{1}{2\varepsilon}\mu\left(\varphi_{i}^{\text{Im}},\Phi_{0}\right)+\frac{1}{2}\mu(\varphi_{i}^{\text{Re}},\varphi_{i}^{\text{Im}})=0$

that $F_{A_{0}}=0$ (resp. $F_{A_{0}}^{+}=0$) for the limiting connection.

Bootstrapping now proceeds in the standard way using the Seiberg-Witten
equations (4.4–4.5) to show

$\displaystyle\|(\psi_{i}^{\text{Im}},a_{i}^{\text{Im}})\|_{L^{k,2}(K_{k})}$
$\displaystyle\leq$
$\displaystyle\frac{C_{k}}{\varepsilon^{2k+1}}\text{Exp}\left(-\frac{c}{\varepsilon}\right)\to
0$
$\displaystyle\|(\psi_{i}^{\text{Re}},a_{i}^{\text{Re}})\|_{L^{k,2}(K_{k})}$
$\displaystyle\to$ $\displaystyle 0$

for every $k\geq 3$ (thus in particular on $K\subseteq K_{k}$), and the
conclusion follows.

To spell out the first few steps for $n=3$ in Case (I), first apply the
elliptic estimate for the operator $\mathbb{d}$ of (5.4), which in this case
is independent of $A$ with $(k,p)=(1,6)$. The second equation (4.5) shows

$\displaystyle\|a_{i}\|_{L^{1,6}(K_{1})}$ $\displaystyle\leq$ $\displaystyle
C_{1,6}\left(\frac{1}{\varepsilon^{2}}\|\mu(\Phi_{0},\psi_{i}^{\text{Im}})+\mu(\psi_{i}^{\text{Re}},\psi_{i}^{\text{Im}})\|_{L^{6}(K_{0})}+\|a_{i}\|_{L^{6}(K_{0})}\right)$
$\displaystyle\leq$ $\displaystyle\frac{C}{\varepsilon^{3}}\
\text{Exp}\left(-\frac{c}{\varepsilon}\right)\to 0.$

This in turn implies $A_{0}\in L^{1,6}$. Likewise, for the spinor, applying
the elliptic estimate for $\not{D}_{\widetilde{A}}$ (for a fixed smooth
background connection $\widetilde{A}$) to the first equation (4.4) yields

$\displaystyle\|\psi^{\text{Im}}_{i}\|_{L^{1,6}(K_{1})}$ $\displaystyle\leq$
$\displaystyle
C_{1,6}\left(\|(\widetilde{A}-A_{0})\psi_{i}^{\text{Im}}\|_{L^{6}(K_{0})}+\|\gamma(a_{i})\psi_{i}^{\text{Re}}\|_{L^{6}(K_{0})}+\|\psi_{i}^{\text{Im}}\|_{L^{6}(K_{0})}\right)$
$\displaystyle\leq$
$\displaystyle\frac{C}{\varepsilon^{3}}\text{Exp}(-\frac{c}{\varepsilon})\to
0$ $\displaystyle\|\psi^{\text{Re}}_{i}\|_{L^{1,6}(K_{1})}$
$\displaystyle\leq$ $\displaystyle
C_{1,6}\left(\|(\widetilde{A}-A_{0})\psi_{i}^{\text{Re}}\|_{L^{6}(K_{0})}+\|\gamma(a_{i})\psi_{i}^{\text{Im}}\|_{L^{6}(K_{0})}+\|\psi_{i}^{\text{Re}}\|_{L^{2}(K_{0})}\right)\to
0$

Repeating the elliptic estimate for the connection now with $(k,p)=(2,2)$ and
using the fact that multiplication induces a bounded map $L^{1,6}\times
L^{1,6}\to L^{1,2}$ one has

$\displaystyle\|a_{i}\|_{L^{2,2}(K_{2})}$ $\displaystyle\leq$ $\displaystyle
C_{1,6}\left(\frac{1}{\varepsilon^{2}}\|\mu(\Phi_{0},\psi_{i}^{\text{Im}})+\mu(\psi_{i}^{\text{Re}},\psi_{i}^{\text{Im}})\|_{L^{1,2}(K_{1})}+\|a_{i}\|_{L^{6}(K_{1})}\right)$
$\displaystyle\leq$
$\displaystyle\frac{C_{2,2}}{\varepsilon^{2}}\left(\|\Phi_{0}\|_{L^{1,6}(K_{1})}\|\psi_{i}^{\text{Im}}\|_{L^{1,6}(K_{1})}+\|\psi_{i}^{\text{Re}}\|_{L^{1,6}(K_{1})}\|\psi_{i}^{\text{Im}}\|_{L^{1,6}(K_{1})}+\|a_{i}\|_{L^{1,2}(K_{1})}\right)$
$\displaystyle\leq$
$\displaystyle\frac{C_{2}}{\varepsilon^{5}}\text{Exp}\left(-\frac{c}{\varepsilon}\right)\to
0.$

The bootstrapping continues in this fashion to obtain the desired bounds in
$L^{k,2}$ for every $k>2$. ∎

######  Remark 6.5.

Theorem 1.3 does not applying in the case of the $\text{ADHM}_{1,2}$ Seiberg-
Witten equations described in Example 4.9. As discussed in remark 1.6 the form
of the non-linear terms in this case does not satisfy the hypotheses of
Corollary 1.2. Specifically, in this case one need not have that
$\mu(\Psi^{\text{Re}},\Psi^{\text{Re}})=0$, thus the differential inequality
3.6 contains an additional term of the form $\langle
q_{1},f(q_{0},q_{0})\rangle$. It seems likely to the author that in this
situation, the techniques of [4] (Section 5) could be extended to bootstrap
convergence to $C^{\infty}_{loc}$ (though without the stronger exponential
convergence statement of Proposition 6.4).

In addition, for the $\text{ADHM}_{1,2}$ Seiberg-Witten equations, the
arguments of Section 3 may be extended to partially deal with the term
$\langle q_{1},f(q_{0},q_{0})\rangle$ in the following way. Using Young’s
inequality and the fact that the Green’s function (2.10) is $L^{2}$-integrable
in dimension 3, Proposition 3.1 can be adapted to prove that diverging spinors
have the form

$\Phi_{i}=\Phi_{0}+\varphi_{i}$

where $\varphi_{i}=O(\varepsilon)$ in $C^{0}_{loc}$. This provides a step in
confirming the asymptotic expansions postulated in [2] (Section 5.3).

##  Appendix A An Extension in $n=3$ Dimensions

This appendix provides a minor strengthening of Corollary 1.2 and Proposition
6.4 to include the case of a family of compact sets $K_{\varepsilon}$
parameterized by $\varepsilon$. These results are employed in Corollary 1.3
and Appendix A of [25] and in [27].

Let $K_{\varepsilon}\Subset\mathcal{Z}_{0}$ be an $\varepsilon$-parameterized
family of compact subsets in the complement of $\mathcal{Z}_{0}$. For a small
constant $c_{1}$ to be chosen momentarily, denote by
$R_{\varepsilon}=c_{1}\cdot\text{dist}(K_{\varepsilon},\mathcal{Z})$. We
restrict to the case that $R_{\varepsilon}\to 0$, else we are in the previous
situation with $K=\bigcup_{\varepsilon}K_{\varepsilon}$. Let
$K_{\varepsilon}^{\prime}$ denote a family of slightly larger compact subsets
such that for some $\varepsilon_{0}>0$, the following conditions are met for
all $\varepsilon<\varepsilon_{0}$:

  1. (1)

For some constant $\kappa_{1}<1$, one has

$\text{dist}(K_{\varepsilon},Y-K_{\varepsilon}^{\prime})\geq\kappa_{1}R_{\varepsilon}.$

  2. (2)

If $y_{0}\in K_{\varepsilon}$ and $y\in B_{R_{\varepsilon}}(y_{0})$ then

$|\Lambda(y)|^{2}\geq\frac{|\Lambda(y_{0})|^{2}}{2}.$

  3. (3)

The bounds

$\frac{\|A_{\varepsilon}\|_{L^{1,3}(K^{\prime}_{\varepsilon})}}{\inf_{K_{\varepsilon}}|\Lambda(y)|}\to
0\hskip 56.9055pt\|A_{\varepsilon}\|_{C^{0}(K^{\prime}_{\varepsilon})}\leq
c_{2}\inf_{K_{\varepsilon}}|\Lambda(y)|$ (A.1)

are satisfied for all $\varepsilon<\varepsilon_{0}$ where $c_{2}<1/8$.

When assumptions (1)–(3) above are satisfied, and $c_{1}$ is chosen
sufficiently small, the following extension of Theorem 1.1 and Corollary 1.2
holds.

######  Corollary A.1.

Let $D_{\varepsilon}$ be a concentrating Dirac operator with fixed degeneracy,
and $Q$ a non-linear operator satisfying the hypotheses of Corollary 1.2.
Suppose that $K_{\varepsilon}\Subset K_{\varepsilon}^{\prime}\Subset
Y-\mathcal{Z}$ are nested family of compact subsets of $Y-\mathcal{Z}_{0}$. If
the family satisfies assumptions (1)–(3) above. Then the conclusion of Theorem
1.1 continues to hold, i.e. there exist constants $C,c$ independent of
$\varepsilon$ and an $\varepsilon_{0}>0$ such that for
$\varepsilon<\varepsilon_{0}$, any solution

$(D+\tfrac{1}{\varepsilon}\mathcal{A})\mathfrak{q}+Q(\mathfrak{q})=0$

satisfies

$\|\pi_{\mathfrak{H}}q\|_{C^{0}(K_{\varepsilon})}\leq\frac{C}{|\text{dist}(K_{\varepsilon},\mathcal{Z})|^{3/2}}\text{Exp}\left(-\frac{\Lambda_{K_{\varepsilon}}c}{\varepsilon}\text{dist}(K_{\varepsilon},\mathcal{Z})\right)\|\mathfrak{q}\|_{L^{1,2}(K^{\prime}_{\varepsilon})}.$

###### Proof.

The proof is identical to Corollary 1.2, except that now for each point
$y_{0}$ the radius of the ball used in Lemma 2.9 depends on $\varepsilon$. In
addition, the second bound of (A.1) is used to obtain (3.5) in this case, and
the first is used to absorb the terms of Lemma 3.2. ∎

To conclude, we note a particular case of interest. Suppose that in Case (I),
$(\mathcal{Z}_{0},A_{0},\Phi_{0})$ is a $\mathbb{Z}_{2}$-harmonic spinor which
is non-degenerate in the sense that there exists a $c_{2}>0$ such that

$|\Phi_{0}|(x)\geq c_{2}\sqrt{\text{dist}(x,\mathcal{Z}_{0})}.$

Then one has $\Lambda_{K_{\varepsilon}}\geq
c_{2}\sqrt{R_{\varepsilon}}=c_{2}\sqrt{c_{1}}\cdot\text{dist}(K_{\varepsilon},\mathcal{Z}_{0})$.
In this case, the Corollary A.1 implies:

######  Corollary A.2.

Suppose $(\mathcal{Z}_{0},A_{0},\Phi_{0})$ is a non-degenerate
$\mathbb{Z}_{2}$-harmonic spinor, and $K_{\varepsilon}=\\{y\ |\
\text{dist}(y,\mathcal{Z}_{0})\geq c_{1}\varepsilon^{2/3}\\}$ is the family of
compact subsets above. Then if the deformation
$(\varphi_{\varepsilon},a_{\varepsilon})$ satisfies (A.3), then

$\|(\varphi^{\text{Im}},a)\|_{C^{0}(K_{\varepsilon})}\leq\frac{C}{|\text{dist}(K_{\varepsilon},\mathcal{Z})|^{3/2}}\text{Exp}\left(-\frac{c_{1}}{\varepsilon}\text{dist}(K_{\varepsilon},\mathcal{Z}_{0})^{3/2}\right)\|(\varphi,a)\|_{L^{1,2}(K_{\varepsilon}^{\prime})}$
(A.2)

∎

Corollary A.2 implies the conclusions asserted in Corollary 1.3 and Appendix B
of [25]. For the situation described there, one has that
$K_{\varepsilon}=\\{y\ |\ \text{dist}(y,\mathcal{Z}_{0})\geq
c_{1}\varepsilon^{2/3-\gamma_{1}}\\}$ for $\gamma<<1$, the hypotheses (1)–(2)
are easily seen to be satisfied. Hypothesis (3) follows from the slightly
stronger assertion that

$\|\nabla\varphi_{\varepsilon}\|_{L^{3}(K_{\varepsilon})}\leq\frac{C}{\varepsilon^{2/3}}\hskip
28.45274pt\text{and}\hskip
28.45274pt\|\varphi_{\varepsilon}\|_{C^{0}(K_{\varepsilon})}\leq\frac{C}{\varepsilon^{2/3}}.$
(A.3)

The results of Appendix B of [25] show that the bounds A.3 are satisfied for
the model solutions used in the gluing construction of [25, 27]. Thus
hypotheses (1)–(3) hold, and Corollary A.2 implies the assertions in Corollary
1.3 and Appendix B of [25].

Corollary A.2 also implies a characteristic length scale that is not obvious
from the convergence statements of Theorem 4.10. In particular, the
exponential decay result applies on families of compact subsets with
$\text{dist}(K_{\varepsilon},\mathcal{Z}_{0})\sim\varepsilon^{2/3}$. For
smaller compact subsets, the exponent factor approaches $1$ and the conclusion
of (A.2) becomes trivial. This suggests that $r=O(\varepsilon^{2/3})$ is a
characteristic length scale for the convergence to a $\mathbb{Z}_{2}$-harmonic
spinor in the case of the two-spinor Seiberg-Witten equations. Indeed, this
length scale naturally appears in the construction of the model solutions used
in the gluing construction of [25, 27]. It is also the same length scale the
appears in the equivalent problem for Hitchin’s equations [20, 7, 8]. A
promising approach to the surjectivity of the gluing problem is to attempt to
dilate this characteristic length to be unit size to extract limiting profiles
of the sequence $(\varphi_{\varepsilon},a_{\varepsilon})$ along the singular
set $\mathcal{Z}_{0}$ and show these necessarily arise from gluing data as in
[25, 27].

##  Appendix B Estimates for the Green’s Function

This appendix proves assertion (iii) in the proof of Lemma 2.9. This is a
consequence of the following:

######  Lemma B.1.

Let $(X,g)$ be a Riemannian manifold of dimension 3 or 4 with bounded
geometry, and take $R_{0}$ less than the injectivity radius of $X$. Let $M>0$,
and for a point $x_{0}\in X$, denote by $G(x_{0},x)$ the Green’s function with
Dirichlet boundary condition on $B_{R_{0}}(x_{0})$, so that

$\begin{cases}\left(-\Delta_{g}-\frac{M^{2}}{\varepsilon^{2}}\right)G(x_{0},x)=\delta_{x_{0}}\
\hskip 28.45274ptx\in B_{R_{0}}\\\ G(x_{0},x)=0\hskip 99.58464ptx\in\partial
B_{R_{0}}\end{cases}$

where $\Delta_{g}$ denotes the (positive-definite) Laplacian of the Riemannian
metric $g$. Then, there exists an $\varepsilon_{0}>0$ depending only on the
geometry of $(X,g)$ and a constant $C_{n}$ depending only on the dimension $n$
such that the bound

$|G(x_{0},x)|\leq\frac{C_{n}}{|x-x_{0}|^{n-2}}\
\text{Exp}\left(-\frac{M}{2\varepsilon}|x-x_{0}|\right)$ (B.1)

holds for $\varepsilon<\varepsilon_{0}$.

###### Proof.

This follows from a comparison principle argument using the Green’s function
on Euclidean space. We prove the lemma in dimension $n=3$; the general case is
the same using the appropriate power of $|x-x_{0}|$.

Let $\Delta_{0}$ denote the Laplacian on $\mathbb{R}^{3}$ with the Euclidean
metric $g_{0}$, and $G_{0}^{M/2}$ denote the Green’s function

$\left(-\Delta_{g}-\frac{M^{2}}{4\varepsilon^{2}}\right)G_{0}^{M/2}=\delta_{0}$

on all of $\mathbb{R}^{3}$. It is a standard fact that

$G_{0}^{M/2}=\frac{C_{3}}{|x-x_{0}|}\text{Exp}\left(-\frac{M}{2\varepsilon}|x-x_{0}|\right).$

Using geodesic normal coordinates on $B_{R_{0}}(x_{0})$, define
$\varphi=G-G_{0}^{M/2}$ where $G=G(x_{0},x)$. Then

$\displaystyle(-\Delta_{g}-\tfrac{M^{2}}{\varepsilon^{2}})\varphi$
$\displaystyle=$
$\displaystyle\delta_{x_{0}}+(\Delta_{0}-\Delta_{g})G_{0}^{M/2}-(\Delta_{0}+\tfrac{M^{2}}{4\varepsilon^{2}})G_{0}^{M/2}-\tfrac{3M^{2}}{4\varepsilon^{2}}G_{0}^{M/2}$
$\displaystyle=$ $\displaystyle
0+(\Delta_{0}-\Delta_{g})G_{0}^{M/2}-\tfrac{3M^{2}}{4\varepsilon^{2}}G_{0}^{M/2}.$

Since $g=g_{0}+O(r^{2})$ in geodesic normal coordinates, for a radially
symmetric function $f(r)$ one has

$(\Delta_{0}-\Delta_{g})f(r)=O(r^{2})\left[\partial_{r}^{2}+\frac{1}{r}\partial_{r}\right]f(r)+O(r)\partial_{r}f(r)=O(r^{2})\partial_{r}^{2}+O(r)\partial_{r}$

where $O(r^{k})$ denotes a quantity bounded by $Cr^{k}$ for a constant $C$
depending only on the geometry of $(X,g)$. Differentiating,

$\displaystyle\partial_{r}G_{0}^{M/2}$ $\displaystyle=$ $\displaystyle
C\left(-\frac{1}{r^{2}}-\frac{M}{2\varepsilon
r}\right)e^{-\frac{M}{2\varepsilon}r}$
$\displaystyle\partial_{r}^{2}G_{0}^{M/2}$ $\displaystyle=$ $\displaystyle
C\left(\frac{2}{r^{3}}+\frac{M}{2\varepsilon}+\frac{M^{2}}{4\varepsilon^{2}r}\right)e^{-\frac{M}{2\varepsilon}r}$

hence

$|(\Delta_{0}-\Delta_{g})G_{0}^{M/2}|\leq
C\left(\frac{1}{r^{3}}+\frac{M}{2\varepsilon}+\frac{M^{2}}{4\varepsilon^{2}r}\right)e^{-\frac{M}{2\varepsilon}r}\leq\frac{3M^{2}}{4\varepsilon^{2}}G_{0}^{M/2}$

once $\varepsilon$ is sufficiently small depending only on $R_{0}$ and the
geometry of $(X,g)$. It follows that $\varphi$ satisfies

$\displaystyle(\Delta_{g}+\tfrac{M^{2}}{\varepsilon^{2}})\varphi$
$\displaystyle\leq$ $\displaystyle 0\hskip 28.45274pt\text{on}\hskip
28.45274ptB_{R_{0}}(x_{0})$ $\displaystyle\varphi$ $\displaystyle\leq$
$\displaystyle 0\hskip 28.45274pt\text{on}\hskip 28.45274pt\partial
B_{R_{0}}(x_{0})$

where the last line follows since $G=0$ on $\partial B_{R_{0}}$ by definition,
and $-G_{0}^{M/2}<0$ everywhere. The maximum principle implies that
$\varphi\leq 0$ on $B_{R_{0}}$ once $\varepsilon$ is sufficiently small, which
yields the desired estimate (B.1). ∎

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