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

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

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

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

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

######  Lemma 6.1.

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

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

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

###### Proof.

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

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

######  Definition 6.2.

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

$\mathbb{H}^{1}_{e}(\mathcal{E}_{\tau}).$$p_{1}^{*}\mathbb{L}^{2}(\mathcal{E}_{\tau})$$\not{\mathbb{D}}(\mathcal{Z},u):=\not{D}_{\mathcal{Z}}u$

The next proposition calculates the linearization of the universal Dirac
operator

$\text{d}\not{\mathbb{D}}_{(\mathcal{Z}_{\tau},\Phi_{\tau})}:L^{2,2}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})\oplus
rH^{1}_{e}(Y-\mathcal{Z}_{\tau};S^{\text{Re}})\ \longrightarrow\
L^{2}(Y-\mathcal{Z}_{\tau};S^{\text{Re}})$ (6.5)

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

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

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

######  Proposition 6.3.

The linearization (6.5) is given by

$\displaystyle\text{d}\not{\mathbb{D}}_{(\mathcal{Z}_{\tau},\Phi_{\tau})}(\eta,\psi)$
$\displaystyle=$ $\displaystyle\mathcal{B}_{\Phi_{\tau}}(\eta)\ +\
\not{D}_{A_{\tau}}\psi$ (6.6)

where

$\mathcal{B}_{\Phi_{\tau}}(\eta)=\left(-\frac{1}{2}\sum_{ij}\dot{g}_{\eta}(e_{i},e_{j})e^{i}.\nabla^{\tau}_{j}+\frac{1}{2}d\text{Tr}_{g_{\tau}}(\dot{g}_{\eta}).+\frac{1}{2}\text{div}_{g_{\tau}}(\dot{g}_{\eta}).+\mathcal{R}(B_{\tau},\eta).\right)\Phi_{\tau}\
\ $ (6.7)

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

###### Proof Sketch..

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

$X=(Y\times[0,1],g_{s\eta}+ds^{2})$ (6.8)

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

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

$\text{d}\not{\mathbb{D}}_{(\mathcal{Z}_{\tau},\Phi_{\tau})}(\eta,0)=\left(\frac{d}{ds}\Big{|}_{s=0}\mathfrak{T}_{g_{\tau}}^{g_{s\eta}}\circ\not{D}_{g_{s\eta}}\circ(\mathfrak{T}_{g_{\tau}}^{g_{s\eta}})^{-1}\right)\Phi_{\tau}$
(6.9)

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

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

######  Theorem 6.4.

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

$\left(\frac{d}{ds}\Big{|}_{s=0}\mathfrak{T}_{g_{0}}^{g_{s}}\circ\not{D}_{g_{s}}\circ(\mathfrak{T}_{g_{0}}^{g_{s}})^{-1}\right)\Psi=-\frac{1}{2}\sum_{ij}\dot{g}_{s}(e_{i},e_{j})e^{i}.\nabla^{g_{0}}_{j}\Psi+\frac{1}{2}d\text{Tr}_{g_{0}}(\dot{g}_{s}).\Psi+\frac{1}{2}\text{div}_{g_{0}}(\dot{g}_{s}).\Psi$
(6.10)

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

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

###  6.2. The Deformation Operator

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

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

$\text{d}\not{\mathbb{D}}_{(\mathcal{Z}_{\tau},\Phi_{\tau})}=\begin{pmatrix}\Pi^{\text{Ob}}\mathcal{B}_{\Phi_{\tau}}&\Lambda(\tau)\pi_{\tau}\\\
\\\
(1-\Pi^{\text{Ob}})\mathcal{B}_{\Phi_{\tau}}&\not{D}_{A_{\tau}}\end{pmatrix}\
:\ \begin{matrix}L^{2,2}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})\\\ \oplus\\\
rH^{1}_{e}\end{matrix}\ \ \longrightarrow\ \ \begin{matrix}\text{\bf
Ob}(\mathcal{Z}_{0})\\\ \oplus\\\ \text{Range}_{\tau}^{\perp}.\end{matrix}$
(6.11)

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

$L^{2,2}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})$$\text{\bf
Ob}(\mathcal{Z}_{\tau})$$L^{2}(\mathcal{Z}_{\tau};\mathcal{S}_{\tau}).$$\Pi^{\text{Ob}}\mathcal{B}_{\Phi_{\tau}}$$\text{ob}_{\tau}^{-1}$$T_{\Phi_{\tau}}$

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

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

$F_{\eta}\left(t,z\right):=(t,z+\chi(r)\eta(t)).$ (6.12)

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

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

######  Theorem 6.5.

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

$T_{\Phi_{\tau}}:L^{1/2,2}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})\longrightarrow
L^{2}(\mathcal{Z}_{\tau};\mathcal{S}_{\tau})$ (6.13)

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

$T_{\Phi_{\tau}}(\eta(t))=\left(-\tfrac{3|\mathcal{Z}_{\tau}|}{2}(\Delta+1)^{-\tfrac{3}{4}}\
\circ\ \mathcal{T}_{\Phi_{\tau}}\circ\frac{d}{dt^{2}}\right)\eta(t)+K\eta(t)$
(6.14)

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

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

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

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

######  Corollary 6.6.

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

###### Proof.

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

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

######  Corollary 6.7.

For any $m\geq 0$, the extension

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

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

$\|\eta\|_{m+1/2,2}\leq C_{m}\
(\|T_{\Phi_{\tau}}(\eta)\|_{{m,2}}+\|\eta\|_{m+1/4,2}).$ (6.15)

∎

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

######  Corollary 6.8.

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

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

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

$\|\eta\|_{m+1/2,2}\leq C_{m}\ \left(\|f\|_{m,2}\ +\
\|\eta\|_{m+1/4,2}\right)\ +\ C_{m}M^{m}\left(\|f\|_{2}\ +\
\|\eta\|_{1/4,2}\right).$ (6.16)

∎

######  Remark 6.9.

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

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

###  6.3. Mode-Dependent Deformations

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

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

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

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

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

$|d\chi|\leq\frac{C}{R_{0}}.$ (6.19)

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

$\chi_{\ell}(r):=\chi(|\ell|r)\chi_{r_{0}}(r).$ (6.20)

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

Using this, define a new family of diffeomorphisms by

$\underline{F}_{\eta}(t,z):=\left(t,z+\underline{\chi}[\eta]\right)$ (6.21)

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

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

######  Corollary 6.10.

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

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

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

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

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

###### Proof Sketch.

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

######  Remark 6.11.

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

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

  * (A)

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

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

  * (B)

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

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

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

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

respectively.

######  Lemma 6.12.

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

  1. (A)

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

$\|r^{\beta}M_{\Phi}(\eta)\|_{L^{2}(N_{r_{0}})}\leq
C\|\eta\|_{w_{A}-(1+\alpha+\beta)}$

where $\|\eta\|_{s}$ denotes the $L^{s,2}$-norm on $\mathcal{Z}_{\tau}$. In
particular, the terms of weight $w_{A}=2$ satisfy

     * •

$\Big{\|}\underline{\chi}[\eta^{\prime\prime}]\sigma_{j}r^{\beta}\Phi\Big{\|}_{L^{2}(N_{r_{0}})}+\Big{\|}\underline{\partial_{a}\chi}[\eta^{\prime}]\sigma_{j}r^{\beta}\Phi\Big{\|}_{L^{2}(N_{r_{0}})}+\Big{\|}\underline{\partial_{ab}\chi}[\eta]\sigma_{j}r^{\beta}\Phi\Big{\|}_{L^{2}(N_{r_{0}})}\leq
C\|\eta\|_{1-\alpha-\beta}$

     * •

$\Big{\|}\underline{\chi}[\eta^{\prime}]\sigma_{j}r^{\beta}\nabla_{b}\Phi\Big{\|}_{L^{2}(N_{r_{0}})}+\Big{\|}\underline{\partial_{a}\chi}[\eta]\sigma_{j}r^{\beta}\nabla_{b}\Phi\Big{\|}_{L^{2}(N_{r_{0}})}\leq
C\|\eta\|_{1-\alpha-\beta}$

where $a,b=t,x,y$.

  2. (B)

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

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

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

     * •

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

     * •

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

###### Proof.

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

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

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

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

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

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

as desired.

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

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

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

for $n\geq 1$. Then

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

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

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

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

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

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

###  6.4. Non-Linear Terms

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

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

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

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

######  Lemma 6.13.

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

$\displaystyle Q_{h_{0}}(\eta,\psi)$ $\displaystyle=$
$\displaystyle\underline{\mathcal{B}}_{\psi}(\eta)\ \ +\ \
M_{\Phi}(\eta,\eta)\ \ +\ \ M_{\psi}(\eta,\eta)\ \ +\ \ F_{\Phi+\psi}(\eta)$

where

  * •

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

  * •

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

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

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

  * •

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

  * •

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

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

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

###### Proof.

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

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

######  Lemma 6.14.

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

  1. (A’)

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

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

  2. (B’)

Likewise,

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

###### Proof.

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

##  7\. Concentrating Local Solutions

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

###  7.1. Hilbert Spaces and Boundary Conditions

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

$\boxed{\lambda(\varepsilon)=\varepsilon^{1/2}}$ (7.1)

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

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

  1. (1)

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

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

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

  2. (2)

For each $\tau\in(-\tau_{0},\tau_{0})$, let $\Phi_{\tau}^{h_{\varepsilon}}$ be
the de-singularized spinor as defined in Definition 4.5 of [41]. The norm
$|\Phi^{h_{\varepsilon}}_{\tau}|$ is a smooth function on $Y$ with the
following properties for constants $c_{1},c_{2}$ (see [41, Sec. 4]):

     * (2a)

For $r\geq c_{1}\varepsilon^{2/3}$, it satisfies
$\big{|}|\Phi^{h_{\varepsilon}}_{\tau}|-|\Phi_{\tau}|\big{|}\leq
C\text{Exp}(-\tfrac{r^{3/2}}{\varepsilon})$.

     * (2b)

For $r\leq\lambda$, it is a monotonically increasing function of $r$ with a
uniform lower bound $|\Phi_{\tau}^{h_{\varepsilon}}|\geq
c_{2}\varepsilon^{1/3}$.

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

######  Definition 7.1.

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

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

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

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

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

######  Lemma 7.2.

There exists a Hilbert subspace

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

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

$(\mathcal{L}_{(\Phi,A)}\ ,\ \Pi^{+}):L^{1,2}(Y^{+})\longrightarrow
L^{2}(Y^{+})\oplus H^{+}_{\Phi_{\tau}}$ (7.4)

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

######  Definition 7.3.

Define the “inside” Hilbert spaces by

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

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

######  Remark 7.4.

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

###  7.2. Concentrating Local Solutions

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

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

$(\Phi_{1},A_{1}):=\chi^{+}\cdot(\Phi^{+}_{\varepsilon,\tau},A^{+}_{\varepsilon,\tau})+(1-\chi^{+})\cdot(\Phi_{\tau},A_{\tau}).$
(7.5)

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

######  Theorem 7.5.

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

  1. (I)

They satisfy

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

where SW denotes the extended, gauge-fixed Seiberg–Witten equations with
respect to $(g_{\tau},B_{\tau})$, and $e_{1},f_{1}$ are error terms obeying

     * (1)

$e_{1}\in\Gamma(S^{\text{Re}})$, and
$\text{supp}(e_{1})\subseteq\text{supp}(d\chi^{+})$

     * (2)

$\|e_{1}\|_{L^{2}(Y)}\leq C\varepsilon^{-1/24+\gamma}.$

     * (3)

$\|f_{1}\|_{L^{2}(Y)}\leq C\varepsilon^{M}$ for $M>10$, and
$\text{supp}(f_{1})\subseteq\text{supp}(d\chi^{+})$.

for $\gamma<<1$. Moreover, the derivatives
$\partial_{\tau}e_{1},\partial_{\tau}f_{1}$ also satisfy (1)–(3).

  2. (II)

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

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

where

$\|(\varphi,a)\|_{H^{1,+}_{\varepsilon,0}}\ +\
\|\partial_{\tau}(\varphi,a)\|_{H^{1,+}_{\varepsilon,0}}\leq
C\varepsilon^{-1/12-\gamma}.$ (7.6)

  3. (III)

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

∎

######  Remark 7.6.

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

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

######  Theorem 7.7.

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

$\mathcal{L}^{+}_{(\Phi_{1},A_{1})}:H^{1,+}_{\varepsilon,\nu}(Y^{+})\longrightarrow
L^{2,+}_{\varepsilon,\nu}(Y^{+})$ (7.7)

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

$\|(\varphi,a)\|_{H^{1,+}_{\varepsilon,\nu}}\leq\frac{C_{\nu}}{\varepsilon^{1/12+\gamma^{\prime}}}\|\mathcal{L}^{+}_{(\Phi_{1},A_{1})}(\varphi,a)\|_{L^{2,+}_{\varepsilon,\nu}}$
(7.8)

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

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

###  7.3. Outside Boundary Conditions

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

$\mathcal{L}_{(\Phi_{0},A_{0})}(\varphi_{1},\varphi_{2},a)=\begin{pmatrix}\not{D}_{A_{0}}&0&0\\\
0&\not{D}_{A_{0}}&\gamma(\\_)\frac{\Phi_{0}}{\varepsilon}\\\
0&\frac{\mu(\\_,\Phi_{0})}{\varepsilon}.&\mathbb{d}\end{pmatrix}\begin{pmatrix}\varphi_{1}\\\
\varphi_{2}\\\ a\end{pmatrix}.$ (7.9)

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

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

######  Definition 7.8.

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

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

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

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

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

######  Lemma 7.9.

There exists a Hilbert subspace

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

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

$(\mathcal{L}^{\text{Im}}_{(\Phi,A)}\ ,\
\Pi^{-}):L^{1,2}(Y^{-},S^{\text{Im}}\oplus\Omega)\longrightarrow
L^{2}(Y^{-};S^{\text{Im}}\oplus\Omega)\oplus H^{-}_{\Phi_{\tau}}$ (7.10)

is Fredholm of Index 0.∎

######  Definition 7.10.

Define the “outside” Hilbert spaces by

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

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

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

######  Lemma 7.11.

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

$\mathcal{L}^{\text{Im}}_{(\Phi_{\tau},A_{\tau})}:H^{1,-}_{\varepsilon,\nu}(Y^{-};S^{\text{Im}}\oplus\Omega)\longrightarrow
L^{2,-}_{\varepsilon,\nu}(Y^{-};S^{\text{Im}}\oplus\Omega)$ (7.11)

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

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

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

###### Proof Sketch.

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

$\mathcal{L}^{\text{Im}}\mathcal{L}^{\text{Im}}\begin{pmatrix}\varphi_{2}\\\
a\end{pmatrix}=\begin{pmatrix}\not{D}_{A_{\tau}}\not{D}_{A_{\tau}}\varphi_{2}\\\
\mathbb{d}\mathbb{d}a\end{pmatrix}+\frac{1}{\varepsilon^{2}}\begin{pmatrix}\gamma(\mu(\varphi_{2},\Phi_{\tau}))\Phi_{\tau}\\\
\mu(\gamma(a)\Phi_{\tau},\Phi_{\tau}))\end{pmatrix}+\frac{1}{\varepsilon}\mathfrak{B}(\varphi_{2},a)$
(7.12)

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

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

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

###  7.4. Exponential and Polynomial Decay

Solutions of

$\displaystyle\mathcal{L}_{(\Phi_{1},A_{1})}^{+}(\varphi,a)\ $
$\displaystyle=$ $\displaystyle g^{+}$ (7.13)

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

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

######  Lemma 7.12.

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

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

###### Proof.

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

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

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

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

######  Lemma 7.13.

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

$\|(\varphi_{2},a)\|_{C^{m}(K_{\varepsilon})}\leq\frac{C}{\varepsilon^{2m+1}r_{K}^{3/2}}\text{Exp}\left(-\frac{cr_{K}^{3/2}}{\varepsilon}\right)\|(\varphi,a)\|_{H^{1,+}_{\varepsilon,0}}$
(7.14)

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

###### Proof.

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

##  8\. Universal Seiberg–Witten Equations

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

###  8.1. Hilbert Bundles

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

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

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

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

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

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

######  Definition 8.1.

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

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

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

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

Finally, define global spaces by

$\mathbb{H}^{1}_{\varepsilon,\nu}:=\mathbb{H}^{1,+}_{\varepsilon,\nu}(\mathcal{E}_{\tau})\oplus\mathbb{H}^{1,-}_{\varepsilon,\nu}(\mathcal{E}_{\tau})\hskip
56.9055pt\mathbb{L}^{2}_{\varepsilon,\nu}:=\Big{\\{}(\mathcal{Z}_{\xi,\tau},\varphi,a)\
|\ \xi\in\mathcal{E}_{\tau}\ ,\
\|(\varphi,a)\|_{L^{2}_{\varepsilon,\nu}(Y)}<\infty\Big{\\}}$ (8.5)

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

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

######  Lemma 8.2.

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

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

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

###### Proof Sketch.

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

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

$S_{g_{\tau}}\otimes L\otimes
E$$S_{g_{\xi}}\otimes\underline{F}_{\xi}^{*}(L\otimes
E)$$\underline{F}_{\xi}^{*}(S_{g_{0}}\otimes L\otimes E)$$S_{g_{\tau}}\otimes
L\otimes
E$$\mathfrak{T}_{g_{\xi}}^{g_{\tau}}$$\mathfrak{S}_{\xi}$$\underline{F}_{\xi}^{*}$

where

  * •

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

  * •

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

  * •

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

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

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

###  8.2. Concentrating Local Families

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

######  Definition 8.3.

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

  1. (A)

The universal family of concentrating local solutions by

$\left(\frac{\Phi_{1}({\varepsilon,\tau,\xi})}{\varepsilon},A_{1}({\varepsilon,\tau,\xi})\right):=\underline{\Upsilon}^{-1}\left(\xi,\left(\frac{\Phi_{1}(\varepsilon,\tau)}{\varepsilon},A_{1}(\varepsilon,\tau)\right)\right)$
(8.8)

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

  2. (B)

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

$\left(\Phi^{h_{\varepsilon}}_{\tau,\xi},A^{h_{\varepsilon}}_{\tau,\xi}\right):=\underline{\Upsilon}^{-1}\left(\xi,(\Phi^{h_{\varepsilon}}_{\tau},A^{h_{\varepsilon}}_{\tau})\right)\hskip
42.67912pt\left(\Phi_{\tau,\xi},A_{\tau,\xi}\right):=\underline{\Upsilon}^{-1}\left(\xi,(\Phi_{\tau},A_{\tau})\right)$
(8.9)

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

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

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

$\text{SW}_{\varepsilon,\tau,\xi}(\varphi,a,\psi,b):=\text{SW}_{\tau}\left(\left(\frac{\Phi_{1}({\varepsilon,\tau,\xi})}{\varepsilon},A_{1}({\varepsilon,\tau,\xi})\right)+\chi^{+}_{\varepsilon}\cdot(\varphi,a)+\chi^{-}_{\varepsilon}\cdot(\psi,b)\right)$
(8.10)

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

######  Definition 8.4.

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

$\mathbb{H}^{1}_{\varepsilon,\nu}({\mathcal{E}_{\tau}})$$p^{*}\mathbb{L}^{2}_{\varepsilon,\nu}({\mathcal{E}_{\tau}})$$\mathbb{SW},\overline{\mathbb{SW}}$

defined by

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

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

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

$-d^{\star}c-i\frac{\langle
i\zeta,{\Phi_{\xi,\tau}^{h_{\varepsilon}}}\rangle}{\varepsilon}=0,$ (8.13)

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

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

######  Corollary 8.5.

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

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

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

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

###### Proof.

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

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

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

$\overline{\mathbb{SW}}:\mathbb{H}^{1}_{\varepsilon,\nu}(\mathcal{E}_{\tau}\times\mathcal{X}_{\tau})\to
p_{1}^{*}\mathbb{L}^{2}_{\varepsilon,\nu}$ (8.14)

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

###  8.3. Universal Linearization

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

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

$\text{d}{\mathbb{SW}}_{h_{1}}:L^{2,2}(\mathcal{Z}_{\tau})\ \oplus\
H^{1,+}_{\varepsilon,\nu}(Y^{+})\ \oplus\ H^{1,-}_{\varepsilon,\nu}(Y^{-})\
\longrightarrow\ L^{2}(Y),$ (8.15)

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

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

######  Proposition 8.6.

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

$\text{d}{\mathbb{SW}}_{h_{1}}(\xi,\varphi,a,\psi,b)=\mathfrak{B}_{h_{1}}(\xi)\
+\ \mathcal{L}_{h_{1}}\left(\chi_{\varepsilon}^{+}(\varphi,a)\ +\
\chi_{\varepsilon}^{-}(\psi,b)\right)$ (8.16)

where

  * •

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

$\mathfrak{B}_{h_{1}}(\xi)=\frac{d}{ds}\Big{|}_{s=0}\left((\mathfrak{T}_{g_{\tau}}^{g_{\xi}(s)})^{-1}\circ\text{SW}_{p_{\xi}(s)}\circ\mathfrak{T}_{g_{\tau}}^{g_{\xi}(s)}\right)\left(\frac{\Phi_{1}}{\varepsilon},A_{1}\right)$
(8.17)

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

  * •

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

###### Proof.

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

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

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

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

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

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

$H^{1}(S_{E}^{g_{\zeta}}\oplus\Omega)$$L^{2}(S_{E}^{g_{\zeta}}\oplus\Omega)$$H^{1}(S_{E}^{g}\oplus\Omega)$$L^{2}(S_{E}^{g}\oplus\Omega)$$H^{1}(S_{E}^{g}\oplus\Omega)$$L^{2}(S_{E}^{g}\oplus\Omega)$$\begin{pmatrix}\text{varying
}g_{\zeta}\\\ \text{fixed
}\mathcal{Z}\end{pmatrix}$$\begin{pmatrix}\text{varying
}\mathcal{Z}_{\zeta}\\\ \text{fixed
}g\end{pmatrix}$$\begin{pmatrix}\text{fixed }S^{g}\\\ \text{fixed
}\mathcal{Z}\end{pmatrix}$$\text{SW}_{p_{\zeta}}$$(\mathfrak{T}_{g_{\zeta}}^{g})^{-1}$$\mathfrak{S}_{\zeta}\circ\underline{F}_{\zeta}^{*}$$\mathbb{SW}$$\mathfrak{S}_{\zeta}\circ\underline{F}_{\zeta}^{*}$$(\mathfrak{T}_{g_{\zeta}}^{g})^{-1}$$\Upsilon_{\zeta}\text{SW}\Upsilon_{\zeta}^{-1}$

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

$\text{SW}\left((\Phi,A)+(\varphi,a)\right)=\text{SW}(\Phi,A)+\mathcal{L}_{(\Phi,A)}(\varphi,a)+Q(\varphi,a),$
(8.19)

the derivative is given by

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

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

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

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

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

######  Proposition 8.7.

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

$\mathfrak{B}_{h_{1}}(\xi)=\begin{pmatrix}\frac{1}{\varepsilon}\underline{\mathcal{B}}_{\Phi_{1}}(\xi)\\\
\underline{\mathcal{D}}_{A_{1}}(\xi)+\underline{\mu}_{\Phi_{1}}(\xi)\end{pmatrix}$
(8.22)

where

  1. (1)

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

$\underline{\mathcal{B}}_{\Phi_{1}}(\xi)=\left(-\frac{1}{2}\sum_{ij}\dot{g}_{\xi}(e_{i},e_{j})e^{i}.\nabla_{j}+\frac{1}{2}d\text{Tr}_{g_{\tau}}(\dot{g}_{\xi}).+\frac{1}{2}\text{div}_{g_{\tau}}(\dot{g}_{\xi}).+\mathcal{R}(B_{\tau},\xi).\right)\Phi_{1}$
(8.23)

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

  2. (2)

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

${\bf\underline{\mathfrak{D}}}_{A_{1}}(\xi)=\left(-\frac{1}{2}\sum_{ij}\dot{g}_{\xi}(e_{i},e_{j})\mathbb{c}(e^{i})\nabla^{\text{LC}}_{j}+\frac{1}{2}\mathbb{c}(d\text{Tr}_{g_{\tau}}(\dot{g}_{\xi}))+\frac{1}{2}\mathbb{c}(\text{div}_{g_{\tau}}(\dot{g}_{\xi}))\right)A_{1}$
(8.24)

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

  3. (3)

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

$\frac{1}{2}\underline{\mu}_{\Phi_{1}}(\xi)=\left(0,-\frac{1}{2}\sum_{jk}\
\frac{\langle
i\dot{g}_{\xi}(e_{j},e_{k})e^{j}.\Phi_{1},\Phi_{1}\rangle}{\varepsilon^{2}}ie^{k}\right)$
(8.25)

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

###### Proof.

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

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

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

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

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

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

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

###  8.4. Non-Linear Terms

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

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

$\displaystyle\mathbb{SW}\left(h\right)$ $\displaystyle=$
$\displaystyle\text{SW}\left(\frac{\Phi_{1}}{\varepsilon},A_{1}\right)\ +\
\text{d}{\mathbb{SW}}_{h_{1}}(h)\ +\ \mathbb{Q}_{h_{1}}(h)$ (8.26)

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

######  Proposition 8.8.

The non-linear term has the form

$\mathbb{Q}_{h_{1}}(h)=Q_{\text{SW}}(\varphi,a)\ +\ \varepsilon
Q_{\Phi_{1}}(\xi,\varphi)\ +\ \varepsilon Q_{A_{1}}(\xi,a)\ +\
Q_{a.\varphi}(\xi,\varphi,a)\ +\ Q_{\mu}(\xi,\varphi,\varphi)$ (8.27)

where

  1. (1)

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

  2. (2)

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

  3. (3)

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

  4. (4)

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

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

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

  5. (5)

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

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

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

###### Proof.

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

##  9\. Relating Deformation Operators

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

###  9.1. Two Deformation Operators

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

$\boxed{\xi(t)=\varepsilon\eta(t).}$ (9.1)

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

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

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

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

######  Lemma 9.1.

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

$\|\Xi^{-}(\eta)\|_{L^{2}_{\varepsilon,\nu}}\leq
C\varepsilon^{11/12-\gamma}\|\eta\|_{3/2+\underline{\gamma}-\nu}$ (9.4)

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

###### Proof.

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

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

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

$\frac{\Phi_{1}}{\varepsilon}=\frac{\Phi_{\tau}}{\varepsilon}+\varphi_{1},$
(9.5)

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

Then

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

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

$\|\varepsilon\mathcal{B}_{\varphi}(\eta)\mathbb{1}^{-}\|_{L^{2,-}_{\varepsilon,\nu}}\leq
C\varepsilon\|\eta\|_{3/2+\underline{\gamma}-\nu}\Big{(}\|\varphi_{1}\|_{rH^{1}_{e}(Y^{-})}\Big{)}\leq
C\varepsilon^{11/12-\gamma}\|\eta\|_{3/2+\underline{\gamma}-\nu}.$ (9.7)

as desired. In the final inequality, we have used that the $rH^{1}_{e}$ and
$H^{1,+}_{\varepsilon}$ norms are comparable for $r\geq\varepsilon^{2/3}$.
This establishes the desired bound for the spinor components.

For the $S^{\text{Im}}\oplus(\Omega^{0}\oplus\Omega^{1})$, components, note
that that the difference from the (flat) limiting connection $A_{\tau}$ and
the imaginary components of $\Phi_{1}$ are exponentially small on
$\text{supp}(\mathbb{1}^{-})$ by Lemma 7.13; exponentially small here meaning
$O(\varepsilon^{-3}\text{Exp}(-\tfrac{1}{\varepsilon^{\gamma}}))$ for
$\gamma>0$. Since $\underline{\mathcal{D}}_{A_{\tau}}$ depends only on the
$\Omega^{0}\oplus\Omega^{1}$-component and $\underline{\mu}_{\Phi_{1}}$ only
depend on the $S^{\text{Im}}$-component by Item (2) of Lemma 3.5 (up to an
exponentially small factor coming from the difference in (2a) in Section 7.1),
the same argument as the spinor components shows these components satisfy
(9.4) with an exponential factor in place of $\varepsilon^{11/12}$, which may
be absorbed once $\varepsilon$ is sufficiently small. ∎

The situation for radii less than the invariant scale, i.e. for $\Xi^{+}$,
stands in contrast to that of Lemma 9.1: there, the two deformation operators
bear no meaningful relation. Since the invariant scale shrinks rapidly as
$\varepsilon\to 0$, however, the norm of either deformation operator in this
region decreases like a fixed positive power of $\varepsilon$ so that
$\Xi^{+}$ is effectively negligible, as the upcoming lemma shows. Since the
weight function $R_{\varepsilon}$ is almost constant (up to a factor of
$\varepsilon^{-\gamma}$) on $\text{supp}(\mathbb{1}^{+})$, the lemma considers
only the unweighted norms.

The proof utilizes the following re-scaling, which plays an essential role in
the proof of Theorem 7.5 (see [41, Sec. 5.3]). There is a re-scaled coordinate
$\rho$ satisfying $c_{1}\varepsilon^{2/3}\leq\rho\leq c_{2}\varepsilon^{2/3}$
for constants $c_{1},c_{2}$ such that de-singularized solutions
$(\Phi^{h_{\varepsilon}},A^{h_{\varepsilon}})$ are given by

$(\Phi^{h_{\varepsilon}},A^{h_{\varepsilon}})=(\varepsilon^{1/3}\Phi^{H}(\rho),A^{H}(\rho))$
(9.8)

where $\Phi^{H},A^{H}$ are fixed, smooth, $\varepsilon$-independent functions
on $\mathcal{Z}_{\tau}\times\mathbb{R}^{2}$ in Fermi coordinates (3.9).
Moreover, $\Phi^{H}\sim\rho^{1/2}$ for $\rho>>1$, and
$A^{H}=f(\rho)\left(\tfrac{dz}{z}-\tfrac{d\overline{z}}{\overline{z}}\right)$,
where $f(\rho)$ vanishes to second order at $\rho=0$. See Section 4 of [41]
for detailed proofs.

######  Lemma 9.2.

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

$\displaystyle\|\Xi^{+}(\eta)\|_{L^{2}}$ $\displaystyle\leq$
$\displaystyle\|\text{d}\mathbb{SW}_{h_{1}}(\xi,0)\mathbb{1}^{+}\|_{L^{2}}+\|\text{d}\not{\mathbb{D}}_{h_{0}}(\eta,0)\mathbb{1}^{+}\|_{L^{2}}$
(9.9) $\displaystyle\leq$ $\displaystyle
C\varepsilon^{-\gamma}\Big{(}\varepsilon^{1/3}\|\eta\|_{1,2}\ +\
\varepsilon^{11/12}\|\eta\|_{3/2+\underline{\gamma},2}\ +\ \
\varepsilon\|\eta\|_{2,2}\ +\
\varepsilon^{19/12}\|\eta\|_{5/2+\underline{\gamma},2}\Big{)}.$ (9.10)

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

###### Proof.

(9.9) is immediate from (9.2) and the triangle inequality, so it suffices to
show both terms on the right side of (9.9) satisfy the desired bound.

To begin with the deformations of the singular Dirac operator, the formula
(6.7) shows

$\|\text{d}\not{\mathbb{D}}_{h_{0}}(\eta,0)\mathbb{1}^{+}\|_{L^{2}}\leq\|(\dot{g}_{\eta})_{ij}e^{i}.\nabla\Phi_{\tau}\|_{L^{2}}+\|d\text{Tr}_{g_{\tau}}(\dot{g}_{\eta}).\Phi_{\tau}\|_{L^{2}}+\|\text{div}_{g_{\tau}}(\dot{g}_{\eta}).\Phi_{\tau}\|_{L^{2}}+\|\mathcal{R}(B_{\tau},\eta).\Phi_{\tau}\|_{L^{2}}.$

where summation is implicit in the first term. Since $\Phi_{\tau}$ is
polyhomogeneous with leading order $r^{1/2}$ by Lemma 4.5, each of these terms
is of Type A in the sense of Lemma 6.12, with weight $w_{A}\leq 2$. Copying
the proof of Item (A) of Lemma 6.12, but now only integrating over the support
of $\mathbb{1}^{+}$ shows that

$\|(\dot{g}_{\eta})_{ij}e^{i}.\nabla\Phi_{\tau}\|_{L^{2}}\leq
C\varepsilon^{1/3-\gamma}\|\eta\|_{1,2}\hskip
42.67912pt\|d\text{Tr}_{g_{\tau}}(\dot{g}_{\eta}).\Phi_{\tau}\|_{L^{2}}+\|\text{div}_{g_{\tau}}(\dot{g}_{\eta}).\Phi_{\tau}\|_{L^{2}}\leq
C\varepsilon^{1-\gamma}\|\eta\|_{2,2}.$

as desired. The term $\mathcal{R}(\mathcal{B}_{\tau},\eta)$ is subsumed by
these because it has weight $w_{A}=1$. This shows the second term of (9.9)
satisfies the desired bound.

By (8.22) and the triangle inequality, the second term of (9.9) is bounded by

$\displaystyle\|\text{d}\mathbb{SW}_{h_{1}}(\xi)\mathbb{1}^{+}\|_{L^{2}}\leq\|\underbrace{\underline{\mathcal{B}}_{\Phi_{1}}(\eta)}_{\text{(I)}}\|_{L^{2}}\
+\
\varepsilon\|\underbrace{\underline{\mathcal{D}}_{A_{1}}(\eta)}_{\text{(II)}}\|_{L^{2}}\
+\
\varepsilon\|\underbrace{\underline{\mu}_{\Phi_{1}}(\eta)}_{\text{(III)}}\|_{L^{2}}.$

We bound each term (I)–(III) separately. Beginning with (I), write
$(\Phi_{1},A_{1})=(\Phi^{h_{\varepsilon}}_{\tau},A^{h_{\varepsilon}}_{\tau})+(\varepsilon\varphi_{1},a_{1})$
where $(\varphi_{1},a_{1})$ are as in Item (2) of Theorem 7.5. The term (I) is
comprised of four subterms (Ia)–(Id) as in (8.23); for each of these there is
a leading order part coming from
$(\Phi^{h_{\varepsilon}},A^{h_{\varepsilon}})$ and a perturbation coming from
$(\varphi_{1},a_{1})$. We begin with the leading order part of (Ia). Omitting
indices and subscripts and writing $N^{+}=\text{supp}(\mathbb{1}^{+})$ for
clarity,

$\displaystyle\|\dot{g}_{\eta}.\nabla\Phi^{h_{\varepsilon}}\|^{2}_{L^{2}}$
$\displaystyle\leq$ $\displaystyle
C\int_{N^{+}}\left(|\underline{\chi}(\eta^{\prime})|^{2}+|\underline{d\chi}(\eta)|^{2}\right)|\nabla\Phi^{h_{\varepsilon}}|^{2}rdrd\theta
dt$ $\displaystyle\leq$ $\displaystyle
C\int_{N^{+}}\left(|\underline{\chi}(\eta^{\prime})|^{2}+|\underline{d\chi}(\eta)|^{2}\right)|\nabla_{\rho}\Phi^{h_{\varepsilon}}|^{2}\rho
d\rho d\theta dt$ $\displaystyle\leq$ $\displaystyle
C\varepsilon^{2/3}\int_{\mathcal{Z}_{\tau}}\left(|\underline{\chi}(\eta^{\prime})|^{2}+|\underline{d\chi}(\eta)|^{2}\right)\int_{\rho\leq\varepsilon^{-\gamma}}|\nabla_{\rho}\Phi^{H}|^{2}\rho
d\rho d\theta dt$ $\displaystyle\leq$ $\displaystyle
C\varepsilon^{2/3-\gamma}\|\eta\|_{1,2}^{2}$

where we have changed variables to the rescaled coordinate $\rho$ (in both the
volume and $\nabla$) and then substituted (9.8). The last inequality follows
from the fact that $\|\underline{d\chi}(\eta)\|_{L^{2}(S^{1})}\leq
C\|\eta^{\prime}\|_{L^{2}(S^{1})}$ (as in the proof of Lemma 6.14), and the
fact that $\Phi^{H}\sim\rho^{1/2}$ for $\rho\geq\text{const}$. The same
argument applies to the terms (Ib)–(Id), except there is an additional factor
of $\varepsilon^{2/3}$ because there is no derivative to rescale, but the norm
$\|\eta\|_{L^{2,2}}$ is needed.

Turning now to the leading order of term (II), which is again comprised of
three sub-terms (IIa)–(IIc) as in (8.24), a similar rescaling argument applies
to show

$\displaystyle\varepsilon^{2}\|\dot{g}_{\eta}\nabla
A^{h_{\varepsilon}}\|_{L^{2}}^{2}$ $\displaystyle\leq$
$\displaystyle\varepsilon^{2}\int_{N^{+}}\left(|\underline{\chi}(\eta^{\prime})|^{2}+|\underline{d\chi}(\eta)|^{2}\right)|\nabla_{\rho}f(\rho)\left(\tfrac{dz}{z}-\tfrac{d\overline{z}}{\overline{z}}\right)|^{2}\rho
d\rho d\theta dt$ $\displaystyle=$
$\displaystyle\varepsilon^{2}\varepsilon^{-4/3}\int_{N^{+}}\left(|\underline{\chi}(\eta^{\prime})|^{2}+|\underline{d\chi}(\eta)|^{2}\right)|G(\rho)|^{2}\rho
d\rho d\theta dt\leq\varepsilon^{2/3}\|\eta\|^{2}_{1,2}$

where we have substituted $\tfrac{1}{z}=\tfrac{\varepsilon^{2/3}}{\rho
e^{i\theta}}$ (and likewise for $\overline{z}$), then used the fact that
$f(\rho)$ vanishes to second order at $\rho=0$ to combine these into a smooth
bounded function $G(\rho)$. The last inequality follows just as in term (I).
Terms (IIb)–(IIc) proceed with the same alterations as (Ib)–(Id).

In both terms (I)–(II) the perturbation terms arising from
$(\varphi_{1},a_{1})$, the same integration as for the leading order terms is
used. First, for terms (Ia) and (IIa), a factor of
$\|\eta\|_{3/2+\underline{\gamma},2}$ can be pulled out, reducing the integral
to the $H^{1,+}_{\varepsilon}$-norm; for the other terms, the same applies to
(Ib–Id) and (IIb)–(IIc) with $\|\eta\|_{5/2+\underline{\gamma},2}$ (and the
weight in the norm gives an extra factor of $\varepsilon^{2/3-\gamma}$ for
these).

The term (III) the terms of (8.25) of the form $\langle
ig_{\xi}.\Phi_{\tau},\varphi_{1}\rangle$ and $\langle
ig_{\xi}.\varphi_{1},\Phi_{\tau}\rangle$ is can be bounded simply by pulling
an $\varepsilon\|\eta\|_{3/2+\underline{\gamma}}$ out of the integral and
using the weight on the $H^{1,+}_{\varepsilon}$-norm involving $\mu$. Finally,
for the correction terms $\langle ig_{\xi}.\varphi_{1},\varphi_{1}\rangle$, a
bound by $\varepsilon^{11/12}\|\eta\|_{3/2+\underline{\gamma}}$ follows from
the interpolation inequality in the upcoming Lemma 9.4. ∎

###  9.2. The Range Component of Deformations

The previous subsection provided bounds on the perturbation term in (9.2);
this subsection bounds the main term $\text{d}\not{\mathbb{D}}(\eta,0)$. The
range component $(1-\Pi_{\tau})\text{d}\not{\mathbb{D}}(\eta,0)$ is the
crucial term affected by the loss of regularity whose significance was
discussed in Subsection 2.5.1 (the boxed term in 2.26).

The key bound is provided by the following lemma. The lemma shows that, due to
the use of the mode-dependent deformations, the derivative
$\text{d}\not{\mathbb{D}}(\eta,0)$ has effective support (Definition 4.6)
equal to the support of the deformation $\eta$ in Fourier space.

######  Proposition 9.3.

Suppose that $\eta\in C^{\infty}(\mathcal{Z}_{\tau},N\mathcal{Z}_{\tau})$
satisfies the following property: there is an $M\in\mathbb{R}$ such that
$\|\eta\|_{m+1/2,2}\leq CM^{m}\|\eta\|_{1/2,2}$ for all $m>0$. Then
deformation operator (6.3) of the universal Dirac operator satisfies

$\|\text{d}\not{\mathbb{D}}_{h_{0}}(\eta,0)\|_{L^{2}_{-\nu}}\leq
CM^{\nu}\|\eta\|_{1/2,2}$ (9.11)

for any weight $\nu>0$. In particular, it is effectively supported where
$r=O(M^{-1})$ in the sense of Definition 4.6.

###### Proof.

By Lemma 4.5, $\Phi_{\tau}$ is polyhomogeneous. In particular, it satisfies
the required bounds (6.22) uniformly in $\tau$, so that each term (6.7) of
$\text{d}_{h_{0}}\not{\mathbb{D}}(\eta,0)$ is a term of Type A with weight
$w_{A}\leq 2$ in the sense of Lemma 6.12. The conclusion follows directly from
applying Item (A) of that lemma with $\beta=-\nu$ and invoking the assumption
on $\eta$. ∎

###  9.3. Non-Linear Bounds

This subsection bounds the non-linear terms in Proposition 8.8.

The statement of the next lemma involves an auxiliary partition of unity
defined as follows. With $\gamma<<1$ being the constant such that
$\mathbb{1}^{+}$ is the indicator function of
$\\{r\leq\varepsilon^{2/3-\gamma}\\}$, let $\zeta^{\pm}$ be a partition of
unity consisting of two logarithmic cut-off functions such that
$|d\zeta^{\pm}|\leq Cr^{-1}$, and $\zeta^{+}=1$ where
$\\{r\leq\varepsilon^{2/3-2\gamma}\\}$ while
$\text{supp}(\zeta^{+})\subseteq\\{r\leq 2\varepsilon^{2/3-\gamma}\\}$. The
purpose of introducing $\zeta^{+}$ is that it is equal to $1$ on a
neighborhood larger than the support of $\mathbb{1}^{+}$ by a factor of
$\varepsilon^{\gamma}$; this extra buffer zone allows the exponential decay of
Lemma 7.13 to take effect on $\text{supp}(\zeta^{-})$ for configurations that
decay away from the support of $\mathbb{1}^{+}$ (see the proof of Corollary
10.8 in Section 10).

######  Lemma 9.4.

Let $\mathbb{Q}=Q_{\text{SW}}+Q_{\Phi_{1}}+Q_{A_{1}}+Q_{a.\varphi}+Q_{\mu}$ be
the non-linear terms from Proposition 8.8. Then these satisfy the following
bounds.

  1. (I)

$Q_{\text{SW}}$ satisfies

$\displaystyle\ \ \ \ \ \|Q_{\text{SW}}(\varphi,a)\zeta^{+}\|\leq
C\varepsilon^{1/3-\gamma}\|(\varphi,a)\|_{H^{1}_{\varepsilon}}^{2}\hskip
42.67912pt$
$\displaystyle\|Q^{\text{Re}}_{\text{SW}}(\varphi,a)\zeta^{-}\|_{L^{2}}\leq
C\varepsilon^{1/4}\|(\varphi^{\text{Im}},a)\|^{2}_{H^{1}_{\varepsilon}}$
$\displaystyle\|Q^{\text{Im}}_{\text{SW}}(\varphi,a)\zeta^{-}\|_{L^{2}}\leq
C\varepsilon^{1/4}\|\varphi^{\text{Re}}\|_{H^{1}_{\varepsilon}}\|(\varphi^{\text{Im}},a)\|_{H^{1}_{\varepsilon}}.$

  2. (II)

Provided $\|\xi\|_{3/2+\underline{\gamma}}\leq 1$, then
$Q_{\Phi_{1}},Q_{A_{1}}$ satisfy

$\displaystyle\|Q_{\Phi_{1}}(\eta,\varphi)\mathbb{1}^{+}\|_{L^{2}}$
$\displaystyle\leq$ $\displaystyle
C\varepsilon^{1-\gamma}\|\eta\|_{3/2+\underline{\gamma}}\left(\|\varphi\|_{H^{1}_{\varepsilon}}\
+\
\varepsilon^{1/3}\|\eta\|_{1}+\varepsilon\|\eta\|_{2}\right)+C\varepsilon^{5/3-\gamma}\|\eta\|_{5/2+\underline{\gamma}}\|\varphi\|_{H^{1}_{\varepsilon}}$
$\displaystyle\|Q_{\Phi_{1}}(\eta,\varphi)\mathbb{1}^{-}\|_{L^{2}}$
$\displaystyle\leq$ $\displaystyle
C\left(\varepsilon\|\eta\|_{3/2+\underline{\gamma}}\|\varphi\|_{H^{1}_{\varepsilon}}\
+\ \varepsilon\|\eta\|_{1/2}\|\eta\|_{3/2+\underline{\gamma}}\right),$

and identically for $Q_{A_{1}}$ with $\|a\|_{H^{1}_{\varepsilon}}$ in place of
$\|\varphi\|_{H^{1}_{\varepsilon}}$.

  3. (III)

Provided $\|\xi\|_{3/2+\underline{\gamma}}\leq 1$, then

$\displaystyle\|Q_{a.\varphi}(\eta,a,\varphi)\|_{L^{2}}$ $\displaystyle\leq$
$\displaystyle
C\varepsilon\|\eta\|_{3/2+\underline{\gamma}}\|Q_{\text{SW}}(\varphi,a)\|_{L^{2}}$
$\displaystyle\|Q_{\mu}(\eta,\varphi)\|_{L^{2}}$ $\displaystyle\leq$
$\displaystyle
C\varepsilon\|\eta\|_{3/2+\underline{\gamma}}\left(\|\varphi\|_{H^{1}_{e}}+\|Q_{\text{SW}}(\varphi,a)\|_{L^{2}}\right).$

###### Proof.

(I) Configurations on $Y$ satisfy the interpolation inequality

$\|u\|_{L^{4}}\leq C\|u\|_{L^{2}}^{1/4}\|u\|_{L^{1,2}}^{3/4}.$ (9.12)

Since $|\zeta^{\pm}|^{2}\leq|\zeta^{\pm}|\leq 1$, and $|d\zeta^{\pm}|\leq
CR_{\varepsilon}^{-1}$ is bounded by the weight on the $L^{2}$-terms in the
$H^{1}_{\varepsilon}$-norm, applying this shows, e.g.

$\displaystyle\|Q_{\text{SW}}(\zeta^{+}\varphi,\zeta^{+}a)\|_{L^{2}}$
$\displaystyle\leq$ $\displaystyle C\|\zeta^{+}\varphi\|_{L^{2}}^{1/4}\
\|\zeta^{+}\varphi\|^{3/4}_{L^{1,2}}\ \|\zeta^{+}a\|^{1/4}_{L^{2}}\
\|\zeta^{+}a\|^{3/4}_{L^{1,2}}$ $\displaystyle\leq$ $\displaystyle
C\cdot\text{max}(\zeta^{+}R_{\varepsilon}^{1/2})\cdot\|(\varphi,a)\|^{2}_{H^{1}_{\varepsilon}}$
$\displaystyle\leq$ $\displaystyle
C\varepsilon^{1/3-\gamma}\|(\varphi,a)\|^{2}_{H^{1}_{\varepsilon}}.$

because the $H^{1}_{\varepsilon}$-norm dominates the $L^{2}$-norm with an
extra weight of $R_{\varepsilon}$. On the support of $\zeta^{-}$, the proof is
the same but the fact that $Q_{\text{SW}}$ has at most one factor in
$S^{\text{Re}}$ means the weight gives an extra factor of
$\varepsilon^{1/4}|\Phi_{\tau}|^{-1/4}R_{\varepsilon}^{1/4}\leq\varepsilon^{1/4}R_{\varepsilon}^{1/8}$
for $Q_{\text{SW}}^{\text{Im}}$ on $\text{supp}(\zeta^{-})$ and
$\varepsilon^{1/2}|\Phi_{\tau}|^{-1/2}<\varepsilon^{1/4}$ for
$Q_{\text{SW}}^{\text{Re}}$.

(II) Follows from the same considerations as Lemmas 9.2 and 9.1 (with
$(\varphi,a)$ replacing $(\varphi_{1},a_{1})$ in 9.5), and employing Lemma
6.14 in place of Lemma 6.12, then invoking the assumption that
$\|\xi\|_{3/2+\underline{\gamma}}<1$.

(III) Using Taylor’s theorem on the terms involving $\xi$, then pulling out a
factor of
$\|\xi\|_{3/2+\underline{\gamma}}=\varepsilon\|\eta\|_{3/2+\underline{\gamma}}$
as in the proof of Lemma 6.14 reduces to terms of the form in (I) or of the
form $\varepsilon^{-1}|\Phi_{\tau}||\varphi|$ which are bounded by the
$H^{1}_{\varepsilon}$-norm directly. ∎

##  10\. Contraction Subspaces

This section and the next complete the bulk of the proof of Theorem 1.6. This
is done by constructing three linear parametrices, to be denoted by
$P_{\xi},P^{+},P^{-}$, one each for the three steps of the cyclic iteration
described in Section 2.4. These three parametrices are combined into a (non-
linear) approximate inverse
$\mathbb{A}:L^{2}\to\mathbb{H}^{1}_{\varepsilon}(\mathcal{E}_{\tau}\times\mathcal{X}_{\tau})$
of $\overline{\mathbb{SW}}$ as in (8.14), which is the non-linear analogue of
(2.23). Then define

$T=\text{Id}-\mathbb{A}\circ\overline{\mathbb{SW}}_{\Lambda},$ (10.1)

where
$\overline{\mathbb{SW}}_{\Lambda}=\overline{\mathbb{SW}}-\chi^{-}\varepsilon^{-1}{\Lambda(\tau)\Phi_{\tau}}$,
so that each successive application of $T$ carries out a cycle of the
alternating iteration (cf. 2.14).

######  Proposition 10.1.

There exist closed subspaces $\mathcal{H}\subseteq
T(\mathbb{H}^{1}_{\varepsilon}(\mathcal{E}_{\tau}\times\mathcal{X}_{\tau})$,
and $\mathfrak{L}\subseteq\mathbb{L}^{2}_{\varepsilon}$, and a closed ball
$\mathcal{C}\subseteq\mathcal{H}$ containing $0$ such that the following hold
for $h\in\mathcal{C}$ and $N\in\mathbb{N}$, provided
$\varepsilon_{0},\tau_{0}$ are sufficiently small.

  1. (A)

The restriction $T:\mathcal{C}\to\mathcal{C}$ is continuous and depends
smoothly on $(\varepsilon,\tau)$.

  2. (B)

$\|\overline{\mathbb{SW}}_{\Lambda}(T^{N}h)\|_{\mathfrak{L}}\leq\delta^{N}\|\overline{\mathbb{SW}}_{\Lambda}(h)\|_{\mathfrak{L}},$

  3. (C)

$\|Th_{1}-Th_{2}\|_{\mathcal{H}}\leq
C\sqrt{\delta}\|h_{1}-h_{2}\|_{\mathcal{H}}$

where $\delta=\varepsilon^{1/48}$. In particular, $T$ is a contraction for
$\varepsilon$ sufficiently small.

Note the proposition implicitly uses the exponential map from Section 6.1 to
conflate an open subset of $T\mathbb{H}^{1}_{\varepsilon}$ with
$\mathbb{H}^{1}_{\varepsilon}$ so that (10.1) makes sense. The remainder of
the current section constructs the spaces $\mathcal{H},\mathfrak{L}$, and
Section 11 constructs the three parametrices to complete the proof of the
proposition. Theorem 1.6 is deduced as a consequence of Proposition 10.1 in
Section 12.

###  10.1. Three Fourier Regimes

The main challenge in the proof of Proposition 10.1 is controlling the loss of
regularity of the deformation operator $\underline{T}_{\Phi_{\tau}}$ from
Section (6.2). This requires careful analysis of three regimes of Fourier
modes, which are dealt with differently; these are associated to three
different length scales on $Y$.

There are two different spaces of sections on $\mathcal{Z}_{\tau}$ with
Fourier decompositions, and these Fourier decompositions are linked to length
scales on $Y$ by two distinct mechanisms. First, the intrinsic concentration
property of the obstruction (recall Proposition 5.3 and Lemma 5.4) links the
length scales to the Fourier modes in $\text{\bf Ob}(\mathcal{Z}_{\tau})\simeq
L^{2}(\mathcal{Z}_{\tau};\mathcal{S}_{\tau})$. Second, by fiat, the use of
mode-dependent deformations (Section 6.3) links length scales on $Y$ to the
Fourier modes of a deformation $\eta\in
L^{2,2}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})$.

The three length scales $r\leq
c\varepsilon^{2/3-\gamma},c\varepsilon^{2/3-\gamma}\leq r\leq
c\varepsilon^{1/2}$ and $r\geq c\varepsilon^{1/2}$ of the inside, the outside,
and the neck regions respectively give rise to three corresponding Fourier
regimes.

######  Definition 10.2.

For a fixed $\gamma<<1$, set

$\displaystyle L^{\text{low}}=\varepsilon^{-(1/2+\gamma)}$ $\displaystyle
L^{\text{med}}=\varepsilon^{-2/3}$

and for $\ell\in\mathbb{Z}$ define

$\displaystyle{\pi}^{\text{low}}(e^{i\ell t})$ $\displaystyle=$
$\displaystyle\begin{cases}e^{i\ell t}\hskip 14.22636pt|\ell|\leq
L^{\text{low}}\\\ 0\hskip 17.07182pt\ \ |\ell|>L^{\text{low}}\end{cases}\hskip
28.45274pt\pi^{\text{med}}(e^{i\ell t})=\begin{cases}e^{i\ell t}\hskip
14.22636ptL^{\text{low}}<|\ell|\leq L^{\text{med}}\\\ 0\hskip
22.76228pt|\ell|>L^{\text{med}},|\ell|\leq L^{\text{low}}\end{cases}$
$\displaystyle\pi^{\text{high}}(e^{i\ell t})$ $\displaystyle=$
$\displaystyle(\text{Id}-\pi^{\text{med}}-\pi^{\text{low}})e^{i\ell t}.$

We write e.g.
$\Psi^{\text{low}}:=\pi^{\text{low}}\circ\text{ob}_{\tau}^{-1}(\Psi)$ as
shorthand for the projections of an obstruction element
$\Psi\in\text{Ob}(\mathcal{Z}_{\tau})$ to the corresponding Fourier regimes.

Lemma 5.4 yields the following statements for the projection operators of
Definition 10.2: for $\varepsilon$ sufficiently small, suppose that
$\psi_{1},\psi_{2}$ satisfy
$\text{dist}(\mathcal{Z}_{\tau},\text{supp}(\psi_{i}))\geq R_{i}$ for $i=1,2$
where $R_{1}=c\varepsilon^{1/2}$ and $R_{2}=c\varepsilon^{2/3-\gamma}$. Then
for any $M\in\mathbb{N}$

$\displaystyle\|(1-\pi^{\text{low}})\circ\text{ob}_{\tau}^{-1}\circ\Pi_{\tau}(\psi_{1})\|_{L^{2}(\mathcal{Z}_{\tau})}$
$\displaystyle\leq$ $\displaystyle
C_{M}\varepsilon^{M}\|\psi_{1}\|_{L^{2}(Y)}$ (10.2)
$\displaystyle\|\pi^{\text{high}}\circ\text{ob}_{\tau}^{-1}\circ\Pi_{\tau}(\psi_{2})\|_{L^{2}(\mathcal{Z}_{\tau})}$
$\displaystyle\leq$ $\displaystyle
C_{M}\varepsilon^{M}\|\psi_{2}\|_{L^{2}(Y)}.$ (10.3)

hold for constants $C_{M}$. Specifically, these are obtained by applying Lemma
5.4 with taking $\gamma=\gamma_{1}$ in the statement of the lemma, where
$\gamma_{1}$ satisfies
$-\tfrac{2}{3}-\gamma=-\tfrac{2}{3}(\tfrac{1}{1-\gamma^{\prime}})$ for
$\gamma$ as in Definition 10.2.

Sections 5.2 and 6.2 described respectively two invertible elliptic operators,
either of which can be used to cancel the obstruction:

$\displaystyle\underline{T}_{\Phi_{\tau}}:L^{1/2,2}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})$
$\displaystyle\longrightarrow$ $\displaystyle
L^{2}(\mathcal{Z}_{\tau};S_{\tau})$ (10.4)
$\displaystyle\text{ob}_{\tau}^{-1}\not{D}:\ \ \ \ \ \ \ \ \ \ \
\mathcal{X}_{\tau}\ \ \ \ \ \ \ \ $ $\displaystyle\longrightarrow$
$\displaystyle L^{2}(\mathcal{Z}_{\tau};S_{\tau}).$ (10.5)

The first is an isomorphism by Corollary 6.10, the second by Lemma 5.6. The
pre-images of the three Fourier regimes in the codomain give rise to three
corresponding regimes in each
$L^{1/2,2}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})$ and $\mathcal{X}_{\tau}$.

(10.4) is used to cancel the low and medium modes, while (10.5) is used to
cancel the high modes. There is an important and delicate balance that must be
struck here. The higher the Fourier mode, the more extreme the loss of
regularity of (10.4) is, thus there is an upper limit on the modes for which
(10.4) may be used to cancel the obstruction, else the alternating iteration
will not converge. Conversely, there is a lower limit to the modes which may
be solved for using (10.5). Because solutions of (10.5) grow across the neck
region rather than decay, the modes solved for this way must be smaller than
the dominant error by at least that growth factor. (10.2) and (10.3) provide
such estimates, but only in the case of spinors with support restricted away
from $\mathcal{Z}_{\tau}$. The support of error terms coming from the cut-off
functions $d\chi^{\pm}$ in the alternating method therefore yield a lower
bound on the range of modes where using (10.5) is permissible 333This lower
bound is dictated by the somewhat arbitrary choice that $d\chi^{\pm}$ are
supported where $r=(\varepsilon^{1/2})$ and $r=O(\varepsilon^{2/3-\gamma})$.
Different choices, however, also change the powers of $\varepsilon$ appearing
elsewhere (e.g. Theorem 7.5). Different choices result in different ranges for
the three regimes in Definition 10.2, but the central issue cannot be avoided.
The key point is that together, the regions enclosed by these upper and lower
bounds cover the entire spectrum in $L^{2}(\mathcal{Z}_{\tau};S_{\tau})$.

Next, we will define a combined space and operator that cancels the
obstruction using (10.4) for the low and medium modes and (10.5) in the high
modes. First, however, one minor alteration is required. To achieve control on
higher Sobolev norms, we replace (10.4) by an auxiliary operator
$\underline{T}^{\circ}_{\tau}$ which is a small perturbation of it,
constructed as follows. In Fermi coordinates (Definition 3.9) and the
accompanying trivialization (Lemma 3.10) on $N_{r_{0}}(\mathcal{Z}_{\tau})$,
smooth objects may be decomposed using Fourier modes in the $t$-direction,
leading to families of Fourier series smoothly parameterized by the normal
coordinates $(x,y)$. Since $\text{d}{\mathbb{D}}_{h_{0}}(\eta,0)$ is supported
in $N_{r_{0}}(\mathcal{Z}_{0})$, we may define $\underline{T}_{\tau}^{\circ}$
by restricting these Fourier modes. Write $g_{\tau}=g_{0}+h(t,x,y)$ where
$g_{0}$ is the product metric in Fermi coordinates, and set

$g^{\circ}=g_{0}+\pi^{\text{low}}(h)\hskip
56.9055pt\Phi_{\tau}^{\circ}:=\pi^{\text{low}}(\Phi_{\tau})$ (10.6)

where $\pi^{\text{low}}$ from Definition 10.2 applied for every fixed
$(x,y)\in D_{r_{0}}$. Let $\underline{T}_{\tau}^{\circ}$ be the deformation
operator defined analogously to $\underline{T}_{\Phi_{\tau}}$ (Theorem 6.5)
using (10.6) in place of $g_{\tau},\Phi_{\tau}$. By construction,
$\underline{T}^{\circ}_{\tau}$ obeys the hypotheses of Corollary (6.8) with
$M=\varepsilon^{-1/2}$, since $g_{\tau},\Phi_{\tau}$ are smooth in the
$t$-direction.

######  Corollary 10.3.

Let $\mathfrak{t}_{\tau}^{\circ}$ be such that
$\underline{T}_{\Phi_{\tau}}=\underline{T}_{\tau}^{\circ}+\mathfrak{t}^{\circ}_{\tau}$.
For any $M>0$, there is a constant $C_{M}$ depending on $M$ such that

$\|\mathfrak{t}^{\circ}_{\tau}(\eta)\|_{2}\leq
C_{M}\varepsilon^{M}\|\eta\|_{1/2,2}.$

In particular for $\varepsilon_{0}$ sufficiently small,
$\underline{T}_{\tau}^{\circ}$ is invertible, and the estimates (6.16) hold
uniformly in $\tau$. In this case, if
$\underline{T}_{\tau}^{\circ}(\eta^{\text{low}})=\Psi^{\text{low}}$ and
$\underline{T}_{\tau}^{\circ}(\eta^{\text{med}})=\Psi^{\text{med}}$, then

$\|\eta^{\text{low}}\|_{m+1/2,2}\leq
C_{m}(L^{\text{low}})^{-m}\|\Psi^{\text{low}}\|_{L^{2}}\hskip
56.9055pt\|\eta^{\text{med}}\|_{m+1/2,2}\leq
C_{m}(L^{\text{med}})^{-m}\|\Psi^{\text{med}}\|_{L^{2}}$ (10.7)

uniformly in $\tau$ for any $m>0$. ∎

We use $\eta^{\text{low}},u^{\text{low}}$ to denote deformations and spinors
whose images under (10.4) and (10.5) fall in the corresponding regimes in
$L^{2}(\mathcal{Z}_{\tau};S_{\tau})$, and likewise for the medium and high
regimes. Note that $\eta^{\text{low}}$ need not precisely have modes
restricted to the low regime in
$L^{2}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})$ since
$\underline{T}_{\tau}^{\circ}$ does not preserve support in Fourier space; for
the purposes of estimates, however, (10.7) says this is effectively true.

######  Definition 10.4.

Let $\mathfrak{W}_{\varepsilon,\tau}\subseteq
L^{1/2,2}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})\oplus\mathcal{X}_{\tau}$ be
the subspace defined as the image of the following composition.

$L^{2}(\mathcal{Z}_{\tau};S_{\tau})$$\begin{matrix}L^{2}(\mathcal{Z}_{\tau};S_{\tau})\\\
\oplus\\\
L^{2}(\mathcal{Z}_{\tau};S_{\tau})\end{matrix}$$\begin{matrix}L^{1/2,2}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})\\\
\oplus\\\
\mathcal{X}_{\tau}.\end{matrix}$$(\pi^{\text{med}},\pi^{\text{high}})$$(\underline{T}_{\tau}^{\circ},\not{D}^{-1}_{A_{\tau}}\text{ob}_{\tau})$

Equip $\mathfrak{W}_{\varepsilon,\tau}$ with the norm

$\|(\eta^{\text{low}},\eta^{\text{med}},u)\|_{\mathfrak{W}}:=\left(\|\eta^{\text{low}}\|_{1/2,2}^{2}\
+\ \varepsilon^{-1/3}\|\eta^{\text{med}}\|_{1/2,2}^{2}\ +\
\varepsilon^{-4/3}\|u\|_{H^{1}_{e}}\right)^{1/2}.$ (10.8)

By construction,
$(\underline{T}_{\tau}^{\circ},\text{ob}_{\tau}^{-1}\not{D}_{A_{\tau}}):\mathfrak{W}_{\varepsilon,\tau}\to
L^{2}(\mathcal{Z}_{\tau};S_{\tau})$ is an isomorphism. The first two
components give rise to a corresponding set of renormalized deformations
$\xi=\varepsilon\eta$.

Here, we may alter $\pi^{\text{med}}$ so that the projection smoothly
interpolates between $0$ and $1$ on the mode with Fourier index $\left\lceil
L^{\text{med}}\right\rceil$. The same applies to the norm between the low and
medium modes. It may therefore be arranged that the family
$\mathfrak{W}_{\varepsilon,\tau}$ form a smooth vector bundle over pairs
$(\varepsilon,\tau)$.

###  10.2. Contraction Subspaces

This subsection defines the subspaces $\mathcal{H},\mathfrak{L}$ whose
existence is asserted in Proposition 10.1. The definitions have two tasks: (1)
finding a closed subspace $\mathcal{H}\subseteq
T\mathbb{H}^{1}_{\varepsilon}(\mathcal{E}_{\tau}\times\mathcal{X}_{\tau})$ on
which the linearization has index 0 (cf. Proposition 2.4), and (2) defining
correctly weighted norms.

The upcoming weighted norms are chosen in hindsight and are necessary for the
map $T$ to satisfy Items (B) and (C) of Proposition 10.1. The proof that a
single application of $T$ reduces the error as in (B) relies the error having
a certain effective support property (recall Definition 4.6): the majority of
its effective support must be clustered where $d\chi^{+}\neq 0$. The weights
in the upcoming norms impose this property (without Difficulty II from Section
2.5, the weighted norms could simply weight the integral over support of
$d\chi^{+}$ differently — the use of effective support should be thought of as
imposing a slight generalization of this).

Recall that
$T\mathbb{H}^{1}_{\varepsilon}(\mathcal{E}_{\tau}\times\mathcal{X}_{\tau})$ is
isomorphic via $\underline{\Upsilon}$ (Lemma 8.2) to a trivial bundle with
fiber $H^{1,+}_{\varepsilon}\oplus H^{1,-}_{\varepsilon}\oplus
L^{2,2}(\mathcal{Z}_{\tau};N\mathcal{Z}_{\tau})\oplus\mathcal{X}_{\tau}$.
Notice that $\mathfrak{W}_{\varepsilon,\tau}\subseteq
T\mathbb{H}^{1}_{\varepsilon}(\mathcal{E}_{\tau}\times\mathcal{X}_{\tau})$
despite the lower regularity in (10.8), since only a finite-dimensional
subspace of deformations is included.

######  Definition 10.5.

Set

$\displaystyle\mathcal{H}^{+}$ $\displaystyle:=$
$\displaystyle\left\\{\mathcal{L}_{(\Phi_{1},A_{1})}^{-1}(g\mathbb{1}^{+})\ \
\Big{|}\ \ g\in L^{2}(Y^{+})\right\\}\subseteq H^{1,+}_{\varepsilon}$
$\displaystyle\mathcal{H}^{-}$ $\displaystyle:=$
$\displaystyle\left\\{\mathcal{L}_{(\Phi_{\tau},A_{\tau})}^{-1}((1-\Pi_{\tau})g\mathbb{1}^{-})\
\ \Big{|}\ \ g\in L^{2}(Y^{-})\right\\}\subseteq H^{1,-}_{\varepsilon}$

where $\mathcal{H}^{-}$ uses the solution in $H^{1,-}_{\varepsilon}$ that is
$L^{2}$-orthgonal to $\Phi_{\tau}$ on $Y^{-}$. Equip these with the norms

$\displaystyle\|(\varphi,a)\|_{\mathcal{H}^{+}}$ $\displaystyle=$
$\displaystyle\varepsilon^{-1/12-\gamma}\|\mathcal{L}_{(\Phi_{1},A_{1})}(\varphi,a)\mathbb{1}^{+}\|_{L^{2}}$
(10.9) $\displaystyle\|(\psi,b)\|_{\mathcal{H}^{-}}$ $\displaystyle=$
$\displaystyle\left(\|\psi^{\text{Re}}\|^{2}_{rH^{1}_{e}}\ +\
\varepsilon^{\nu}\|\psi^{\text{Re}}\mathbb{1}^{-}\|^{2}_{rH^{1}_{-\nu}}\ +\
\varepsilon^{-1/2}\|(0,\psi^{\text{Im}},b)\|_{H^{1,-}_{\varepsilon}}\right)^{1/2}.$
(10.10)

where $\nu=\nu^{-}=\tfrac{1}{2}-\gamma^{-}$.

######  Definition 10.6.

Define

$\displaystyle\mathcal{H}_{\varepsilon,\tau}$ $\displaystyle:=$
$\displaystyle\mathcal{H}^{+}\oplus\mathcal{H}^{-}\oplus\mathfrak{W}_{\varepsilon,\tau}\oplus\mathbb{R}\hskip
51.21504pt\mathfrak{L}_{\varepsilon,\tau}:=L^{2}(Y)$

and equip these with the norms

$\displaystyle\|(h_{1},h_{2},\eta,\mu)\|_{\mathcal{H}}$ $\displaystyle:=$
$\displaystyle\left(\|h_{1}\|_{\mathcal{H}^{+}}^{2}\ +\
\|h_{2}\|_{\mathcal{H}^{-}}^{2}\ +\ \|\eta\|_{\mathfrak{W}}^{2}\ +\
\varepsilon^{-2}|\mu|^{2}\right)^{1/2}\vspace{.5cm}$ (10.11)
$\displaystyle\vskip 6.0pt plus 2.0pt minus
2.0pt\|\mathfrak{e}\|_{\mathfrak{L}}$ $\displaystyle:=$
$\displaystyle\Big{(}\varepsilon^{-2/12-2\gamma}\|\mathfrak{e}\mathbb{1}^{+}\|^{2}_{L^{2}}\
+\ \|\mathfrak{e}^{\text{Re}}\mathbb{1}^{-}\|^{2}_{L^{2}}\ +\
\varepsilon^{\nu}\|\mathfrak{e}^{\text{Re}}\mathbb{1}^{-}\|^{2}_{L^{2}_{-\nu}}\vskip
3.0pt plus 1.0pt minus 1.0pt$ (10.13) $\displaystyle\ \ \ +\ \
\varepsilon^{-1/3}\|\mathfrak{e}^{\text{Im}}\mathbb{1}^{-}\|^{2}_{L^{2}}\ \ +\
\
\varepsilon^{-1/3}\|\pi^{\text{med}}\Pi_{\tau}(\mathfrak{e})\|^{2}_{L^{2}}\Big{)}^{1/2}$

where $\mathfrak{e}=(\mathfrak{e}^{\text{Re}},\mathfrak{e}^{\text{Im}})\in
L^{2}(S^{\text{Re}})\oplus L^{2}(S^{\text{Im}}\oplus\Omega)$. These families
of spaces form smooth Banach vector bundles over pairs $(\varepsilon,\tau)$
for $\varepsilon,\tau$ sufficiently small.

Note that the weighted terms dictate that the $\nu$-weighted term in the norms
is larger than the unweighted term by at most a factor of
$\varepsilon^{-\nu/2}$, despite the fact that $\sup
r^{-\nu}=\varepsilon^{-\nu(2/3-\gamma)}$ on $\text{supp}(\mathbb{1}^{-})$.
This shows that configurations in $\mathfrak{L}$ have
$\varepsilon^{1/2}$-effective support. Note also that Theorem 7.5 implies that
the pullback (10.9) is indeed a norm.

The following lemma shows the linearized Seiberg–Witten equations are bounded
on the above spaces with operator norm independent of $\varepsilon$ (up to a
factor of $\gamma)$. This assertion is non-trivial: it means that the new
norms control the loss of regularity of the deformation operator. Indeed, if
one simply uses the $L^{1/2,2},H^{1,\pm}_{\varepsilon}$ and $L^{2}$-norms, the
operator norm is only bounded by $\varepsilon^{-4/3}$.

######  Lemma 10.7.

There is a constant $C$ independent of $\varepsilon,\tau$ such that the
linearization
$\text{d}\overline{\mathbb{SW}}_{h_{1}}:\mathcal{H}\to\mathfrak{L}$ satisfies

$\|\text{d}\overline{\mathbb{SW}}_{h_{1}}(h)\|_{\mathfrak{L}}\leq
C\varepsilon^{-\gamma}\|h\|_{\mathcal{H}}$ (10.14)

for some $\gamma<<1$.

###### Proof.

By Proposition 8.7, the linearization on $h=(\xi,\varphi,a,\psi,b,\mu,u)$
where $\xi=\varepsilon\eta$ may be written in the trivialization of Lemma 8.2
as

$\displaystyle\text{d}\overline{\mathbb{SW}}_{h_{1}}(h)$ $\displaystyle=$
$\displaystyle\tfrac{1}{\varepsilon}\underline{\mathcal{B}}_{\Phi_{1}}(\xi)\
+\ \mathcal{L}_{(\Phi_{1},A_{1})}(\mathfrak{p})\ +\
\mu\chi^{-}\frac{\Phi_{\tau}}{\varepsilon}$ (10.15) $\displaystyle=$
$\displaystyle\text{d}\not{\mathbb{D}}_{h_{0}}(\eta,0)+\Xi(\eta)+\mathcal{L}_{(\Phi_{1},A_{1})}(\mathfrak{p})\
+\ \mu\chi^{-}\frac{\Phi_{\tau}}{\varepsilon}.$ (10.16)

where $\Xi$ is as in (9.2), and
$\mathfrak{p}=\chi^{+}(\varphi,a)+\chi^{-}(\psi+u,b)$ where $(\varphi,a)\in
H^{1,+}_{\varepsilon}$, $(\psi,b)\in H^{1,-}_{\varepsilon}$ and
$u\in\mathcal{X}_{\tau}$. The proof now consists of four steps.

Step 1 ($(\varphi,a)$ terms): Abbreviating
$\mathcal{L}_{(\Phi_{1},A_{1})}=\mathcal{L}$, and using the definition of
$\mathcal{H}^{+}$,

$\displaystyle\|\mathcal{L}(\chi^{+}(\varphi,a))\|_{\mathfrak{L}}$
$\displaystyle=$
$\displaystyle\varepsilon^{-1/12-\gamma}\|\mathcal{L}(\varphi,a)\mathbb{1}^{+}\|_{L^{2}}+\|d\chi^{+}\varphi^{\text{Re}}\|_{L^{2}}+\
\varepsilon^{\nu}\|d\chi^{+}\varphi^{\text{Re}}\|_{L^{2}_{-\nu}}$
$\displaystyle\ +\
\varepsilon^{-1/3}\|d\chi^{+}(\varphi^{\text{Im}},a)\|_{L^{2}}\ +\
\varepsilon^{-1/3}\|\pi^{\text{med}}\Pi_{\tau}(d\chi^{+}\varphi^{\text{Re}})\|_{L^{2}}\vskip
6.0pt plus 2.0pt minus 2.0pt$ $\displaystyle\leq$
$\displaystyle\|(\varphi,a)\|_{\mathcal{H}^{+}}\
+\varepsilon^{1/24-\gamma}\|\mathcal{L}(\varphi,a)\mathbb{1}^{+}\|_{L^{2}}\ +\
\|d\chi^{+}\varphi^{\text{Re}}\|_{L^{2}}$ $\displaystyle
O(\varepsilon^{-3}e^{\varepsilon^{-\gamma}})\|(\varphi,a)\|_{H^{1,+}}\ +\
O(\varepsilon^{M})\|(\varphi,a)\|_{L^{2}}$ $\displaystyle\leq$
$\displaystyle\|(\varphi,a)\|_{\mathcal{H}^{+}}.$

where the five terms are bounded respectively as follows. The first by the
definition (10.9) of the $\mathcal{H}^{+}$-norm, the second is reduced to the
case of the first by Lemma 7.12, and the third is identical to the second
since $d\chi^{+}$ is supported where $r=O(\varepsilon^{1/2})$. The fourth is
exponentially small by Lemma (7.13) and Theorem 7.5 because the latter shows
$\|-\|_{L^{2}}\leq\|-\|_{H^{1,+}_{\varepsilon}}\leq\|-\|_{\mathcal{H}^{+}}$,
and the fifth is polynomially small by (10.2).

Step 2 ($(\psi,b)$ terms): Write
$\mathcal{L}_{(\Phi_{1},A_{1})}=\mathcal{L}_{(\Phi_{0},A_{0})}+K_{1}$. The
boundedness of the $\mathcal{L}_{(\Phi_{0},A_{0})}(\chi^{-}(\psi+u,b))$ terms
now follows similarly to the above, using the decay result of Lemma 4.7 in
place of Lemma 7.12, and the boundedness of (7.11) and (4.11) for both
$\nu=0,\nu^{-}$. In this, the bound
$\|\psi^{\text{Re}}+u\|_{rH^{1}_{\varepsilon}}\leq 2\|h\|_{\mathcal{H}}$ is
used, which comes from the $\varepsilon^{-4/3}$-weight in (10.8),,

The only minor difference from Step 1 is for the $\pi^{\text{med}}$-term in
the $\mathfrak{L}$-norm: this is zero on $(\psi,b)$ by definition, and on $u$
is dominated by the factor of $\varepsilon^{-4/3}$ in the
$\|u\|_{\mathfrak{W}}$ norm. Finally, $K_{1}$ is exponentially small on
$\text{supp}(\mathbb{1}^{+})$ by Lemma 7.13 (applied with
$K_{\varepsilon}=\\{r\geq\varepsilon^{2/3-\gamma/2}\\}$).

Step 3 ($\mu$ term): The term $\chi^{-}\varepsilon^{-1}\mu\Phi_{\xi,\tau}$ is
obviously bounded by $\|\mu\|_{\mathcal{H}}$ because the $\varepsilon^{-2}$
weight cancels the $\varepsilon$ in the denominator, and $\Phi_{\tau}\in
L^{2}\cap L^{2}_{-\nu}$. The $\pi^{\text{med}}$-term is smaller, because
$\|\pi^{\text{med}}(\chi^{-}\Phi_{\tau})\|_{L^{2}}=\|\pi^{\text{med}}((1-\chi^{-})\Phi_{\tau})\|_{L^{2}}\leq
C\varepsilon$ using Lemma 4.5.

Step 4 (deformation terms): Proceeding now to the terms involving
$\eta=\varepsilon^{-1}\xi$, Corollary 10.3 provides bounds on the higher
Sobolev norms of $\eta\in\mathfrak{W}$. Using these, Proposition 9.3 applied
with $\nu=0,\nu^{-}$ and Proposition 9.1 show that

$\displaystyle\|\text{d}\not{\mathbb{D}}_{h_{0}}(\eta,0)\mathbb{1}^{-}\|_{L^{2}}\
$ $\displaystyle\leq$ $\displaystyle C\|\eta\|_{1/2}$ (10.17)
$\displaystyle\|\text{d}\not{\mathbb{D}}_{h_{0}}(\eta,0)\mathbb{1}^{-}\|_{L^{2}_{-\nu}}$
$\displaystyle\leq$ $\displaystyle
C(\varepsilon^{-\nu(1/2+\gamma)}\|\eta^{\text{low}}\|_{1/2}+C\varepsilon^{-2\nu/3}\|\eta^{\text{med}}\|_{1/2})\leq
C\varepsilon^{-\nu/2-\gamma}\|\eta\|_{\mathfrak{W}}$ (10.18)
$\displaystyle\|\Xi^{-}(\eta)\|_{L^{2}}$ $\displaystyle\leq$ $\displaystyle
C\varepsilon^{11/12-\gamma}(\varepsilon^{-1/2}\|\eta^{\text{low}}\|_{1/2}+\varepsilon^{-2/3}\|\eta^{\text{med}}\|_{1/2})\leq
C\varepsilon^{5/12-\gamma}\|\eta\|_{\mathfrak{W}}.$ (10.19)

Moreover, since
$r^{-\nu}\leq\varepsilon^{-2\nu/3}\leq\varepsilon^{-\nu/2}\varepsilon^{-\nu/6}$
on $\text{supp}(\mathbb{1}^{-})$, (10.19) shows that

$\varepsilon^{\nu/2}\|\Xi^{-}(\eta)\|_{L^{2}_{-\nu}}\leq
C\varepsilon^{4/12-\gamma}\|\eta\|_{\mathfrak{W}}$ (10.20)

as well. Next, Proposition 9.2 (and the bounds of Corollary 10.3 again) shows
that

$\displaystyle\|\text{d}\not{\mathbb{D}}_{h_{0}}(\eta,0)\mathbb{1}^{+}+\Xi^{+}(\eta)\|_{L^{2}}$
$\displaystyle\leq$ $\displaystyle
C\varepsilon^{-\gamma}\Big{(}\varepsilon^{1/3}\|\eta\|_{1}\ +\
\varepsilon^{11/12}\|\eta\|_{3/2+\underline{\gamma}}\ +\ \
\varepsilon\|\eta\|_{2}\ +\
\varepsilon^{19/12}\|\eta\|_{5/2+\underline{\gamma}}\Big{)}$ (10.21)
$\displaystyle\leq$ $\displaystyle
C\varepsilon^{1/12-\gamma}\|\eta^{\text{low}}\|_{1/2}+C\|\eta^{\text{med}}\|_{1/2}$
$\displaystyle\leq$ $\displaystyle
C\varepsilon^{1/12-\gamma}\|\eta\|_{\mathfrak{W}}.$ (10.22)

Together, (10.17)–(10.22) show that all but the $\pi^{\text{med}}$-term in the
$\mathfrak{L}$-norm are bounded for $\text{d}\not{\mathbb{D}}_{h_{0}}+\Xi$.

For this final term, one has

$\|\pi^{\text{med}}\Pi_{\tau}\text{d}_{h_{0}}\not{\mathbb{D}}(\eta,0)\mathbb{1}^{-}\|_{L^{2}}\
\leq\
\|\pi^{\text{med}}\Pi_{\tau}\text{d}_{h_{0}}\not{\mathbb{D}}(\eta,0)\|_{L^{2}}\
+\
\|\pi^{\text{med}}\Pi_{\tau}\text{d}_{h_{0}}\not{\mathbb{D}}(\eta,0)\mathbb{1}^{+}\|_{L^{2}}$
(10.23)

by the triangle inequality, and we bound each of these terms separately.

For the first term of (10.23),

$\displaystyle\varepsilon^{-1/6}\|\pi^{\text{med}}\Pi_{\tau}\text{d}_{h_{0}}\not{\mathbb{D}}(\eta,0)\|_{L^{2}}$
$\displaystyle\leq$
$\displaystyle\varepsilon^{-1/6}\|\pi^{\text{med}}\underline{T}_{\tau}^{\circ}(\eta)\|_{L^{2}}\
+\ \varepsilon^{-1/6}\|\mathfrak{t}_{\tau}^{\circ}(\eta)\|_{L^{2}}$
$\displaystyle\leq$ $\displaystyle
C\|\eta^{\text{med}}\|_{\mathfrak{W}}+O(\varepsilon^{M})\|\eta\|_{\mathfrak{W}}$

by Definition (10.4) and Lemma 10.3.

For the second term of (10.23), split
$\eta=\eta^{\text{low}}+\eta^{\text{med}}$; the portion involving
$\eta^{\text{med}}$ is obviously bounded because the same $\varepsilon^{-1/6}$
weight appears in (10.8) and in the $\mathfrak{L}$-norm. Thus it suffices to
consider the $\eta^{\text{low}}$ terms. Write
$\text{d}\not{\mathbb{D}}_{h_{0}}=\text{d}^{\circ}\not{\mathbb{D}}+\mathfrak{d}^{\circ}$
where the latter are formed using the metrics of Lemma 10.3. Just as in that
lemma, $\|\mathfrak{d}\|\leq C\varepsilon^{M}$ and so the terms involving the
latter are negligible. Recalling Proposition 5.3 now, $\pi^{\text{med}}$ is
calculated by the sequence of inner-products
$\langle\Psi_{\ell},\text{d}^{\circ}\not{\mathbb{D}}(\eta^{\text{low}})\mathbb{1}^{+}\rangle_{{\mathbb{C}}}$
for $L^{\text{low}}<|\ell|\leq L^{\text{med}}$. The second and third bullet
points of Proposition 5.3 show that the $\zeta_{\ell},\xi_{\ell}$ terms of
$\Psi_{\ell}$ are smaller by a factor of $|\ell|^{-1}>\varepsilon^{-1/2}$,
which dominates the $\varepsilon^{-1/6}$ weight. For the leading order terms
$\chi\Psi_{\ell}^{\circ}$, the restriction of Fourier modes on
$\text{d}^{\circ}\not{\mathbb{D}}$ and Cauchy-Schwartz imply

$\displaystyle\langle\chi\Psi_{\ell}^{\circ},\text{d}^{\circ}\not{\mathbb{D}}(\eta^{\text{low}})\mathbb{1}^{+}\rangle_{{\mathbb{C}}}$
$\displaystyle=$ $\displaystyle 0\hskip 187.78836pt|\ell|\geq 2L^{\text{low}}$
$\displaystyle\langle\chi\Psi_{\ell}^{\circ},\text{d}^{\circ}\not{\mathbb{D}}(\eta^{\text{low}})\mathbb{1}^{+}\rangle_{{\mathbb{C}}}$
$\displaystyle\leq$ $\displaystyle
C\|\Psi_{\ell}^{\circ}\|_{L^{2}(\mathbb{1}^{+})}\|\text{d}^{\circ}\not{\mathbb{D}}(\eta^{\text{low}})\|_{L^{2}(\mathbb{1}^{+})}\hskip
28.45274ptL^{\text{low}}\leq|\ell|\leq 2L^{\text{low}}$ $\displaystyle\leq$
$\displaystyle
C\varepsilon^{1/6-2\gamma}\varepsilon^{1/12-\gamma}\|\eta^{\text{low}}\|_{1/2}$

where the final line directly integrates (5.1) over
$\text{supp}(\mathbb{1}^{+})$, and invokes the second bound on the second term
of (9.9) in Proposition 9.2, which applies equally well to
$\text{d}^{\circ}\not{\mathbb{D}}$. Returning to (10.23), the above combine to
show

$\|\pi^{\text{med}}\Pi_{\tau}\text{d}\not{\mathbb{D}}_{h_{0}}(\eta,0)\mathbb{1}^{-}\|_{L^{2}}\leq
C\|\eta\|_{\mathfrak{W}}$

as desired. ∎

Lemma 9.4 may also be used to bound the nonlinear terms in terms of the
$\mathcal{H},\mathfrak{L}$-norms.

######  Corollary 10.8.

Suppose that $h_{N}\in\mathcal{H}$ satisfies $\|h_{N}\|_{\mathcal{H}}\leq
C\varepsilon^{-1/20}$. Then there is a $C>0$ such that

  1. (A)

$\|\mathbb{Q}(h)\|_{\mathfrak{L}}\leq
C\varepsilon^{2/12-\gamma}\|h\|_{\mathcal{H}}^{2}$,

  2. (B)

$\|\text{d}\mathbb{Q}_{h_{N}}(h)\|_{\mathfrak{L}}\leq
C\varepsilon^{1/12-\gamma}\|h\|_{\mathcal{H}}$

  3. (C)

If $\|h_{1}\|_{\mathcal{H}},\|h_{2}\|_{\mathcal{H}}\leq C\varepsilon^{-1/20}$,
then $\|\mathbb{Q}(h_{1})-\mathbb{Q}(h_{2})\|_{\mathfrak{L}}\leq
C\varepsilon^{1/12-\gamma}\|h_{1}-h_{2}\|_{\mathcal{H}}$

hold uniformly in $\varepsilon,\tau$.

###### Proof.

Let $h=(\xi,\varphi,a,\psi,b,\mu,u)$. Using that