# The Decoupled Haydys-Witten Equations
and a Weitzenböck Formula

theoremdefinition theorem propositiondefinition proposition
corollarydefinition corollary lemmadefinition lemma paragraph section

The Decoupled Haydys-Witten Equations
and a Weitzenböck Formula

Michael<EMAIL_ADDRESS>
Institute for Mathematics, Heidelberg University, Im Neuenheimer Feld 205,
Heidelberg, Germany.

#### Abstract.

The Haydys-Witten equations are partial differential equations on five-
dimensional Riemannian manifolds that are equipped with a non-vanishing vector
field $v$. Conjecturally, their solutions determine the Floer differential in
a gauge-theoretic approach to Khovanov homology. This article introduces a
certain decoupled version of the Haydys-Witten equations, a specialization of
the Haydys-Witten equations that exhibits a Hermitian Yang-Mills structure.
These equations exist whenever the vector bundle defined by the orthogonal
complement of $v$ admits an almost Hermitian structure. We investigate the
relation between the full Haydys-Witten equations and their decoupled version
on manifolds with poly-cylindrical ends and boundaries, and find conditions
under which the Haydys-Witten equations reduce to the decoupled equations.
This relies on a Weitzenböck-like formula that shows that the difference
between the full Haydys-Witten equations and the decoupled equations is
governed by the asymptotic behaviour of solutions near boundaries and non-
compact ends. Regarding the analysis near boundaries, we provide a detailed
analysis of the polyhomogeneous expansion of Haydys-Witten solutions with
twisted Nahm pole boundary conditions, generalizing work of Siqi He in the
untwisted case. The corresponding analysis at non-compact ends relies on a
vanishing theorem by Nagy-Oliveira that was recently generalized by the
author.

##  1 Introduction

Let $(M^{5},g)$ be a five-manifold with poly-cylindrical ends, where ends may
be located at either finite or infinite geodesic distance. Assume $M^{5}$
admits a non-vanishing unit vector field $v\/$ and that the subbundle $\ker
g(v,\cdot)\subset TM^{5}$ admits an almost Hermitian structure. This means
that there is an almost complex structure $J:\ker g(v,\cdot)\to\ker
g(v,\cdot)$ that is compatible with the metric, i.e.
$g(J\cdot,J\cdot)=g(\cdot,\cdot)$.

In this article we investigate the Haydys-Witten equations on manifolds
$(M^{5},g,v,J)$. The existence of $J$ provides a specialization of the
equations that we call _decoupled_ Haydys-Witten equations. Crucially, the
decoupled version of the equations exhibits a Hermitian Yang-Mills structure,
which provides additional tools in solving the equations. This structure
becomes most apparent in the $4\mathcal{D}$-formulation of the Haydys-Witten
equations, an extension of Witten’s $3\mathcal{D}$-formulation of the extended
Bogomolny equations (EBE) [Wit11], which is introduced below and used
throughout the introduction. The main contribution of this article consists in
working out conditions under which the Haydys-Witten equations reduce to the
decoupled version.

Curiously, in the context of Witten’s gauge theoretic approach to homological
knot invariants [Wit11], manifolds are generally equipped with the additional
structure $(g,v,J)$. In that situation one considers the Haydys-Witten
equations on five-manifolds of the form $M^{5}=\mathbb{R}_{s}\times
X^{3}\times\mathbb{R}_{y}^{+}$, equipped with a product metric $g$, and sets
$v=\partial_{y}$. The subbundle $\ker g(v,\cdot)$ is then simply the tangent
space of $\mathbb{R}_{s}\times X^{3}$, which always admits an almost complex
structure, as it is an open and orientable four-manifold. Conjecturally, when
$X^{3}=\mathbb{R}^{3}$ or $S^{3}$ and in the presence of a magnetically
charged knot $K\subset X^{3}$, the homology groups obtained from
$\theta$-Kapustin-Witten solutions (stationary Haydys-Witten solutions) at
$s\to\pm\infty$ modulo Haydys-Witten instantons reproduces a knot invariant
known as Khovanov homology. Hence, the results presented here may offer a
fresh perspective on the gauge theoretic approach to Khovanov homology.

In the following we briefly introduce the $4\mathcal{D}$-formulation of the
Haydys-Witten equations and the decoupled version of the equations, see
section 2 for a more detailed, global discussion. For this, let $G$ denote a
compact Lie group, $G_{\mathbb{C}}$ its complexification, $E\to M^{5}$ a
$G$-principal bundle, and $E_{\mathbb{C}}$ the associated
$G_{\mathbb{C}}$-principal bundle. Furthermore, let $\mathcal{A}(E)$ denote
gauge connections and write $\Omega^{2}_{v,+}(M^{5})$ for Haydys’ self-dual
two-forms with respect to $v$ [Haydys2010].

Given a pair
$(A,B)\in\mathcal{A}(E)\times\Omega^{2}_{v,+}(M^{5},\operatorname{ad}E)$ and
an almost complex structure $J$, one can locally define four differential
operators $\mathcal{D}_{\mu}$ that act on sections of
$\operatorname{ad}E_{\mathbb{C}}$. To that end, consider normal coordinates
$(x^{i},y)_{i=0,1,2,3}$ near a point $p$, chosen in such a way that
$v=\partial_{y}$ and that $J$ takes the canonical form with respect to the
coordinate vector fields $\partial_{i}$ at $p$. In these coordinates,
$B=\sum_{a,b,c=1}^{3}\phi_{a}(dx^{0}\wedge
dx^{a}+\tfrac{1}{2}\epsilon_{abc}dx^{b}\wedge dx^{c})$. The four differential
operators are defined by

$\displaystyle\mathcal{D}_{0}$ $\displaystyle=\nabla^{A}_{0}+i\nabla^{A}_{1}$
$\displaystyle\mathcal{D}_{1}$ $\displaystyle=\nabla^{A}_{2}+i\nabla^{A}_{3}$
(1) $\displaystyle\mathcal{D}_{2}$
$\displaystyle=\nabla^{A}_{y}-i[\phi_{1},\cdot]$
$\displaystyle\mathcal{D}_{3}$
$\displaystyle=[\phi_{2},\cdot]-i[\phi_{3},\cdot]$ (2)

There is a complex conjugation, induced from
$\operatorname{ad}E_{\mathbb{C}}$, that we denote by
$\overline{\mathcal{D}_{\mu}}$. Furthermore, $G_{\mathbb{C}}$-valued gauge
transformations act on the operators by conjugation, i.e.
$\mathcal{D}_{\mu}\mapsto g\mathcal{D}_{\mu}g^{-1}$.

In this formulation, the Haydys-Witten equations
${\mathop{\mathbf{HW}}{}_{v\,\/}}(A,B)=0$ are given by

$\displaystyle[\overline{\mathcal{D}_{0}},\overline{\mathcal{D}_{i}}]-\tfrac{1}{2}\varepsilon_{ijk}[\mathcal{D}_{j},\mathcal{D}_{k}]$
$\displaystyle=0\ ,\qquad i=1,\ldots,3\ ,$ (3)
$\displaystyle\sum_{\mu=0}^{3}[\mathcal{D}_{\mu},\overline{\mathcal{D}_{\mu}}]$
$\displaystyle=0\ .$ (4)

A typical approach in solving equations of this type is to utilize their
symmetry properties. Since there is an action by complex gauge transformation,
one natural idea is to use a Donaldson-Uhlenbeck-Yau type approach, where one
first extracts some underlying holomorphic data from
$G_{\mathbb{C}}$-invariant parts of the equations, and subsequently hopes to
find a complex gauge transformation that ensures also the remaining equations
are satisfied.

Unfortunately, although the Haydys-Witten equations are invariant under
$G$-valued gauge transformations and the action lifts naturally to
$G_{\mathbb{C}}$, neither (3) nor (4) are invariant under
$G_{\mathbb{C}}$-valued gauge transformations. There is, however, a subset of
solutions for which the three equations in (3) decompose into their
$G_{\mathbb{C}}$-invariant parts. This is given by solutions that satisfy the
following equations:

$\displaystyle\begin{split}[\mathcal{D}_{\mu},\mathcal{D}_{\nu}]&=0\
,\qquad\mu,\nu=0,\ldots,3\ ,\\\
\sum_{\mu=0}^{3}[\mathcal{D}_{\mu},\overline{\mathcal{D}_{\mu}}]&=0\
.\end{split}$ (5)

We refer to (5) as _decoupled Haydys-Witten equations_ and denote them by
${{\mathop{\mathbf{dHW}}{}_{v,J\,\/}}(A,B)=0}$. A global version of these
equations is provided in section 2. The commutativity equations are
$G_{\mathbb{C}}$-invariant and can be interpreted as a complex moment map
condition in a hyper-Kähler reduction, while the remaining equation provides
the real moment map condition. Put differently, the decoupled equations
exhibit a Hermitian Yang-Mills structure, such that a Donaldson-Uhlenbeck-Yau
type approach and other powerful tools become available.

Clearly, whenever ${\mathop{\mathbf{dHW}}{}_{v,J\,\/}}(A,B)=0$, then
${\mathop{\mathbf{HW}}{}_{v\,\/}}(A,B)=0$. Here, we prove that in certain
situations the reverse is true, such that the Haydys-Witten equations reduce
to the decoupled Haydys-Witten equations. This is controlled, on the one hand
by the (asymptotic) geometry, and on the other hand by the asymptotic
behaviour of the fields $(A,B)$, at boundaries and cylindrical ends of
$M^{5}$. Crucially, we need to assume that  (a) boundaries of $M^{5}$ are flat
and $(A,B)$ satisfies Nahm pole boundary conditions with knot singularities,
and  (b) non-compact ends of $M^{5}$ are asymptotically locally Euclidean
(ALE) or flat (ALF) gravitational instantons and $(A,B)$ approaches a finite
energy solution of the $\theta$-Kapustin-Witten equations.  For the sake of
brevity, we omit further technical conditions for now and instead refer to
assumptions (A1) - (A4) in section 6 for details. With that understood, our
main result is as follows (this is section 6).

######  Theorem A.

Let $G=SU(2)$, $M^{5}$ a manifold with poly-cylindrical ends, $v$ a non-
vanishing vector field that approaches ends at a constant angle, and $J$ an
almost Hermitian structure on $\ker g(v,\cdot)$. Assume
${\mathop{\mathbf{HW}}{}_{v\,\/}}(A,B)=0$ and that assumptions (A1) \- (A4)
are satisfied, then ${\mathop{\mathbf{dHW}}{}_{v,J\,\/}}(A,B)=0$.

The proof of Theorem A is based on a Weitzenböck formula of the form

$\displaystyle\int_{M^{5}}\left\lVert{\mathop{\mathbf{HW}}{}_{v\,\/}}(A,B)\right\rVert^{2}=\int_{M^{5}}\left\lVert{\mathop{\mathbf{dHW}}{}_{v,J\,\/}}(A,B)\right\rVert^{2}+\int_{M^{5}}d\chi$
(6)

From this it’s clear that whenever ${\mathop{\mathbf{HW}}{}_{v\,\/}}(A,B)=0$
and any boundary contributions (including contributions from non-compact ends)
in $\int_{M^{5}}d\chi$ vanish, then also
${\mathop{\mathbf{dHW}}{}_{v,J\,\/}}(A,B)=0$. The key insights of this article
lie in determining conditions under which all boundary contributions vanish,
if one imposes Nahm pole boundary conditions at finite distances and
asymptotically stationary solutions at infinity. Due to the two different
flavours of boundary conditions, we rely on two distinct facts: Elliptic
regularity of the Nahm pole boundary conditions, on the one hand, and a
vanishing theorem for solutions of $\theta$-Kapustin-Witten solutions on ALE
and ALF spaces, on the other. Let us shortly explain how these two facts
appear in the proof.

First, the Nahm pole boundary conditions for $(A,B)$ state, in particular,
that at order $y^{-1}$ the fields satisfy the extended Bogomolny equations
(EBE), which in the $4\mathcal{D}$ formalism correspond to

$\displaystyle\mathcal{D}_{0}$ $\displaystyle=0\ ,$
$\displaystyle[\mathcal{D}_{i},\mathcal{D}_{j}]$ $\displaystyle=0\qquad
i,j=1,\ldots,3\ ,$
$\displaystyle\sum_{i=1}^{3}[\mathcal{D}_{i},\overline{\mathcal{D}_{i}}]$
$\displaystyle=0\ .$ (7)

This means that the leading order terms are already solutions of the decoupled
Haydys-Witten equations. As will be discussed in detail below, elliptic
regularity of the Haydys-Witten equations states that deviations from the EBE-
solutions can only appear at order $y^{1+\delta}$, for some $\delta>0$
[Mazzeo2013a, MW17, He18]. This, in turn, implies that $\chi$ only involves
terms of order $y^{\delta}$. As a consequence, any contributions to $\int
d\chi$ from boundaries with Nahm pole boundary conditions vanish.

We expect that Theorem A is also generally true in the presence of knot
singularities. Unfortunately, the twisted knot singularity solutions are only
known implicitly (by a continuation argument) [Dim22], such that extracting
information from elliptic regularity is difficult. Although we currently have
no proof for this extension of the result, we include a discussion of the
relevant boundary conditions and state a necessary condition that is known to
be satisfied in the untwisted case.

Second, and perhaps more surprisingly, the boundary terms also vanish at
asymptotic ends when the fields approach stationary solutions of the Haydys-
Witten equations, or equivalently solutions of the $\theta$-Kapustin-Witten
equations, with finite energy. This relies on a vanishing theorem that was
originally conjectured by Nagy and Oliveira [NO21] and for which a proof
appeared recently in [Ble23]. The vanishing theorem states that for a finite
energy solution of the $\theta$-Kapustin-Witten equations on an ALE or ALF
gravitational instanton $A$ is flat, $\phi$ is $\nabla^{A}$-parallel, and
$[\phi\wedge\phi]=0$.

This article is structured as follows. In section 2 we summarize Haydys’
geometry and the Haydys-Witten equations, define the decoupled Haydys-Witten
equations, and establish the promised Weitzenböck formula. In section 3 we
further specify the basic geometric setting that is necessary to specify
boundary conditions and which is used in evaluating the integral of the exact
term in the Weitzenböck formula. A key step in this article is an
investigation of the polyhomogeneous expansion of a twisted Nahm pole solution
of the Haydys-Witten equations, which is presented in section 4. Subsequently,
in section 5, we determine the asymptotic behaviour of $\chi$ at the various
boundaries and ends, filling in the details of the remaining boundary
conditions as we go. Finally, in section 6, we bring everything together and
show that in certain situations the boundary term in (6) vanishes, which
immediately implies Theorem A.

#### Acknowledgements

This work is funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy EXC 2181/1 -
390900948 (the Heidelberg STRUCTURES Excellence Cluster).

##  2 The Decoupled Haydys-Witten Equations and a Weitzenböck Formula

In this section, we introduce the global version of the Haydys-Witten
equations and their decoupled version, and establish a Weitzenböck formula
that relates it to the full Haydys-Witten equations.

Let $(M^{5},g)$ be a complete Riemannian manifold. Consider a principal
$G$-bundle $E\to M^{5}$ for some compact Lie group $G$, write $\mathcal{A}(E)$
for the space of gauge connections, and denote by $\operatorname{ad}E$ the
adjoint bundle associated to the Lie algebra $\mathfrak{g}$ of $G$.

Assume $M^{5}$ admits a non-vanishing unit vector field $v\/$ and write
${\eta=g(v,\cdot)\in\Omega^{1}(M)}$ for its dual one-form. Since $v$ is non-
vanishing, the distribution $\ker\eta\subset TM^{5}$ is regular, i.e. a vector
subbundle, of rank 4. Observe that the pointwise linear map

$\displaystyle
T_{\eta}:\Omega^{2}(M)\to\Omega^{2}(M),\quad\omega\mapsto\star_{5}(\omega\wedge\eta)$
(8)

has eigenvalues $\\{-1,0,1\\}$, such that $\Omega^{2}(M)$ decomposes into the
corresponding eigenspaces:

$\displaystyle\Omega^{2}(M)=\Omega^{2}_{v,-}(M)\oplus\Omega^{2}_{v,0}(M)\oplus\Omega^{2}_{v,+}(M)$
(9)

We use the notation $\omega^{+}$ to denote the part of $\omega$ that lies in
$\Omega^{2}_{v,+}(M)$.

Consider a pair
$(A,B)\in\mathcal{A}(E)\times\Omega^{2}_{v,+}(\operatorname{ad}E)$, consisting
of a gauge connection $A$ and a self-dual two form $B$. Let $\nabla^{A}$
denote the covariant derivative on $\Omega^{2}_{v,+}(\operatorname{ad}E)$
induced by $A$ and the Levi-Civita connection, and define
$\delta_{A}^{+}:\Omega^{2}_{v,+}(M,\operatorname{ad}E)\to\Omega^{1}(M,\operatorname{ad}E)$
by the composition $\delta_{A}^{+}B=-g^{\mu\nu}\iota_{\mu}\nabla^{A}_{\nu}B$.
The Haydys-Witten equations are defined by:

$\displaystyle\begin{split}F_{A}^{+}-\sigma(B,B)-\nabla_{v}^{A}B&=0\ ,\\\
\imath_{v}\ F_{A}-\delta_{A}^{+}B&=0\ .\end{split}$ (10)

Assume now that there is an almost complex structure $J$ on the vector bundle
$\ker\eta\to M^{5}$ that is compatible with the metric, i.e.
$g(J\cdot,J\cdot)=g(\cdot,\cdot)$. Note that $J$ lifts, via the tensor product
$J\otimes J$, to a map on $\Omega^{2}_{v,+}(M^{5})$. At each point this map
has eigenvalues $\\{+1,-1,-1\\}$. To see this, let $p\in M^{5}$ and
$(x^{\mu},y)_{\mu=0,\ldots,3}$ be coordinates in a neighbourhood of $p$, such
that $v=\partial_{y}$ and chosen in such a way that $dx^{\mu}$ is a canonical
basis of $J$, i.e. $Jdx^{0}=dx^{1}$ and $Jdx^{2}=dx^{3}$. The two-forms
$e_{i}=dx^{0}\wedge dx^{i}+\frac{1}{2}\epsilon_{ijk}dx^{j}\wedge dx^{k}$,
$i=1,2,3$, are a local basis of $\Omega^{2}_{v,+}$. One easily sees that
$Je_{1}=+e_{1}$ and $Je_{2/3}=-e_{2/3}$.

The decoupled Haydys-Witten equations arise if one sets to zero the
$-1$-eigenparts of the expressions in the Haydys-Witten equations (10) that
contain $B$, independently from the remaining terms:

######  Definition 2.1 (Decoupled Haydys-Witten Equations).

$\displaystyle\tfrac{1+J}{2}\left(\sigma(B,B)+\nabla^{A}_{v}B\right)$
$\displaystyle=F_{A}^{+}$
$\displaystyle\tfrac{1-J}{2}\left(\sigma(B,B)+\nabla^{A}_{v}B\right)$
$\displaystyle=0$ (11) $\displaystyle\delta_{A}^{+}\tfrac{1+J}{2}B$
$\displaystyle=\imath_{v}F_{A}$ $\displaystyle\delta_{A}^{+}\tfrac{1-J}{2}B$
$\displaystyle=0$ (12)

We denote the associated differential operator by
${{\mathop{\mathbf{dHW}}{}_{v,J\,\/}}}$.

In these equations the part of the $B$-terms that is located in the negative
eigenspaces is decoupled from the gauge curvature, hence the name. A direct
calculation in coordinates shows that the decoupled Haydys-Witten equations
are locally equivalent to (5) in the $4\mathcal{D}$-formalism.

To find a relation between ${\mathop{\mathbf{HW}}{}_{v\,\/}}$ and
${\mathop{\mathbf{dHW}}{}_{v,J\,\/}}$, split the terms in the Haydys-Witten
equations that involve $B$ into their positive and negative eigenparts with
respect to $J$. To simplify notation, write $J^{\pm}:=\tfrac{1\pm J}{2}$ as
shorthand for the projection to the $\pm 1$-eigenspaces of $J$ in
$\Omega^{2}_{v,+}$.

Denote by $\star$ the Hodge star operator and equip
$\Omega^{k}(M^{5},\operatorname{ad}E)$ with the density-valued inner product
${\langle a,b\rangle=\operatorname{Tr}a\wedge\star b}$. Upon integration this
provides the usual $L^{2}$-product $\langle
a,b\rangle_{L^{2}(W)}=\int_{W^{n}}\langle a,b\rangle$ on
$\Omega^{k}(W^{n},\operatorname{ad}E)$. The (density-valued) $L^{2}$-norm of
the Haydys-Witten operator can be rewritten as follows

$\displaystyle\left\lVert{\mathop{\mathbf{HW}}{}_{v\,\/}}(A,B)\right\rVert^{2}$
$\displaystyle=\left\lVert
F_{A}^{+}-\left(J^{+}+J^{-}\right)\left(\sigma(B,B)+\nabla^{A}_{v}B\right)\right\rVert^{2}+\left\lVert\imath_{v}F_{A}-\delta_{A}^{+}\left(J^{+}+J^{-}\right)B\right\rVert^{2}$
(13)
$\displaystyle=\left\lVert{\mathop{\mathbf{dHW}}{}_{v,J\,\/}}(A,B)\right\rVert^{2}-2\left\langle
F_{A}^{+},J^{-}\left(\sigma(B,B)+\nabla^{A}_{v}B\right)\right\rangle-2\left\langle\imath_{v}F_{A}-\delta_{A}^{+}J^{+}B,\delta_{A}^{+}J^{-}B\right\rangle$
(14)

Here we used that the $\pm 1$ eigenspaces of $J$ in $\Omega^{2}_{v,+}$ are
orthogonal with respect to $\langle\cdot,\cdot\rangle$ to remove one of the
mixed terms on the right hand side. By an explicit calculation in a basis one
further finds that the extra terms on the right hand side are in fact total
derivatives:

$\displaystyle d\operatorname{Tr}\left(F_{A}\wedge J^{-}B\right)$
$\displaystyle=\left\langle
F_{A}^{+},\nabla^{A}_{v}J^{-}B\right\rangle+\left\langle\imath_{v}F_{A},\delta_{A}^{+}J^{-}B\right\rangle$
(15) $\displaystyle d\operatorname{Tr}\left(\delta_{A}^{+}J^{+}B\wedge
J^{-}B\wedge\eta\right)$ $\displaystyle=\left\langle
F_{A}^{+},J^{-}\sigma(B,B)\right\rangle-\left\langle\delta_{A}^{+}J^{+}B,\delta_{A}^{+}J^{-}B\right\rangle$
(16)

For the first equation one again uses orthogonality of the different subspaces
of $\Omega^{2}$ with respect to $\langle\cdot,\cdot\rangle$ and the Bianchi
identity $d_{A}F_{A}=0$. The second equation follows similarly after some
reordering of derivatives.

With these identifications we arrive at the following.

######  Lemma .

There is a Weitzenböck formula adapted to the decoupled Haydys-Witten
equations

$\displaystyle\int_{M^{5}}\left\lVert{\mathop{\mathbf{HW}}{}_{v\,\/}}(A,B)\right\rVert^{2}$
$\displaystyle=\int_{M^{5}}\left(\left\lVert{\mathop{\mathbf{dHW}}{}_{v,J\,\/}}(A,B)\right\rVert^{2}+d\chi\right)$
(17)

where the exact term $\chi=\chi_{1}+\chi_{2}$ is given by

$\displaystyle\chi_{1}$ $\displaystyle=-2\operatorname{Tr}\left(F_{A}\wedge
J^{-}B\right)$ (18) $\displaystyle\chi_{2}$
$\displaystyle=-2\operatorname{Tr}\left(\delta_{A}^{+}J^{+}B\wedge
J^{-}B\wedge\eta\right)$ (19)

##  3 Poly-Cylindrical Ends and Boundary Conditions

In Witten’s approach to Khovanov homology, one considers manifolds of the form
$\mathbb{R}_{s}\times X^{3}\times\mathbb{R}^{+}_{y}$, which possess both a
boundary and non-compact ends. Moreover, the non-vanishing vector field
$v=\partial_{y}$ differentiates between the non-compact directions
$\mathbb{R}_{s}$ and $\mathbb{R}^{+}_{y}$, by way of the glancing angles
$\theta=\pi/2$ and $\beta=0$, respectively. This structure is well-described
by manifolds with poly-cylindrical ends and boundaries.

In the literature, poly-cylindrical ends are typically assumed to be located
at geodesic infinity. In this case poly-cylindrical manifolds are identified
with the interior of a manifold with corners that is equipped with a metric,
such that any boundary point lies at infinity and the metric is of product
type within some small tubular neighborhood. Sometimes the definition is
further relaxed by assuming that the metric on the manifold with corners is an
_exact $b$-metric_ in the sense of Melrose [Mel95, Mel96]. In this latter case
the metric on $M$ approaches a product metric exponentially fast, or put
differently, its asymptotic expansion in $\\{x^{k}\\}_{k\in\mathbb{Z}}$
vanishes to all orders. We expect that our results can be generalized to this
situation, though we don’t further pursue this here.

In our situation, we also want to include boundaries and corners at finite
distance. Therefore, when we talk about a (poly-)cylindrical end of a manifold
we always take this to encompass boundaries (and corners) at both, finite and
infinite geodesic distance. We will now set up notation that will be used in
the rest of this article and provide definitions for the underlying geometry.
After that, we also provide a first description of the relevant boundary
conditions.

Let $M$ be a manifold with corners. For any $p\in M$ choose a chart $(U,\phi)$
with $\phi(x)=p$ and define the _depth_ of $p$ by the number of components of
$x$ that are zero. This coincides with the number of boundary faces of $M$
that contain $p$ and is independent of the choice of chart. Define the
depth-$k$ stratum of $M$ to be the points

$\displaystyle S^{k}(M):=\left\\{\ p\in
M\mathrel{}\middle|\mathrel{}\operatorname{depth}(p)=k\ \right\\}$ (20)

The interior of $M$ corresponds to $S^{0}(M)$, while all boundary faces of
codimension one are contained in the boundary stratum $\partial M:=S^{1}(M)$,
and more generally points that lie on corners of codimension $k$ are collected
in $S^{k}(M)$. Clearly $M=\sqcup_{k=0}^{n}S^{k}(M)$, providing a
stratification of $M$. Each stratum $S^{k}(M)$ is a manifold of dimension
$n-k$, since corners of higher codimension are explicitly excluded.

The boundary stratum is a disjoint union of connected components $\partial
M=\sqcup_{i\in I\/}\partial_{i}M$, where for simplicity we assume that $I$ is
finite. Let us emphasize that each boundary face $\partial_{i}M$ is an open
manifold without boundary. We typically denote a boundary defining function of
the boundary face $\partial_{i}M$ by $s_{i}$.

######  Definition 3.1.

A manifold with poly-cylindrical ends is a complete Riemannian manifold
$(M,g)$ that is diffeomorphic to a submanifold (with corners) of a compact
manifold with corners $M_{0}$, equipped with a metric $g_{0}$ that pulls back
to $g$, and such that the following conditions are satisfied: For each
boundary component $\partial_{i}M_{0}$ there exists a tubular neighbourhood
$[0,\epsilon)_{s_{i}}\times\partial_{i}M_{0}$, on which $g_{0}$ is either

$\displaystyle g_{0}=ds_{i}^{2}+h_{\partial_{i}M}\qquad\text{or}\qquad
g_{0}=\frac{ds_{i}^{2}}{s_{i}^{2}}+h_{\partial_{i}M}$ (21)

where $h_{\partial_{i}M}$ is a complete metric on $\partial_{i}M$. Moreover,
this is compatible at corners, i.e. there exists a neighbourhood
$[0,\epsilon)^{m}\times X^{n-m}$ of each connected component of $S^{k}(M)$,
where

$\displaystyle g_{0}=\sum_{k\in K}\frac{ds_{k}^{2}}{s_{k}^{2}}+\sum_{i\in
I\setminus K}ds_{i}^{2}+h_{X^{n-m}}$ (22)

######  Example.

Let $M^{5}=\mathbb{R}_{s}\times X^{3}\times\mathbb{R}^{+}_{y}$, together with
the product metric $g=ds^{2}+g_{X^{3}}+dy^{2}$. Consider the compact manifold
with corners $M_{0}=[-1,1]_{s_{0}}\times X^{3}\times[0,1]_{y_{0}}$, equipped
with
$g_{0}=\big{(}\frac{ds_{0}}{1-\left|s_{0}\right|}\big{)}^{2}+h_{X^{3}}+\big{(}\frac{dy_{0}}{1-y_{0}}\big{)}^{2}$.
Then the map that takes $s_{0}\mapsto
s=-\operatorname{sgn}(s_{0})\log(1-\left|s_{0}\right|)$ and $y_{0}\mapsto
y=-\log(1-y_{0})$ defines an isometry between the submanifold
$U=(-1,1)_{s_{0}}\times X^{3}\times[0,1)_{y_{0}}$ and $(M^{5},g)$.

A poly-cylindrical manifold admits a convenient compact exhaustion by a family
of manifolds with corner $M_{\epsilon}$ that mirrors the poly-cylindrical
structure of $M$. To make this precise, note that $h:=\left(\prod_{k\in
K}s_{k}^{2}\right)\cdot g_{0}:TM_{0}\times TM_{0}\to\mathbb{R}$ defines a
Riemannian metric on $M_{0}$. Let $d_{h}(x,y)$ denote the induced distance
function and define $M_{\epsilon}=\left\\{\ x\in
M_{0}\mathrel{}\middle|\mathrel{}d_{h}(x,\partial_{i}M_{0})\geq\epsilon\
\right\\}$. We equip $M_{\epsilon}$ with the restriction of $g$, making it
into a compact Riemannian manifold with poly-cylindrical ends that approaches
$M$ for $\epsilon\to 0$. We will use the manifolds $M_{\epsilon}$ to
regularize the integrals in the Weitzenböck formula section 2. In particular,
since $M_{\epsilon}$ is compact, the following version of Stokes’ theorem
holds.

######  Theorem  (Stokes’ Theorem on Manifolds with Corners [Whi57]).

Let $M_{\epsilon}$ be a compact manifold with corners, then

$\displaystyle\int_{M_{\epsilon}}d\chi=\sum_{i\in
I}\int_{\partial_{i}M_{\epsilon}}\chi\ ,$ (23)

where the sum on the right hand side is over all boundary faces of codimension
one.

Each of the boundaries of $M_{\epsilon}$ can be thought of as an
“$\epsilon$-displacement” of the corresponding end of $M$ into the interior.
By this we mean that for each end $[0,1)_{s}\times W$ of $M$, the
corresponding end of $M_{\epsilon}$ can be identified with an embedding of
$[\epsilon,1]_{s}\times W\hookrightarrow[0,1)_{s}\times W$. At non-compact
ends, the distance between points with $s=\epsilon$ and $s=1$ is finite and
proportional to $\epsilon^{-1}$, while points at $s=0$ reside at infinity.

From now on let $(M^{5},g)$ denote a poly-cylindrical five-manifold and write
$(M_{0},g_{0})$ for a suitable ambient manifold. Throughout, we take $\partial
M^{5}$ to include both, its boundary components and non-compact cylindrical
ends, i.e. we identify the boundary faces $\partial_{i}M^{5}$ with the
corresponding boundary components of $M_{0}$.

Let $[0,\epsilon)_{s}\times W^{4}$ be a tubular neighbourhood of a cylindrical
end and denote the inward-pointing unit-normal vector field by $u$. In case of
a boundary this means that $u=\partial_{s}$, while for a non-compact end we
have $u=s\partial_{s}$. We say $v$ approaches a cylindrical end at constant
angle if there is a tubular neighbourhood111This is equivalent to an
asymptotic equivalence of $g(u,v)-\cos\theta\sim 0$ as $s_{i}\to 0$. It seems
reasonable that this condition can be slightly weakened to
$g(u,v)\sim\cos\theta$. on which the incidence angle $g(u,v)=\cos\theta$ is
constant. This angle determines the natural boundary conditions at a given
cylindrical end.

We will now give a first overview of the relevant boundary conditions, first
for boundaries and then for non-compact ends. The classes of boundary
conditions serve as organizational structure for the rest of this article. A
detailed discussion, in particular for the Nahm pole and knot singularity
model solutions, will be given later, see section 4 and subsection 5.2,
respectively.

At a boundary with incidence angle $g(u,v)=\cos\beta$, $\beta\neq\pi/2$, we
impose $\beta$-twisted Nahm pole boundary conditions, which fix the maximal
rate of growth of $(A,B)$ as one approaches the boundary. The condition
specifies that $(A,B)$ are asymptotic to a model configuration
$(A^{\rho,\beta},B^{\rho,\beta})$ of order $\mathcal{O}(y^{-1})$. The model
configuration is associated to a choice of Lie algebra homomorphism
$\rho:\mathfrak{su}(2)\to\mathfrak{g}$.

It is also possible to include knot singularities of weight
$\lambda\in\Gamma_{\mathrm{char}}^{\vee}$ in the co-character lattice of
$\mathfrak{g}$. In that case one considers the geometric blowup of a surface
$\Sigma_{K}\subset\partial M$, which introduces an additional boundary
$\partial_{K}M\simeq\Sigma_{K}\times H^{2}$, where $H^{2}$ denotes the two-
dimensional hemisphere. While the incidence angle of $v$ at this new boundary
can no longer be constant, we assume that $v$ maintains a constant glancing
angle $\theta$ with $\Sigma_{K}$. At the knot boundary $\partial_{K}M$, the
fields must then be asymptotic to another model configuration
$(A^{\lambda,\theta},B^{\lambda,\theta})$ of order $\mathcal{O}(R^{-1})$,
where $R$ denotes the radial distance to $\Sigma_{K}$.

The Haydys-Witten pair $(A,B)$ satisfies $\beta$-twisted Nahm pole boundary
conditions at $\partial M$, with knot singularity of weight
$\lambda\in\Gamma_{\mathrm{char}}^{\vee}$ along $\Sigma_{K}$, if for some
$\epsilon>0$

  1. (i)

near $\partial_{0}M$:
$(A,B)=\big{(}A^{\rho,\beta},\;A^{\rho,\beta}\big{)}+\mathcal{O}\big{(}y^{-1+\epsilon}\big{)}$

  2. (ii)

near $\partial_{K}M$:
$(A,B)=(A^{\lambda,\theta},B^{\lambda,\theta})+\mathcal{O}(R^{-1+\epsilon})$

Consider now the case of a non-compact end $[0,\epsilon)_{s}\times W^{4}$ with
incidence angle $g(u,v)=\cos\theta$. Natural boundary conditions at non-
compact ends are given by stationary ($s$-invariant) solutions of the
underlying differential equations. It turns out that the specialization to
$s$-invariant Haydys-Witten equations is given either by the Vafa-Witten
equations or the $\theta$-Kapustin-Witten, depending on the value of $\theta$:

$\displaystyle{\mathop{\mathbf{HW}}{}_{v\,\/}}(A,B)\stackrel{{\scriptstyle
s\text{-inv.}}}{{\rightsquigarrow}}\left\\{\begin{array}[]{ll}{\mathop{\mathbf{VW}}{}_{\,\/}}(\tilde{A},\tilde{B},C)&\theta\equiv
0\pmod{\pi}\,,\\\ {\mathop{\mathbf{KW}}{}_{\theta\,\/}}(\tilde{A},\phi)&\text{
else}\end{array}\right.$ (26)

If $\theta\neq 0\pmod{\pi}$, there must be a non-vanishing vector field $w$ on
$W^{4}$, such that $v=\cos\theta u+\sin\theta w$. Moreover, we can define an
orthogonal vector field $v^{\perp}=-\sin\theta u+\cos\theta w$. With this
notation, the four-dimensional fields above are given in terms of $(A,B)$ as
follows

$\displaystyle\tilde{A}$ $\displaystyle:=i^{\ast}A\
,\qquad\begin{array}[]{ll}\tilde{B}=i^{\ast}B\ ,\ C=A_{s}\\\\[3.99994pt]
\phi:=i^{\ast}\left(\imath_{v^{\perp}}B+A_{s}\wedge
w^{\flat}\right)\end{array}$ (29)

We will consider stationary solutions that either satisfy a finite energy
condition, or that exhibit a $\beta=(\pi/2-\theta)$-twisted Nahm pole at one
of the adjacent corners of $M^{5}$.

We arrange the ends of $M^{5}$, in accordance with these boundary conditions,
into the following four classes:

  1. (1)

Nahm pole boundaries $\partial_{\mathrm{NP}}M$,

  2. (2)

Knot boundaries $\partial_{K}M$,

  3. (3)

Kapustin-Witten ends with finite energy $\partial_{\text{KW}}M$, and

  4. (4)

Kapustin-Witten ends with Nahm poles, denoted by $\partial_{\text{NP-KW}}M$.

Let us shortly exemplify this in connection with the gauge theoretic approach
to Khovanov homology. In this situation one considers the five-manifold
$M^{5}=\mathbb{R}_{s}\times X^{3}\times\mathbb{R}^{+}_{y}$ together with the
vector field $v=\partial_{y}$. In presence of a knot $K\subset X^{3}$, one
then moves to the geometric blowup $[M^{5};\Sigma_{K}]$, where
$\Sigma_{K}=\mathbb{R}_{s}\times K$. The blown-up manifold has two types of
boundary components. First, a Nahm pole boundary $\partial_{\text{NP}}M$,
corresponding to those original boundary points at $y=0$ that are not part of
the $\Sigma_{K}$. Second, a knot boundary $\partial_{K}M^{5}$, given by the
points of the unit normal bundle over the surface $\Sigma_{K}$. $M^{5}$ also
has several cylindrical ends. A Kapustin-Witten end appears at $y\to\infty$,
where the angle between $v$ and the inward pointing unit normal vector
$-\partial_{y}$ is $\theta=0$, such that the fields approach a solution of the
Vafa-Witten equations (hence, this should really be called a Vafa-Witten end).
Kapustin-Witten ends with additional Nahm poles at a corner of $M^{5}$ appear
at $s\to\pm\infty$, where fields approach a Kapustin-Witten solutions with a
Nahm pole located at the corner $\\{y=0,s=\pm\infty\\}$.

######  Remark.

In case (3) the fields $(A,B)$ approach a stationary, finite energy solution
of the Kapustin-Witten equations, such that one can determine the behaviour of
$\chi$ with help of the vanishing theorem for $\theta$-Kapustin-Witten
solutions. In the situation of (4) we similarly ask for a stationary solution,
but the fields exhibit a Nahm pole at an adjacent corner. In this situation
the fields diverge and, in particular, do not have finite energy.

##  4 Polyhomogeneous Expansion of Twisted Nahm Pole Solutions

In this section we investigate the asymptotic behaviour of a Haydys-Witten
solution $(A,B)$ in the vicinity of a Nahm pole boundary
$\partial_{\mathrm{NP}}M$, i.e. under the assumption that $(A,B)$ satisfy
twisted Nahm pole boundary conditions without knot singularities. The goal of
this analysis is to extract the leading order behaviour of $F_{A}$ and
$\delta_{A}^{+}J^{+}B$ near a Nahm pole boundary, which is later used to
determine the asymptotics of $\chi$.

The analysis goes back to an essentially identical discussion for the
$\theta=\pi/2$ version of the Kapustin-Witten equations with untwisted Nahm
pole boundary conditions by Mazzeo and Witten in [Mazzeo2013a]. There, a
Weitzenböck formula on $\mathbb{R}^{3}\times\mathbb{R}^{+}$ very similar to
section 2 – in fact a more restrictive one – was established and then utilized
to determine the analytic properties of the Nahm pole boundary conditions in
the first place. While Mazzeo and Witten already touched on the leading and
constant orders for general four-manifolds, Siqi He has later expanded on the
constraints that arise for subleading orders on general four-manifolds and
unveiled a deep relation to the geometry of the boundary [He18].

To set the stage, let $U=W^{4}\times[0,1)_{y}$ be a tubular neighbourhood of a
Nahm pole boundary $\partial_{\mathrm{NP}}M$. Write $u=\partial_{y}$ for the
inward-pointing, unit normal vector field. As always, assume that the
incidence angle determined by $g(u,v)=\cos\beta$ is constant on all of $U$.
Whenever $\beta\neq 0$ there is a non-vanishing unit vector field $w$ on
$W^{4}$, such that $v=\sin\beta w+\cos\beta u$.

Let us shortly recall some additional details about the Nahm pole boundary
conditions on $\partial_{\mathrm{NP}}M$, see [Mazzeo2013a, MW17]. Let
$\phi_{\rho}\in\Omega^{2}_{v,+}(U,\operatorname{ad}E)$ be some fixed boundary
configuration associated to the Nahm pole boundary conditions on $U$. This
means that it satisfies $\phi_{\rho}-\sigma(\phi_{\rho},\phi_{\rho})=0$, such
that its components $(\phi_{\rho})_{i}=\mathfrak{t}_{i}$, $i=1,2,3$, span an
$\mathfrak{su}(2)$ subalgebra inside of $\mathfrak{g}$. There is a unique
corresponding two-form that satisfies
$\phi_{\rho}^{\tau}+\sigma(\phi_{\rho}^{\tau},\phi_{\rho}^{\tau})=0$. Here
$\tau=(132)$ denotes an anticyclic permutation that acts on components by
$(\phi_{\rho}^{\tau})_{i}=(\phi_{\rho})_{\tau(i)}$ and is related to a
reversal of orientation on $\Omega^{2}_{v,+}(U)$. If $\beta\neq 0$, there is
an isomorphism by which we can view elements of $\Omega^{2}_{v,+}(U)$ as the
one-forms dual to a rank 3 subbundle $\Delta_{(u,v)}^{\perp}\subset TU$. Here,
$\Delta_{(u,v)}^{\perp}$ is the orthogonal complement of the subbundle spanned
by $u$ and $v$. With that in mind, the Nahm pole boundary condition states
that as $y\to 0$, $A$ and $B$ are asymptotic to

$\displaystyle A^{\rho,\beta}=\sin\beta\;\frac{\phi_{\rho}^{\tau}}{y}\ ,\qquad
B^{\rho,\beta}=\cos\beta\;\frac{\phi_{\rho}}{y}\ .$ (30)

Near a Nahm pole, the Haydys-Witten equations are amended by the following
gauge fixing condition

$\displaystyle
d_{A^{0}}^{\star_{5}}(A-A^{0})+\sigma(B^{\mathrm{NP}},B-B^{\mathrm{NP}})$
$\displaystyle=0\ ,$ (31)

making the equations into an elliptic system of differential equations.

Ultimately, we want to investigate the leading order of
$\chi_{1}=-2\operatorname{Tr}(F_{A}\wedge J^{-}B)$ and
$\chi_{2}=-2\operatorname{Tr}(\delta_{A}^{+}J^{+}B\wedge J^{-}B\wedge\eta)$
near $\partial_{\mathrm{NP}}M$. Since this involves derivatives and products
of $A$ and $B$, we need to study the subleading orders in a polyhomogeneous
expansion of $A$ and $B$ as $y\to 0$.

Write $A=y^{-1}\sin\beta\ \phi_{\rho}^{\tau}+\omega+a$ and $B=y^{-1}\cos\beta\
\phi_{\rho}+b$, where by assumption of the Nahm pole boundary condition
$a,b\in\mathcal{O}(y^{-1+\epsilon})$, while $\omega$ is a gauge connection on
the restriction of $E$ to $W^{4}$, smoothly extended into the bulk. If $(A,B)$
is a solution of the Haydys-Witten equations that satisfies Nahm pole boundary
conditions, then an elliptic regularity theorem Mazzeo and Witten
[Mazzeo2013a] states that there exist polyhomogeneous expansions

$\displaystyle a$
$\displaystyle\sim\sum_{(\alpha,k)\in\Delta_{0}}a_{\alpha,k}(x^{\mu})\;y^{\alpha}(\log
y)^{k}\ ,$ $\displaystyle b$
$\displaystyle\sim\sum_{(\alpha,k)\in\Delta_{0}}b_{\alpha,k}(x^{\mu})\;y^{\alpha}(\log
y)^{k}$ $\displaystyle(y$ $\displaystyle\to 0)$ (32)

where the indicial set $\Delta_{0}\subset\mathbb{C}\times\mathbb{N}$ is
bounded from the left by $1$ (in particular $a_{0,k}=b_{0,k}=0$).

Plugging $A=y^{-1}\sin\beta\ \phi_{\rho}^{\tau}+\omega+a$ and
$B=y^{-1}\cos\beta\ \phi_{\rho}+b$ into the Haydys-Witten equations then
determines a sequence of constraints on the functions $a_{\alpha,k}$ and
$b_{\alpha,k}$. For our purposes it will suffice to determine $\chi$ up to
$\mathcal{O}(y^{\delta})$, $\delta>0$, so we only need to consider an
expansion up to order $y^{2+\delta}$. We will see below that for generic
values of $\beta$ only the following terms in the expansion are non-zero.

$\displaystyle a$ $\displaystyle=a_{1,1}\ y\log y+a_{1,0}\ y+a_{2,1}\
y^{2}\log y+a_{2,0}\ y^{2}+\mathcal{O}(y^{5/2})$ (33) $\displaystyle b$
$\displaystyle=b_{1,1}\ y\log y+b_{1,0}\ y+b_{2,1}\ y^{2}\log y+b_{2,0}\
y^{2}+\mathcal{O}(y^{5/2})$ (34)

Moreover, the coefficient functions satisfy additional constraints that link
them to the Riemannian curvature of the manifold. In particular, the $\log
y$-terms vanish if sectional curvature is flat in any plane that contains $w$.

To make this more precise, we need to establish analogues of the tools that
were used in the discussion of indicial roots in [Mazzeo2013a] and [He18].
First, the fibers of $\operatorname{ad}E$ decompose under the action of
$(\phi_{\rho})_{i}=\mathfrak{t}_{i}\in\mathfrak{su}(2)_{\mathfrak{t}}$ as a
direct sum of spin $j$ representations. We denote the associated vector
bundles by $V_{j}$, $j\in J$. Since we consider regular Nahm pole conditions,
only positive integers $j$ appear; in fact for $G=SU(N)$ one finds
$J=\\{1,\ldots,N-1\\}$ [Mazzeo2013a]. Second, there is an action by rotations
$\mathfrak{so}(3)\simeq\mathfrak{su}(2)_{\mathfrak{s}}$ on
$\Omega^{2}_{v,+}(W^{4})$ and $(\Delta_{(u,v)}^{\perp})^{\ast}$. For fixed
$j$, the bundles $\Omega^{2}_{v,+}(W^{4},\operatorname{ad}E)$ and
$\operatorname{Hom}(\Delta_{(u,v)}^{\perp},V_{j})\subset\Omega^{1}(W^{4},\operatorname{ad}E)$
thus further decomposes under a combined $\mathfrak{su}(2)_{\mathfrak{f}}$
action, generated by $\mathfrak{f}_{i}=\mathfrak{t}_{i}+\mathfrak{s}_{i}$,
into a direct sum of bundles $V_{j}^{-}$, $V_{j}^{0},$ and $V_{j}^{+}$ of
total spin $j-1$, $j$ and $j+1$, respectively. To simplify notation, we will
furthermore write $V_{j}^{w}:=C^{\infty}(W^{4},V_{j})\ w^{\flat}$, where we
let $\mathfrak{su}(2)_{\mathfrak{s}}$ acts trivially. All in all, the relevant
$\operatorname{ad}E$-valued differential forms decompose as

$\displaystyle\begin{split}\Omega^{1}(W^{4},\operatorname{ad}E)&=\bigoplus_{j\in
J}\bigg{(}\ V_{j}^{w}\oplus V_{j}^{-}\oplus V_{j}^{0}\oplus V_{j}^{+}\
\bigg{)}\\\ \Omega^{2}_{v,+}(W^{4},\operatorname{ad}E)&=\bigoplus_{j\in
J}\bigg{(}V_{j}^{-}\oplus V_{j}^{0}\oplus V_{j}^{+}\bigg{)}\end{split}$ (35)

This decomposition is helpful, because the leading order terms
$A^{\mathrm{NP}}=y^{-1}\sin\beta\phi_{\rho}^{\tau}$ and
$B^{\mathrm{NP}}=y^{-1}\cos\beta\phi_{\rho}$ are invariant under the action of
$\mathfrak{su}(2)_{\mathfrak{f}}$ and, as a consequence, the constraints for
$a_{\alpha,k}$ and $b_{\alpha,k}$ decompose into their $V_{j}^{\eta}$-valued
components. However, there is an additional subtlety regarding invariance of
the Nahm pole terms under the action of $\mathfrak{su}(2)_{\mathfrak{f}}$. To
see this, let rotations $\mathfrak{su}(2)_{\mathfrak{s}}$ act on
$\Omega^{2}_{v,+}$ with the “standard” orientation on $(e_{1},e_{2},e_{3})$
and assume this basis is identified with
$(dx^{1},dx^{2},dx^{3})\in(\Delta_{(u,v)}^{\perp})^{\ast}$. Then
$\phi_{\rho}=\sum_{i}\mathfrak{t}_{i}e_{i}$ or equivalently
$\sum_{i}\mathfrak{t}_{i}dx^{i}$ is an element of the trivial representation
$V_{1}^{-}\subset\Omega^{2}_{v,+}(W^{4},\operatorname{ad}E)\simeq\operatorname{Hom}(\Delta_{(u,v)}^{\perp},\operatorname{ad}E)$,
such that $B^{\mathrm{NP}}$ is invariant. Unfortunately, under this
identification
$\phi_{\rho}^{\tau}=\sum_{i}\mathfrak{t}_{\tau(i)}dx^{i}=\mathfrak{t}_{1}dx^{1}+\mathfrak{t}_{2}dx^{3}+\mathfrak{t}_{3}dx^{2}$
is not invariant, since orientation reversal
$\phi_{\rho}\mapsto\phi_{\rho}^{\tau}$ does not preserve $V_{1}^{-}$. To
remedy this, we need rotations to act on the $\Delta_{(u,v)}^{\perp}$-part of
$A$ with the opposite orientation $(dx^{1},dx^{3},dx^{2})$. With that choice
$\phi_{\rho}^{\tau}$ is an element of the trivial representation
$\tilde{V}_{1}^{-}$, with respect to the analogous (but different)
decomposition

$\displaystyle\Omega^{1}(W^{4},\operatorname{ad}E)=\bigoplus_{j\in
J}\left(\tilde{V}_{j}^{w}\oplus\tilde{V}_{j}^{-}\oplus\tilde{V}_{j}^{0}\oplus\tilde{V}_{j}^{+}\right)\
.$ (36)

$A^{\mathrm{NP}}$ becomes invariant under $\mathfrak{su}(2)_{\mathfrak{f}}$
only with that choice. We denote the restriction of a given differential form
$\alpha$ to one of the subspaces $V_{j}^{\eta}$ or $\tilde{V}_{j}^{\eta}$ in
this decomposition by $(\alpha)^{j,\eta}$ or $(\alpha)^{\widetilde{j,\eta}}$,
respectively.

######  Example.

In a local basis $(e_{i})_{i=1,2,3}$ of
$\Omega^{2}_{v,+}(U,\operatorname{ad}E)$ the subbundles
$V_{1}^{\eta}\subset\Omega^{2}_{v,+}(W^{4},\operatorname{ad}E)$ are given by:

$\displaystyle V_{1}^{-}$
$\displaystyle=\operatorname{span}\left\\{\mathfrak{t}_{1}e_{1}+\mathfrak{t}_{2}e_{2}+\mathfrak{t}_{3}e_{3}\right\\}\
,$ (37) $\displaystyle V_{1}^{0}$
$\displaystyle=\operatorname{span}\left\\{\mathfrak{t}_{i}e_{j}-\mathfrak{t}_{j}e_{i}\right\\}_{i\neq
j}\ ,$ (38) $\displaystyle V_{1}^{+}$
$\displaystyle=\operatorname{span}\left\\{\mathfrak{t}_{1}e_{1}-\mathfrak{t}_{2}e_{2},\
\mathfrak{t}_{1}e_{1}-\mathfrak{t}_{3}e_{3},\
\mathfrak{t}_{i}e_{j}+\mathfrak{t}_{j}e_{i}\right\\}_{i\neq j}\ .$ (39)

Substituting $e_{i}$ by $dx^{i}$ provides the corresponding basis under the
isomorphism of $\Omega^{2}_{v,+}(W^{4},\operatorname{ad}E)$ and
$\operatorname{Hom}(\Delta_{(u,v)}^{\perp},\operatorname{ad}E)\subset\Omega^{1}(W^{4},\operatorname{ad}E)$.
This is the correct decomposition for the action of
$\mathfrak{su}(2)_{\mathfrak{f}}$ on $B$ and its expansions $\phi_{\rho}$,
$b_{\alpha,k}$. In contrast, for the decomposition of
$\Omega^{1}(W^{4},\operatorname{ad}E)$ with respect to the reverse
orientation, a corresponding basis of $\tilde{V}_{1}^{\eta}$,
$\eta\in\\{-,0,+\\}$, is obtained from the one above by replacing $e_{i}$ with
$dx^{\tau(i)}$. For example
$\tilde{V}_{1}^{-}=\operatorname{span}\left\\{\mathfrak{t}_{1}dx^{1}+\mathfrak{t}_{3}dx^{2}+\mathfrak{t}_{2}dx^{3}\right\\}$.
This is the correct basis for the action of $\mathfrak{su}(2)_{\mathfrak{f}}$
on $A$ and its expansions $\phi_{\rho}^{\tau}$, $a_{\alpha,k}$.

The upcoming lemma and its proof specify the algebraic constraints the Haydys-
Witten equations put on the coefficient functions in the polyhomogeneous
expansion up to order $y^{2}$. See [He18] for a similar and more extensive
investigation of these constraints in the case of $\pi/2$-Kapustin-Witten
solutions with Nahm pole boundary conditions.

As becomes clear during the proof, we have to exclude a finite set of angles
at which certain spin-$j$ modes of $\imath_{w}a$, i.e. the components
$(a_{\alpha,k})^{j,w}$, become free parameters. Explicitly, these values are
given by:

$\displaystyle\cos 2\beta\in\left\\{-\tfrac{2j+3}{(j+1)^{2}},\
\tfrac{2j-1}{j^{2}},\ -\tfrac{3j^{2}-4j+3}{j(j+1)^{2}},\
-\tfrac{3j^{2}+2j-4}{j^{2}(j+1)}\right\\}_{j\in J}$ (40)

The statement of the following lemma holds whenever $\beta$ is not of that
form.

######  Lemma .

Assume $A=y^{-1}\sin\beta\ \phi_{\rho}^{\tau}+\omega+a$ and
$B=y^{-1}\cos\beta\ \phi_{\rho}+b$ are a solution of the Haydys-Witten
equations on $W^{4}\times\mathbb{R}^{+}_{y}$ with respect to the non-vanishing
vector field $v=\sin\beta w+\cos\beta\partial_{y}$. Then $\omega$ pulls back
under $\phi_{\rho}$ to the Levi-Civita connection on $TW^{4}$ and
$\nabla^{\omega}_{w}\phi_{\rho}=0$. Furthermore, if $\beta$ is generic, then
$a_{1,k}=b_{1,k}=a_{2,k}=b_{2,k}=0$ for all $k\geq 2$. The solution is smooth
up to the boundary (i.e. $a_{1,1}=b_{1,1}=0$), if and only if the Riemannian
curvature satisfies
$(\cos\beta\imath_{w}F_{\omega}+\imath_{w}\star_{4}F_{\omega})\raisebox{-0.86108pt}{$|$}_{V_{1}^{+}/\tilde{V}_{1}^{0}}=0$.
Finally, if $F_{\omega}=0$, the polyhomogeneous expansion up to order
$y^{2+\delta}$, $\delta>0$, reduces to

$\displaystyle a$ $\displaystyle=y\sin\beta\
(C^{1,+})^{\tau}+y^{2}\left(\sin\beta\ C^{2,+}+\cos\beta\
D^{1,-}\right)^{\tau}+\mathcal{O}(y^{2+\delta})$ (41) $\displaystyle b$
$\displaystyle=y\cos\beta\ C^{1,+}+y^{2}\left(\cos\beta\ C^{2,+}-\sin\beta\
D^{1,-}\right)+\mathcal{O}(y^{2+\delta})$ (42)

where $C^{1,+}\in V_{1}^{+}/\tilde{V}_{1}^{0}$, $D^{1,-}\in V_{1}^{-}$, and
$C^{2,+}\in V_{2}^{+}/\tilde{V}_{2}^{0}$ remain unconstrained.

######  Corollary .

Denote by $i_{y}:W^{4}\hookrightarrow W^{4}\times\mathbb{R}^{+}_{y}$ inclusion
at $y$. For generic $\beta$ and if $F_{\omega}=0$, the total field strength
pulls back to

$\displaystyle i_{y}^{\ast}(F_{A})={}$
$\displaystyle\phantom{{}+{}}y^{-2}\left(\sin^{2}\beta\
[\phi_{\rho}^{\tau}\wedge\phi_{\rho}^{\tau}]\right)+y^{0}\left(\sin^{2}\beta\
[\phi_{\rho}^{\tau}\wedge(C^{1,+})^{\tau}]\right)$ (43)
$\displaystyle+y^{1}\left(\sin\beta\
d_{\omega}(C^{1,+})^{\tau}+\sin^{2}\beta[\phi_{\rho}^{\tau}\wedge(C^{2,+})^{\tau}]+\sin\beta\cos\beta\
[\phi_{\rho}^{\tau}\wedge(D^{1,-})^{\tau}]\right)+\mathcal{O}(y^{1+\delta})$
(44)

Meanwhile, the $dx^{2}$\- and $dx^{3}$\- components of $\delta_{A}^{+}J^{+}B$
are specified by

$\displaystyle i_{y}^{\ast}(\delta_{A}^{+}J^{+}B)\propto{}$
$\displaystyle\phantom{{}+{}}y^{-2}\sin\beta\cos\beta\left([(\phi_{\rho}^{\tau})_{3},(\phi_{\rho})_{1}]dx^{2}-[(\phi_{\rho}^{\tau})_{2},(\phi_{\rho})_{1}]dx^{3}\right)$
(45) $\displaystyle+y^{0}\sin\beta\cos\beta\ \begin{aligned}
\Big{(}&\left([(\phi_{\rho}^{\tau})_{3},(C^{1,+})_{1}]+[(C^{1,+})^{\tau}_{3},(\phi_{\rho})_{1}]\right)dx^{2}\\\
-&\left([(\phi_{\rho}^{\tau})_{2},(C^{1,+})_{1}]+[(C^{1,+})^{\tau}_{2},(\phi_{\rho})_{1}]\right)dx^{3}\Big{)}\end{aligned}$
(46) $\displaystyle+y^{1}\begin{aligned} \Big{(}&\left(\cos\beta\
\nabla^{\omega}_{3}(C^{1,+})_{1}+\sin\beta\cos\beta\
[(\phi_{\rho}^{\tau})_{3},(C^{2,+})_{1}]-\sin^{2}\beta\
[(\phi_{\rho}^{\tau})_{3},(D^{1,-})_{1}]\right)dx^{2}\\\ -&\left(\cos\beta\
\nabla^{\omega}_{2}(C^{1,+})_{1}+\sin\beta\cos\beta[(\phi_{\rho}^{\tau})_{2},(C^{2,+})_{1}]-\sin^{2}\beta\
[(\phi_{\rho}^{\tau})_{2},(D^{1,-})_{1}]\right)dx^{3}\Big{)}\end{aligned}$
(47) $\displaystyle+\mathcal{O}(y^{1+\delta})$ (48)

The rest of this section is occupied with the proof of section 4.

###### Proof.

The proof is analogous to the analysis in [He18], which covers the case
$\beta=0$. Hence, assume that $\beta\neq 0$, in which case there is an
isomorphism
$\Omega^{2}_{v,+}(U,\operatorname{ad}E)\simeq\operatorname{Hom}(\Delta_{(u,v)}^{\perp},\operatorname{ad}E)$.
We use this equivalence to first re-express the Haydys-Witten equations (10)
purely in terms of one-forms. This simplifies the evaluation of the formulas
later on.

Fix a gauge in which $A_{y}=0$ and choose as reference connection
$A^{0}=y^{-1}\sin\beta\ \phi_{\rho}^{\tau}+\omega$, where $\omega$ is the
pullback of a connection on $E\to W^{4}$. Note that when we view $B$ as an
element of
$\operatorname{Hom}(\Delta_{(u,v)}^{\perp},\operatorname{ad}E)\subset\Omega^{1}(W^{4}\times\mathbb{R}^{+}_{y},\operatorname{ad}E)$,
it does not have a $dy$-component: in coordinates $(s,x^{i},y)_{i=1,2,3}$ with
$v=\sin\beta\partial_{s}+\cos\beta\partial_{y}$, the two-form
${B=\sum_{i=1}^{3}\phi_{i}e_{i}}$ is identified with the one-form
$\sum_{i=1}^{3}\phi_{i}dx^{i}$. Since the coefficients $\phi_{\rho}^{\tau}$,
$\phi_{\rho}$, $a_{\alpha,k}$ and $b_{\alpha,k}$ are independent of $y$ and as
they don’t have $dy$-components, we can view them as elements of
$\Omega^{1}(W^{4},\operatorname{ad}E)$. Motivated by this, we slightly abuse
notation in the upcoming version of the Haydys-Witten equations (49) - (52)
and treat $A$ and $B$ as one-forms on $W^{4}\times\\{y\\}$. (Specifically, we
implicitly take the pullback and drop any $dy$-components in $F_{A}$ and
$d_{A}B$.) With that understood, the Haydys-Witten equations can be brought
into the form

$\displaystyle\partial_{y\/}A$
$\displaystyle=-\imath_{w}\left(d_{A}B-\sin\beta\ \star_{4}(F_{A}-[B\wedge
B])-\cos\beta\ \star_{4}d_{A}B\right)$ (49) $\displaystyle\partial_{y\/}B$
$\displaystyle=\phantom{\mathrel{-}}\imath_{w}\left(F_{A}+\cos\beta\
\star_{4}(F_{A}-[B\wedge B])-\sin\beta\ \star_{4}d_{A}B\right)$ (50)
$\displaystyle\partial_{y\/}(\imath_{w}A)$
$\displaystyle=\phantom{\mathrel{-}}d_{A}^{\star_{4}}B$ (51) $\displaystyle 0$
$\displaystyle=\phantom{\mathrel{-}}d_{\omega}^{\star_{4}}a+\sin\beta\star_{4}[\phi_{\rho}^{\tau}\wedge\star_{4}y^{-1}a]+\cos\beta\star_{4}[\phi_{\rho}\wedge\star_{4}y^{-1}b]$
(52)

Plugging the polyhomogeneous expansions of $A$ and $B$ into the terms that
appear in these equations yields:

$\displaystyle F_{A}={}$ $\displaystyle y^{-2}\big{(}\sin^{2}\beta\
[\phi_{\rho}^{\tau}\wedge\phi_{\rho}^{\tau}]+\sin\beta\
\phi_{\rho}^{\tau}\wedge dy\big{)}\ +\ y^{-1}\big{(}\sin\beta\
d_{\omega}\phi_{\rho}^{\tau}\big{)}\ +\ y^{0}\big{(}F_{\omega}\big{)}$ (53)
$\displaystyle{}+\sum_{(\alpha,k)\geq 0}y^{\alpha}(\log
y)^{k}\left(d_{\omega}a_{\alpha,k}+\sin\beta[\phi_{\rho}^{\tau}\wedge
a_{\alpha+1,k}]-(\alpha+1)a_{\alpha+1,k}\wedge dy-(k+1)a_{\alpha+1,k+1}\wedge
dy\right)$ (54) $\displaystyle[B\wedge B]={}$
$\displaystyle{}y^{-2}\left(\cos^{2}\beta\
[\phi_{\rho}\wedge\phi_{\rho}]\right)+\sum_{(\alpha,k)\geq 0}y^{\alpha}(\log
y)^{k}\left(\cos\beta[\phi_{\rho}\wedge b_{\alpha+1,k}]\right)$ (55)
$\displaystyle d_{A}B={}$ $\displaystyle y^{-2}\left(\sin\beta\cos\beta\
[\phi_{\rho}^{\tau}\wedge\phi_{\rho}]+\cos\beta\ \phi_{\rho}\wedge
dy\right)+y^{-1}\left(\cos\beta\ d_{\omega}\phi_{\rho}\right)$ (56)
$\displaystyle+\sum_{(\alpha,k)\geq 0}y^{\alpha}(\log
y)^{k}\begin{multlined}\left(d_{\omega}b_{\alpha,k}+\sin\beta\
[\phi_{\rho}^{\tau}\wedge b_{\alpha+1,k}]+\cos\beta\ [\phi_{\rho}\wedge
a_{\alpha+1,k}]\right.\\\ \left.-(\alpha+1)b_{\alpha+1,k}\wedge
dy-(k+1)b_{\alpha+1,k+1}\wedge
dy\right)\end{multlined}\left(d_{\omega}b_{\alpha,k}+\sin\beta\
[\phi_{\rho}^{\tau}\wedge b_{\alpha+1,k}]+\cos\beta\ [\phi_{\rho}\wedge
a_{\alpha+1,k}]\right.\\\ \left.-(\alpha+1)b_{\alpha+1,k}\wedge
dy-(k+1)b_{\alpha+1,k+1}\wedge dy\right)$ (59) $\displaystyle
d_{A}^{\star_{4}}B={}$ $\displaystyle{}y^{-1}\left(\cos\beta\
d_{\omega}^{\star_{4}}\phi_{\rho}\right)+\sum_{(\alpha,k)\geq
0}y^{\alpha}(\log
y)^{k}\begin{multlined}\left(d_{\omega}^{\star_{4}}b_{\alpha,k}+\sin\beta\
\star_{4}[\phi_{\rho}^{\tau}\wedge\star_{4}b_{\alpha+1,k}]\right.\\\
\left.-\cos\beta\
\star_{4}[\phi_{\rho}\wedge\star_{4}a_{\alpha+1,k}]\right)\end{multlined}\left(d_{\omega}^{\star_{4}}b_{\alpha,k}+\sin\beta\
\star_{4}[\phi_{\rho}^{\tau}\wedge\star_{4}b_{\alpha+1,k}]\right.\\\
\left.-\cos\beta\ \star_{4}[\phi_{\rho}\wedge\star_{4}a_{\alpha+1,k}]\right)$
(62)

Here we write $(\alpha,k)\geq 0$ for pairs of exponents with
$\operatorname{Re}\alpha\geq 0$. Using $(\partial_{y}A)_{i}=-(F_{A})_{iy}$ and
$(\partial_{y}B)_{i}=-(d_{A}B)_{iy}$, we can now read off the contributions to
the Haydys-Witten equations from each of these expressions, order by order in
$y^{\alpha}(\log y)^{k}$.

$\mathcal{O}(y^{-2})$

At order $\mathcal{O}(y^{-2})$ only equations (49) and (50) are non-trivial.
Using that
$\imath_{w}(\star_{4}[\phi_{\rho}^{\tau}\wedge\phi_{\rho}])=\tfrac{1}{2}\imath_{w}(\star_{4}[\phi_{\rho}^{\tau}\wedge\phi_{\rho}^{\tau}]+\star_{4}[\phi_{\rho}\wedge\phi_{\rho}])$,
the Haydys-Witten equations then state:

$\displaystyle\phi_{\rho}^{\tau}$
$\displaystyle=-\imath_{w}\left(\star_{4}[\phi_{\rho}^{\tau}\wedge\phi_{\rho}^{\tau}]\right)$
(63) $\displaystyle\phi_{\rho}$
$\displaystyle=\phantom{\mathrel{-}}\imath_{w}\left(\star_{4}[\phi_{\rho}\wedge\phi_{\rho}]\right)$
(64)

This is satisfied, since under the isomorphism
$\Omega^{2}_{v,+}(U,\operatorname{ad}E)=\operatorname{Hom}(\Delta_{(u,v)}^{\perp},\operatorname{ad}E)$
the expression $\imath_{w}(\star_{4}[\phi_{\rho}\wedge\phi_{\rho}])$ is
identified with $\sigma(\phi_{\rho},\phi_{\rho})$. By construction of the Nahm
pole boundary condition $\phi_{\rho}-\sigma(\phi_{\rho},\phi_{\rho})=0$, while
the same is true for $\phi_{\rho}^{\tau}$ with the opposite sign.

$\mathcal{O}(y^{-1})$

The first additional constraint arises at $\mathcal{O}(y^{-1})$, where we find

$\displaystyle 0$
$\displaystyle=\phantom{\mathrel{-}}\imath_{w}\left(\cos\beta\
d_{\omega}\phi_{\rho}-\sin^{2}\beta\
\star_{4}d_{\omega}\phi_{\rho}^{\tau}-\cos^{2}\beta\
\star_{4}d_{\omega}\phi_{\rho}\right)$ (65) $\displaystyle 0$
$\displaystyle=-\imath_{w}\left(\sin\beta\
d_{\omega}\phi_{\rho}^{\tau}+\sin\beta\cos\beta\
\star_{4}d_{\omega}\phi_{\rho}^{\tau}-\sin\beta\cos\beta\
\star_{4}d_{\omega}\phi_{\rho}\right)$ (66) $\displaystyle 0$
$\displaystyle=\cos\beta\ d_{\omega}^{\star_{4}}\phi_{\rho}$ (67)

These equations are satisfied if the tensor222Here, as always,
$\nabla^{\omega}$ is the product connection on
${\Omega^{1}(W^{4}\times\mathbb{R}^{+}_{y},\operatorname{ad}E)}$ that is
induced by the Levi-Civita connection and the gauge connection $\omega$.
$\nabla^{\omega}\phi_{\rho}$ vanishes. This means that $\phi_{\rho}$
intertwines the Levi-Civita and gauge connection on $TW^{4}$: On the one hand,
$\omega$ must pull back under
$\phi_{\rho}:\Delta_{(u,v)}^{\perp}\to\operatorname{ad}E$ to the Levi-Civita
connection on $\Delta_{(u,v)}^{\perp}$. This also implies that the restriction
of $\omega$ to $\Delta_{(u,v)}^{\perp}\subset TW^{4}$ is valued in
$\rho(\mathfrak{su}(2))=V_{1}\subset\operatorname{ad}E$. On the other hand,
the $w^{\flat}$-component of $\omega$ must be such that
$\nabla^{\omega}_{w}\phi_{\rho}=0$, i.e. $\phi_{\rho}$ extends along $w$ by
parallel transport. Since the Levi-Civita connection acts trivially on
$\operatorname{ad}E$, it preserves $V_{1}\subset\operatorname{ad}E$ and we
find that also $\imath_{w}\omega\in V_{1}$. We note that
$\imath_{w}\star_{4}F_{\omega}$ and $\imath_{w}F_{\omega}$ are then elements
of $V_{1}^{-}\oplus V_{1}^{0}\oplus V_{1}^{+}$ and are determined by the
Riemannian curvature of $W^{4}$. Furthermore, the exterior covariant
derivative satisfies the following mapping properties (cf. [Hen12, p.5]):

$\displaystyle\begin{split}\imath_{w}\star_{4}d_{\omega}&:V_{j}^{w}\to 0\
,\quad V_{j}^{-}\to V_{j}^{0}\ ,\quad V_{j}^{0}\to V_{j}^{-}\oplus
V_{j}^{0}\oplus V_{j}^{+}\ ,\quad V_{j}^{+}\to V_{j}^{0}\oplus V_{j}^{+}\\\
\imath_{w}d_{\omega}&:V_{j}^{w}\to V_{j}^{-}\oplus V_{j}^{0}\oplus V_{j}^{+}\
,\quad V_{j}^{\eta}\to V_{j}^{\eta},\ \eta\in\\{-,0,+\\}\end{split}$ (68)

With a view at the expansions of $F_{A}$ and $d_{A}B$, it becomes clear that
these identities will be helpful in the upcoming analysis, starting at order
$\mathcal{O}(y^{0}(\log y)^{k})$.

$\mathcal{O}(y^{0}(\log y)^{k})$

Examining the higher order expansions of the right hand side of (49) - (52),
one finds that the equations will from now on involve terms of the form
$\imath_{w}\star_{4}[\phi_{\rho}\wedge\cdot]$ and its analogue with respect to
$\phi_{\rho}^{\tau}$. These are global versions of the spin-spin operator
$\mathfrak{J}:=\mathfrak{s}\cdot\mathfrak{t}$, as they act pointwise by
$\sum_{a=1}^{3}\mathfrak{s}_{a}\otimes\mathfrak{t}_{a}$. Indeed, taking into
account that rotations act on $a_{\alpha,k}$ and $b_{\alpha,k}$ with opposite
orientation one finds

$\displaystyle\mathfrak{J}a_{\alpha,k}$
$\displaystyle=-\imath_{w}\star_{4}[\phi_{\rho}^{\tau}\wedge a_{\alpha,k}]$
$\displaystyle\mathfrak{J}b_{\alpha,k}$
$\displaystyle=\imath_{w}\star_{4}[\phi_{\rho}\wedge b_{\alpha,k}]$ (69)

The vector bundles $V_{j}^{\eta}$, $\eta\in\\{-,0,+\\}$, are eigenspaces of
the spin-spin operator with eigenvalues $j+1$, $1$, and $-j$. Meanwhile,
$\mathfrak{J}$ sends $V_{j}^{w}$ to zero. We will also use that
$\imath_{w}[\phi_{\rho}\wedge\cdot]$ maps $V_{j}^{w}\to V_{j}^{0}$ and
$V_{j}^{\eta}\to 0$, $\eta\in\\{-,0,+\\}$.

Equipped with this, we can now proceed to the constraints that arise for the
tower of $\log y$ terms at order $y^{0}$. Hence, consider equations (49) -
(52) at $\mathcal{O}\big{(}(\log y)^{k}\big{)}$, where $k$ is the largest
integer such that $(1,k)\in\Delta_{0}$. The existence of such a maximal value
is part of the defining properties of a polyhomogeneous expansion. For the
moment assume that $k\geq 2$, we will treat the cases $k=1$ and $k=0$ in the
end.

We now introduce the following linear combinations of $V_{j}^{-}\oplus
V_{j}^{0}\oplus V_{j}^{+}$-parts of $a_{\alpha,k}$ and $b_{\alpha,k}$:

$\displaystyle c_{\alpha,k}$ $\displaystyle:=\sin\beta\
a_{\alpha,k}^{\tau}+\cos\beta\ b_{\alpha,k}\ ,$ (70) $\displaystyle
d_{\alpha,k}$ $\displaystyle:=\cos\beta\ a_{\alpha,k}^{\tau}-\sin\beta\
b_{\alpha,k}\ .$ (71)

Note, in particular, that $c_{\alpha,k}$ and $d_{\alpha,k}$ do not include the
component $(a_{1,k})^{j,w}$. Appropriate linear combinations of (51) and (52)
then state

$\displaystyle\sin\beta\ \imath_{w}a_{1,k}$ $\displaystyle=\cos\beta\
\star_{4}[\phi_{\rho}\wedge\star_{4}c_{1,k}]+\sin\beta\
\star_{4}[\phi_{\rho}\wedge\star_{4}d_{1,k}]$ (72) $\displaystyle\cos\beta\
\imath_{w}a_{1,k}$ $\displaystyle=\sin\beta\
\star_{4}[\phi_{\rho}^{\tau}\wedge\star_{4}c_{1,k}]-\cos\beta\
\star_{4}[\phi_{\rho}^{\tau}\wedge\star_{4}d_{1,k}]$ (73)

These equations can be brought in a slightly more helpful form. Note that the
operator $[\phi_{\rho}\wedge\star_{4}[\phi_{\rho}\wedge\star_{4}\,\cdot\ ]]$
annihilates $V_{j}^{\pm}$ and acts on $V_{j}^{0}$ as
$\sum_{i}[\mathfrak{t}_{i},[\mathfrak{t}_{i},\cdot]]$, which is just (the
negative of) the quadratic Casimir on $V_{j}$. Applying
$[\phi_{\rho}\wedge\cdot]$ to the equations thus simplifies the operation on
the right hand side to a simple multiplication by $-j(j+1)$. In complete
analogy
$[\phi_{\rho}^{\tau}\wedge\star_{4}[\phi_{\rho}^{\tau}\wedge\star_{4}\,\cdot\
]]$ annihilates $\tilde{V}_{j}^{\pm}$ and acts on $\tilde{V}_{j}^{0}$ by
multiplication with $-j(j+1)$. This results in (76) and (77) below. Including
appropriate linear combinations of (49) and (50), the coefficient functions in
the polyhomogeneous expansion have to be compatible with the following four
conditions.

$\displaystyle c_{1,k}+\mathfrak{J}c_{1,k}$
$\displaystyle=\phantom{\mathrel{-}}\sin\beta\cos\beta\
\left(\imath_{w}[\phi_{\rho}^{\tau}\wedge
a_{1,k}]-\imath_{w}[\phi_{\rho}\wedge a_{1,k}]\right)$ (74) $\displaystyle
d_{1,k}-\mathfrak{J}d_{1,k}$ $\displaystyle=-\cos^{2}\beta\
\imath_{w}[\phi_{\rho}^{\tau}\wedge a_{1,k}]-\sin^{2}\beta\
\imath_{w}[\phi_{\rho}\wedge a_{1,k}]$ (75) $\displaystyle
j(j+1)\left(\cos\beta\ c_{1,k}+\sin\beta\ d_{1,k}\right)^{j,0}$
$\displaystyle=\phantom{\mathrel{-}}\sin\beta\ \imath_{w}[\phi_{\rho}\wedge
a_{1,k}]$ (76) $\displaystyle j(j+1)\left(\sin\beta\ c_{1,k}-\cos\beta\
d_{1,k}\right)^{\widetilde{j,0}}$
$\displaystyle=\phantom{\mathrel{-}}\cos\beta\
\imath_{w}[\phi_{\rho}^{\tau}\wedge a_{1,k}]$ (77)

Observe that the terms on the left only depend on $c_{1,k}$ and $d_{1,k}$ and
thus only on the $V_{j}^{-}\oplus V_{j}^{0}\oplus V_{j}^{+}$ components of
$a_{1,k}$ and $b_{1,k}$. Meanwhile, the right hand side only depends on the
remaining $V_{j}^{w}$ component $(a_{1,k})^{j,w}$. Moreover, all terms on the
right are necessarily located in $V_{j}^{0}+\tilde{V}_{j}^{0}$ (fiberwise sum
of subspaces). To solve the equations we distinguish between the following six
cases:

  1. (I)

$V_{j}^{-}/\tilde{V}_{j}^{0}$

  2. (II)

$V_{j}^{+}/\tilde{V}_{j}^{0}$

  3. (III)

$V_{j}^{-}\cap\tilde{V}_{j}^{0}$

  4. (IV)

$V_{j}^{+}\cap\tilde{V}_{j}^{0}$

  5. (V)

$V_{j}^{0}/\tilde{V}_{j}^{0}$

  6. (VI)

$V_{j}^{0}\cap\tilde{V}_{j}^{0}$

In fact, a combination of the metric and Killing form provides an inner
product on each $V_{j}^{\eta}$, such that we can (and do) identify the
quotient spaces with the orthogonal complement of $\tilde{V}_{j}^{0}$ inside
of $V_{j}^{\eta}$ (but reflecting this in the notation only adds unnecessary
complexity). In the upcoming paragraphs we will write
$\alpha\raisebox{-0.86108pt}{$|$}_{\mathrm{(I)}}$ to denote restriction to the
subspace specified in case (I), et cetera. Let us note at this point that
$j=1$ is slightly special, since $V_{1}^{-}\cap\tilde{V}_{1}^{0}=\\{0\\}$.

When we restrict (74) - (77) to either of the subspaces in (I) or (II), all
terms on the right hand side vanish. In particular, (74) and (75) reduce to
simple eigenvalue equations. Since $\mathfrak{J}$ acts on $V_{j}^{-}$ with
eigenvalue $j+1$ (which is strictly larger than $1$), we immediately find
$c_{1,k}\raisebox{-0.86108pt}{$|$}_{\mathrm{(I)}}=0=d_{1,k}\raisebox{-0.86108pt}{$|$}_{\mathrm{(I)}}$
for any $j$. Similarly, since $\mathfrak{J}$ acts on $V_{j}^{+}$ with
eigenvalue $-j$, we get $c_{1,k}\raisebox{-0.86108pt}{$|$}_{\mathrm{(II)}}=0$,
except perhaps when $j=1$, while
$d_{1,k}\raisebox{-0.86108pt}{$|$}_{\mathrm{(II)}}=0$ in any case. For the
$j=1$ component of $c_{1,k}\raisebox{-0.86108pt}{$|$}_{\mathrm{(II)}}$ we can
rely on the analogue of (74) at the subleading order $\mathcal{O}\big{(}(\log
y)^{k-1}\big{)}$, which states

$\displaystyle kc_{1,k}+c_{1,k-1}+\mathfrak{J}c_{1,k-1}$
$\displaystyle=\sin\beta\cos\beta\ \left(\imath_{w}[\phi_{\rho}^{\tau}\wedge
a_{1,k-1}]-\imath_{w}[\phi_{\rho}\wedge a_{1,k-1}]\right)$ (78)

Upon restriction to $V_{1}^{+}/\tilde{V}_{1}^{0}$ this reduces to
$kc_{1,k}\raisebox{-0.86108pt}{$|$}_{\mathrm{(II)}}=0$.

Restriction to the subspaces in (III) and (IV) works out slightly differently,
because there the restriction of ${\imath_{w}[\phi_{\rho}^{\tau}\wedge
a_{1,k}]}$ doesn’t vanish (though we still have vanishing of
$\imath_{w}[\phi_{\rho}\wedge a_{1,k}]$). Concentrate on case (III) first,
where $J$ has eigenvalue $j+1$. We find that (74) and (75) determine
${c_{1,k}\raisebox{-0.86108pt}{$|$}_{\mathrm{(III)}}=\frac{\sin\beta\cos\beta}{j+2}(\imath_{w}[\phi_{\rho}^{\tau}\wedge
a])^{j,-}}$ and
${d_{1,k}\raisebox{-0.86108pt}{$|$}_{\mathrm{(III)}}=\frac{\cos^{2}\beta}{j}(\imath_{w}[\phi_{\rho}^{\tau}\wedge
a])^{j,-}}$. However, when plugging this into (77), we get the following
consistency condition:

$\displaystyle
j(j+1)\left(\frac{\sin^{2}\beta\cos\beta}{j+2}-\frac{\cos^{3}\beta}{j}\right)(\imath_{w}[\phi_{\rho}^{\tau}\wedge
a_{1,k}])^{j,-}=-\cos\beta(\imath_{w}[\phi_{\rho}^{\tau}\wedge
a_{1,k}])^{j,-}$ (79)

This can only be satisfied if $\cos 2\beta=-\frac{2j+3}{(j+1)^{2}}$ or if
$(\imath_{w}[\phi_{\rho}^{\tau}\wedge a_{1,k}])^{j,-}=0$. Interestingly, for
any $j\geq 2$ there exists a single value of $\beta\in(0,\pi/2)$ for which the
first equation is satisfied. Therefore, whenever $\beta$ is related to one of
the spins $j$ in this way, both $(c_{1,k})^{j,-}$ and $(d_{1,k})^{j,-}$ are
given as stated above by a multiple of $(\imath_{w}[\phi_{\rho}^{\tau}\wedge
a_{1,k}])^{j,-}$. For generic values of $\beta$, however, we find that
$(\imath_{w}[\phi_{\rho}^{\tau}\wedge a_{1,k}])^{j,-}$ vanishes, and thus also
$c_{1,k}\raisebox{-0.86108pt}{$|$}_{\mathrm{(III)}}=d_{1,k}\raisebox{-0.86108pt}{$|$}_{\mathrm{(III)}}=0$.

For case (IV), recall that $J$ acts with eigenvalue $-j$ on $V_{j}^{+}$. If
$j=1$, equation (74) directly yields ${\imath_{w}[\phi_{\rho}^{\tau}\wedge
a_{1,k}]\raisebox{-0.86108pt}{$|$}_{\mathrm{(IV)}}=0}$, such that
$d_{1,k}\raisebox{-0.86108pt}{$|$}_{\mathrm{(IV)}}=0$ by (75) and
$c_{1,k}\raisebox{-0.86108pt}{$|$}_{\mathrm{(IV)}}=0$ by (77). Otherwise, for
$j\geq 2$, equations (74) and (75) specify
$c_{1,k}\raisebox{-0.86108pt}{$|$}_{\mathrm{(IV)}}=\frac{\sin\beta\cos\beta}{1-j}\imath_{w}[\phi_{\rho}^{\tau}\wedge
a_{1,k}]\raisebox{-0.86108pt}{$|$}_{\mathrm{(IV)}}$ and
$d_{1,k}\raisebox{-0.86108pt}{$|$}_{\mathrm{(IV)}}=-\frac{\cos^{2}\beta}{1+j}\imath_{w}[\phi_{\rho}^{\tau}\wedge
a_{1,k}]\raisebox{-0.86108pt}{$|$}_{\mathrm{(IV)}}$. Plugging these
expressions into (77) now leads to the condition $\cos
2\beta=\frac{2j-1}{j^{2}}$ if $j\geq 2$. In this condition, we can also allow
$j=1$ as input, since the corresponding solution is $\beta=0$, which we
exclude anyway. If $\beta$ coincides with one of these special values,
$(c_{1,k})^{j,+}$ and $(d_{1,k})^{j,+}$ are determined as above in terms of
$(\imath_{w}[\phi_{\rho}^{\tau}\wedge a_{1,k}])^{j,+}$. For generic values of
$\beta$, we have $(\imath_{w}[\phi_{\rho}^{\tau}\wedge
a_{1,k}])^{j,+}=c_{1,k}\raisebox{-0.86108pt}{$|$}_{\mathrm{(IV)}}=d_{1,k}\raisebox{-0.86108pt}{$|$}_{\mathrm{(IV)}}=0$

Case (V) is comparatively simple. Observe that
$\imath_{w}[\phi_{\rho}^{\tau}\wedge
a_{1,k}]\raisebox{-0.86108pt}{$|$}_{\mathrm{(V)}}=0$ and that $J$ acts with
eigenvalue $1$. This means that (75) immediately provides
$\imath_{w}[\phi_{\rho}\wedge
a_{1,k}]\raisebox{-0.86108pt}{$|$}_{\mathrm{(V)}}=0$, such that (74) states
$c_{1,k}\raisebox{-0.86108pt}{$|$}_{\mathrm{(V)}}=0$ and as a consequence (76)
yields $d_{1,k}\raisebox{-0.86108pt}{$|$}_{\mathrm{(V)}}=0$.

Finally, we arrive at case (VI), where neither
$\imath_{w}[\phi_{\rho}^{\tau}\wedge
a_{1,k}]\raisebox{-0.86108pt}{$|$}_{\mathrm{(VI)}}$ nor
$\imath_{w}[\phi_{\rho}\wedge
a_{1,k}]\raisebox{-0.86108pt}{$|$}_{\mathrm{(VI)}}$ vanish. Let us first
consider generic values of $\beta$, where we already know that
$(\imath_{w}[\phi_{\rho}^{\tau}\wedge a_{1,k}])^{j,\pm}=0$. Using
${(\imath_{w}[\phi_{\rho}\wedge
a_{1,k}])^{\tau}}={\imath_{w}[\phi_{\rho}^{\tau}\wedge a_{1,k}]}$ and that
acting with $\tau$ twice is the identity map, we can then deduce that
$\imath_{w}[\phi_{\rho}\wedge a_{1,k}]$ decomposes into a sum of
$\tau$-eigenvectors inside of $V_{j}^{0}\cap\tilde{V}_{j}^{0}$ with
eigenvalues $\pm 1$. Plugging this decomposition into (74) - (77) then shows
that there can only be non-trivial solutions if $j=1$ and $\beta=\pi/2$. Since
we excluded $\beta=\pi/2$ already in the definition of the Nahm pole boundary
condition, we find that $(a_{1,k})^{j,w}=(c_{1,k})^{j,0}=(d_{1,k})^{j,0}=0$
for all $j$.

Still in case $(VI)$, but when $\beta$ is one of the special values $\cos
2\beta=-\frac{2j+3}{(j+1)^{2}}$ or $\frac{2j-1}{j^{2}}$, it is no longer
assured that $(\imath_{w}[\phi_{\rho}^{\tau}\wedge a_{1,k}])^{j,\pm}$
vanishes. As a result we can’t decompose it into $\pm$-eigenparts of $\tau$
within $V_{j}^{0}\cap\tilde{V}_{j}^{0}$. In this situation equations (74) -
(77) instead provide $(c_{1,k})^{j,0}=-\tan\beta\ \imath_{w}[\phi_{\rho}\wedge
a_{1,k}]$ and $(d_{1,k})^{j,0}=\imath_{w}[\phi_{\rho}\wedge a_{1,k}]$.

By way of an intermediate conclusion, it can thus be stated that for generic
$\beta$ and $k\geq 2$, the functions $a_{1,k}$ and $b_{1,k}$ vanish.
Meanwhile, if $\beta$ is one of the (finitely many) special angles determined
by $\cos 2\beta=-\frac{2j+3}{(j+1)^{2}}$ or $\frac{2j-1}{j^{2}}$, their spin
$j$ components are not necessarily zero and are determined – via
$a_{1,k}^{\tau}=\sin\beta\ c_{1,k}+\cos\beta\ d_{1,k}$ and $b_{1,k}=\cos\beta\
c_{1,k}-\sin\beta d_{1,k}$ – by the following expressions

$\displaystyle\begin{split}c_{1,k}&=\frac{\sin\beta\cos\beta}{j+2}\left(\imath_{w}[\phi_{\rho}^{\tau}\wedge
a_{1,k}]\right)^{j,-}-\tan\beta\ \left(\imath_{w}[\phi_{\rho}\wedge
a_{1,k}]\right)^{j,0}-\frac{\sin\beta\cos\beta}{j-1}\left(\imath_{w}[\phi_{\rho}^{\tau}\wedge
a_{1,k}]\right)^{j,+}\\\
d_{1,k}&=\frac{\cos^{2}\beta}{j}\left(\imath_{w}[\phi_{\rho}^{\tau}\wedge
a_{1,k}]\right)^{j,-}+\left(\imath_{w}[\phi_{\rho}\wedge
a_{1,k}]\right)^{j,0}-\frac{\cos^{2}\beta}{j+1}\left(\imath_{w}[\phi_{\rho}^{\tau}\wedge
a_{1,k}]\right)^{j,+}\end{split}$ (80)

The lower order terms $a_{1,k^{\prime}}$ and $b_{1,k^{\prime}}$, $0\leq
k^{\prime}<k$ are then determined by induction. Let us stress that the
possible values of $j\geq 2$ are determined by the (finitely many) spins $j\in
J$ that appear in the decomposition of $\operatorname{ad}E$ under the action
of $\mathfrak{su}(2)_{\mathfrak{s}}$.

As an aside, note that all components are determined by the free parameters
$\\{(a_{1,k})^{j,w},\ (a_{1,k-1})^{j,w},\ldots\\}$, which describe the
$y$-dependence of $\imath_{w}A$, i.e. the $w^{\flat}$-component of the gauge
connection. While our current analysis cannot determine the maximal power of
$(\log y)^{k}$, this is in general controlled by the geometry and topology of
$E\to M^{5}$. For example, if the boundary is of the form
$W^{4}=\mathbb{R}\times X^{3}$ and $w$ is the vector field parallel to the
real line, then upon dimensional reduction $\imath_{w}A$ would be identified
with the $dy$-component of a one-form $\phi$ that satisfies the Kapustin-
Witten equations. In that situation a well-known vanishing theorem states that
$\imath_{w}A$ vanishes if $A+i\phi$ approaches an irreducible flat connection
at $y\to\infty$. In that way, the regularity of twisted Nahm pole solutions is
further controlled by global data, away from the boundary.

The arguments above break down at $k=1$, where in case (II) it is no longer
possible to deduce that the $V_{1}^{+}/\tilde{V}_{1}^{0}$ part of $c_{1,1}$
vanishes. This is because for this component we needed to rely on the
subleading order of (49) and (50). For
$c_{1,1}\raisebox{-0.86108pt}{$|$}_{\mathrm{(II)}}$ this means that we have to
look at the equations at order $\mathcal{O}(1)$, which receive an additional
contribution from $F_{\omega}$:

$\displaystyle c_{1,1}+c_{1,0}+\mathfrak{J}c_{1,0}$
$\displaystyle=\phantom{\mathrel{-}}\cos\beta\
\imath_{w}F_{\omega}+\imath_{w}\star_{4}F_{\omega}+\sin\beta\cos\beta\left(\imath_{w}[\phi_{\rho}^{\tau}\wedge
a_{1,0}]-\imath_{w}[\phi_{\rho}\wedge a_{1,0}]\right)$ (81) $\displaystyle
d_{1,0}-\mathfrak{J}d_{1,0}$ $\displaystyle=-\sin\beta\
\imath_{w}F_{\omega}^{\tau}-\cos^{2}\beta\ \imath_{w}[\phi_{\rho}^{\tau}\wedge
a_{1,0}]-\sin^{2}\beta\ \imath_{w}[\phi_{\rho}\wedge a_{1,0}]$ (82)

Since the terms that contain $F_{\omega}$ are located in $V_{1}^{-}\oplus
V_{1}^{0}\oplus V_{1}^{+}$, only these parts of $c_{1,1}$, $c_{1,0}$ and
$d_{1,0}$ are affected by curvature contributions. Note that the special
values of $\beta$ are related to spins $j\geq 2$, so there are no additional
contributions to the $V_{1}^{-}\oplus V_{1}^{0}\oplus V_{1}^{+}$ part of the
equations. By restriction of the $\mathcal{O}(1)$ equations to $V_{1}^{-}$,
$V_{1}^{0}$, and $V_{1}^{+}$ (remembering that
$V_{1}^{-}\cap\tilde{V}_{1}^{0}=\\{0\\}$) we thus obtain

$\displaystyle\begin{split}c_{1,1}=&\ \left(\cos\beta\
\imath_{w}F_{\omega}+\imath_{w}\star_{4}F_{\omega}\right)\raisebox{-0.86108pt}{$|$}_{V_{1}^{+}/\tilde{V}_{1}^{0}}\\\\[10.00002pt]
c_{1,0}=&\phantom{{}+{}}\tfrac{1}{3}\left(\cos\beta\
\imath_{w}F_{\omega}+\imath_{w}\star_{4}F_{\omega}\right)^{1,-}\\\
&+\tfrac{1}{2}\left(\cos\beta\
\imath_{w}F_{\omega}+\imath_{w}\star_{4}F_{\omega}+\sin\beta\cos\beta\left(\imath_{w}[\phi_{\rho}^{\tau}\wedge
a_{1,0}]-\imath_{w}[\phi_{\rho}\wedge a_{1,0}]\right)\right)^{1,0}\\\
&-\tfrac{1}{2}\left(\cos\beta\ \imath_{w}F_{\omega}^{\tau}+\sin\beta\cos\beta\
\imath_{w}[\phi_{\rho}^{\tau}\wedge
a_{1,0}]\right)\raisebox{-0.86108pt}{$|$}_{V_{1}^{+}\cap\tilde{V}_{1}^{0}}+C^{1,+}\\\\[10.00002pt]
d_{1,0}=&\ \left(\sin\beta\
\imath_{w}F_{\omega}^{\tau}\right)^{1,-}+\tfrac{1}{2}\left(\imath_{w}[\phi_{\rho}\wedge
a_{1,0}]-\cot\beta\ c_{1,0}\right)^{1,0}-\tfrac{1}{2}\left(\sin\beta\
\imath_{w}F_{\omega}^{\tau}+\cos^{2}\beta\ \imath_{w}[\phi_{\rho}^{\tau}\wedge
a_{1,0}]\right)^{1,+}\end{split}$ (83)

Here, $C^{1,+}\in V_{1}^{+}/\tilde{V}_{1}^{0}$ is some undetermined
“integration constant” that cannot be fixed by the current analysis. In
contrast, $\imath_{w}[\phi_{\rho}\wedge a_{1,0}]$ and
$\imath_{w}[\phi_{\rho}^{\tau}\wedge a_{1,0}]$ are further restricted by the
$\mathcal{O}(1)$ versions of (76) and (77) (which remain unchanged) and
consequently depend explicitly on $F_{\omega}$. Indeed, if the $F_{\omega}$
contributions to the $\mathcal{O}(1)$ equations vanish, our earlier arguments
show that, apart from $c_{1,0}=C^{1,+}$, all coefficient functions vanish.

Finally, if $\beta$ is one of the special angles where $\cos
2\beta=-\frac{2j+3}{(j+1)^{2}}$ or $\frac{2j-1}{j^{2}}$, then the results for
$c_{1,1}$, $c_{1,0}$ and $d_{1,0}$ differ only by additionally including the
terms induced by (80), which only appear for spins $j\neq 1$.

The main take away of this discussion is that, for generic $\beta$ and in
absence of curvature contributions, the functions $a_{1,k}$ and $b_{1,k}$
vanish for all $k\geq 1$, while they are proportional to $C^{1,+}\in
V_{1}^{+}/\tilde{V}_{1}^{0}$ for $k=0$.

$\mathcal{O}(y^{1}(\log y)^{k})$
For the constraints at order $\mathcal{O}(y(\log y)^{k})$ the analysis is very
similar. The main difference is that the equations now incorporate terms of
the form $d_{\omega}c_{1,k}$. As before, let $k$ be the largest integer for
which $(2,k)\in\Delta_{0}$. The Haydys-Witten equations (49) - (52) now become

$\displaystyle\left(2c_{2,k}+\mathfrak{J}c_{2,k}\right)-\imath_{w}d_{\omega}d_{1,k}^{\tau}-\phantom{(}\imath_{w}\star_{4}d_{\omega}a_{1,k}\phantom{){}^{\tau}}\
$ $\displaystyle=\phantom{\mathrel{-}}\sin\beta\cos\beta\
\left(\imath_{w}[\phi_{\rho}^{\tau}\wedge
a_{2,k}]-\imath_{w}[\phi_{\rho}\wedge a_{2,k}]\right)$ (84)
$\displaystyle\left(2d_{2,k}-\mathfrak{J}d_{2,k}\right)+\imath_{w}d_{\omega}c_{1,k}^{\tau}-\left(\imath_{w}\star_{4}d_{\omega}b_{1,k}\right){}^{\tau}$
$\displaystyle=-\cos^{2}\beta\ \imath_{w}[\phi_{\rho}^{\tau}\wedge
a_{2,k}]-\sin^{2}\beta\ \imath_{w}[\phi_{\rho}\wedge a_{2,k}]$ (85)
$\displaystyle j(j+1)\left(\cos\beta\ c_{2,k}+\sin\beta\ d_{2,k}\right)^{j,0}$
$\displaystyle=\phantom{\mathrel{-}}\sin\beta\ \imath_{w}[\phi_{\rho}\wedge
a_{2,k}]$ (86) $\displaystyle j(j+1)\left(\sin\beta\ c_{2,k}-\cos\beta\
d_{2,k}\right)^{\widetilde{j,0}}$
$\displaystyle=\phantom{\mathrel{-}}\cos\beta\
\imath_{w}[\phi_{\rho}^{\tau}\wedge a_{2,k}]$ (87)

We have again arranged this such that the right hand side is contained in
$V_{j}^{0}+\tilde{V}_{j}^{0}$. Moreover, we know that for generic values of
$\beta$ and any $k\geq 2$, the functions $c_{1,k}$ and $d_{1,k}$ vanish
completely, such that the derivatives on the left hand side disappear. In that
situation essentially the same arguments as before show that
$(a_{2,k})^{j,w}=c_{2,k}=d_{2,k}=0$, as long as we additionally exclude the
special angles determined by $\cos 2\beta=-\frac{3j^{2}-4j+3}{j(j+1)^{2}}$ or
$-\frac{3j^{2}+2j-4}{j^{2}(j+1)}$.

In carrying out these arguments, we need to rely on the subleading equations
at order $\mathcal{O}(y(\log y)^{k-1})$ again. Here, the subleading equations
provide conditions for the $V_{2}^{+}/\tilde{V}_{2}^{0}$ part of $c_{2,k}$ and
the $V_{1}^{-}/\tilde{V}_{1}^{0}$ ($=V_{1}^{-}$) part of $d_{2,k}$, which are
projected out from the equations at order $\mathcal{O}(y(\log y)^{k})$.
Specifically, the equation for the $c$-terms at order $\mathcal{O}(y(\log
y)^{k-1})$ is

$\displaystyle\begin{multlined}\left(kc_{2,k}+2c_{2,k-1}+\mathfrak{J}c_{2,k-1}\right)-\imath_{w}d_{\omega}d_{1,k-1}^{\tau}-\imath_{w}\star_{4}d_{\omega}a_{1,k-1}\\\
=\sin\beta\cos\beta\left(\imath_{w}[\phi_{\rho}^{\tau}\wedge
a_{2,k-1}]-\imath_{w}[\phi_{\rho}\wedge
a_{2,k-1}]\right)\end{multlined}\left(kc_{2,k}+2c_{2,k-1}+\mathfrak{J}c_{2,k-1}\right)-\imath_{w}d_{\omega}d_{1,k-1}^{\tau}-\imath_{w}\star_{4}d_{\omega}a_{1,k-1}\\\
=\sin\beta\cos\beta\left(\imath_{w}[\phi_{\rho}^{\tau}\wedge
a_{2,k-1}]-\imath_{w}[\phi_{\rho}\wedge a_{2,k-1}]\right)$ (90)

The $V_{2}^{+}/\tilde{V}_{2}^{0}$ part of this equation simply states
$kc_{2,k}\raisebox{-0.86108pt}{$|$}_{\mathrm{(II)}}=0$ for all $k\geq 1$ (for
$k=1,2$ notice that $c_{1,k}$ and $a_{1,k}$ are elements of $V_{1}^{-}\oplus
V_{1}^{0}\oplus V_{1}^{+}$). Similarly, the equation for the $d$ terms at
order $\mathcal{O}(y(\log y)^{k-1})$ states

$\displaystyle\begin{multlined}\left(kd_{2,k}+2d_{2,k-1}-\mathfrak{J}d_{2,k-1}\right)+\imath_{w}d_{\omega}c_{1,k-1}^{\tau}-(\imath_{w}\star_{4}d_{\omega}b_{1,k-1})^{\tau}\\\
=-\cos^{2}\beta\ \imath_{w}[\phi_{\rho}\wedge a_{2,k-1}]-\sin^{2}\beta\
\imath_{w}[\phi_{\rho}^{\tau}\wedge
a_{2,k-1}]\end{multlined}\left(kd_{2,k}+2d_{2,k-1}-\mathfrak{J}d_{2,k-1}\right)+\imath_{w}d_{\omega}c_{1,k-1}^{\tau}-(\imath_{w}\star_{4}d_{\omega}b_{1,k-1})^{\tau}\\\
=-\cos^{2}\beta\ \imath_{w}[\phi_{\rho}\wedge a_{2,k-1}]-\sin^{2}\beta\
\imath_{w}[\phi_{\rho}^{\tau}\wedge a_{2,k-1}]$ (93)

For $k>2$ the $V_{1}^{-}$ part of this equation simply states
$(kd_{2,k})^{1,-}=0$. For $k=2$ we instead have
$(2d_{2,2}^{\tau}+\imath_{w}d_{\omega}c_{1,1}-\imath_{w}\star_{4}d_{\omega}b_{1,1})^{1,-}=0$.
Note that $b_{1,1}=\cos\beta\ c_{1,1}$ is an element of
$V_{1}^{+}/\tilde{V}_{1}^{0}$. By comparison with the mapping properties (68),
we conclude that also $(d_{2,2})^{1,-}=0$.

The equations at order $\mathcal{O}(y(\log y))$ and $\mathcal{O}(y)$ involve
derivatives of $c_{1,1}$, $c_{1,0}$ and $d_{1,0}$, which are all contained in
$V_{1}^{-}\oplus V_{1}^{0}\oplus V_{1}^{+}$, so only these parts of $c_{2,1}$,
$d_{2,1}$, $c_{2,0}$ and $d_{2,0}$ may be non-zero. By restricting the
equations to the various possible subspaces (and since
$V_{1}^{-}\cap\tilde{V}_{1}^{0}=\\{0\\}$) one obtains

$\displaystyle c_{2,1}=$
$\displaystyle\phantom{{}+{}}\tfrac{1}{3}\left(\sin\beta\
\imath_{w}\star_{4}d_{\omega}c_{1,1}+\sin\beta\cos\beta\
(\imath_{w}[\phi_{\rho}^{\tau}\wedge a_{2,1}]-\imath_{w}[\phi_{\rho}\wedge
a_{2,1}])\right)^{1,0}$ (94)
$\displaystyle+\phantom{\tfrac{1}{3}}\left(\sin\beta\
\imath_{w}\star_{4}d_{\omega}c_{1,1}+\sin\beta\cos\beta\
\imath_{w}[\phi_{\rho}^{\tau}\wedge a_{2,1}]\right)^{1,+}$ (95) $\displaystyle
d_{2,1}=$
$\displaystyle\phantom{{}+{}}\left(-\imath_{w}d_{\omega}c_{1,0}^{\tau}+(\imath_{w}\star_{4}d_{\omega}b_{1,0})^{\tau}\right)^{1,-}$
(96)
$\displaystyle+\left(-\imath_{w}d_{\omega}c_{1,1}^{\tau}+(\imath_{w}\star_{4}d_{\omega}b_{1,1})^{\tau}-\cos^{2}\beta\
\imath_{w}[\phi_{\rho}^{\tau}\wedge a_{2,1}]-\sin^{2}\beta\
\imath_{w}[\phi_{\rho}\wedge a_{2,1}\right)^{1,0}$ (97)
$\displaystyle+\tfrac{1}{3}\left(-\imath_{w}d_{\omega}c_{1,1}^{\tau}+(\imath_{w}\star_{4}d_{\omega}b_{1,1})^{\tau}-\sin^{2}\beta\
\imath_{w}[\phi_{\rho}^{\tau}\wedge a_{2,1}]\right)^{1,+}$ (98)

Since the $\mathcal{O}(y\log y)$ terms are non-zero, they appear in the
$\mathcal{O}(y)$ equations and, accordingly, in the following result for
$c_{2,0}$ an $d_{2,0}$:

$\displaystyle c_{2,0}=$
$\displaystyle\phantom{{}+{}}\tfrac{1}{4}\left(\imath_{w}d_{\omega}d_{1,0}^{\tau}+\imath_{w}\star_{4}d_{\omega}a_{1,0}\right)^{1,-}$
(99)
$\displaystyle+\tfrac{1}{3}\left(\imath_{w}d_{\omega}d_{1,0}^{\tau}+\imath_{w}\star_{4}d_{\omega}a_{1,0}+\sin\beta\cos\beta\
\left(\imath_{w}[\phi_{\rho}^{\tau}\wedge
a_{1,0}]-\imath_{w}[\phi_{\rho}\wedge a_{1,0}]\right)-c_{2,1}\right)^{1,0}$
(100)
$\displaystyle+\phantom{\tfrac{1}{3}}\left(\imath_{w}d_{\omega}d_{1,0}^{\tau}+\imath_{w}\star_{4}d_{\omega}a_{1,0}+\sin\beta\cos\beta\
\imath_{w}[\phi_{\rho}^{\tau}\wedge a_{1,0}]-c_{2,1}\right)^{1,+}+C^{2,+}$
(101) $\displaystyle d_{2,0}=$
$\displaystyle\phantom{+}D^{1,-}+\left(-\imath_{w}d_{\omega}c_{1,0}^{\tau}+\left(\imath_{w}\star_{4}d_{\omega}b_{1,0}\right){}^{\tau}-\cos^{2}\beta\
\imath_{w}[\phi_{\rho}^{\tau}\wedge a_{2,0}]-\sin^{2}\beta\
\imath_{w}[\phi_{\rho}\wedge a_{2,0}]-c_{2,1}\right)^{1,0}$ (102)
$\displaystyle+\tfrac{1}{3}\left(-\imath_{w}d_{\omega}c_{1,0}^{\tau}+\left(\imath_{w}\star_{4}d_{\omega}b_{1,0}\right){}^{\tau}-\cos^{2}\beta\
\imath_{w}[\phi_{\rho}^{\tau}\wedge a_{2,0}]-c_{2,1}\right)^{1,+}$ (103)

Here, $C^{2,+}\in V_{2}^{+}/\tilde{V}_{2}^{0}$ and $D^{1,-}\in V_{1}^{-}$
remain undetermined. While we do not reproduce explicit formulas, it is easy
to see that $\imath_{w}[\phi_{\rho}\wedge a_{2,k}]$ and
$\imath_{w}[\phi_{\rho}^{\tau}\wedge a_{2,k}]$ are fully determined by the
bottom two Haydys-Witten equations.

The main conclusion of this part is that for generic $\beta$ and in absence of
curvature contributions to the $\mathcal{O}(1)$ equations (such that
$a_{1,k}=b_{1,k}=0$ for $k\geq 1$) all components of $c_{2,k}$ and $d_{2,k}$
vanish, except for the functions $C^{2,+}$ and $D^{1,-}$ that appear at $k=0$.
∎

##  5 Asymptotics of the Boundary Term

We now determine the asymptotic behaviour of $\chi=\chi_{1}+\chi_{2}$ at
cylindrical ends of $M^{5}$, where we recall from section 2 that

$\displaystyle\chi_{1}$ $\displaystyle=-2\operatorname{Tr}\left(F_{A}\wedge
J^{-}B\right)\ ,$ (104) $\displaystyle\chi_{2}$
$\displaystyle=-2\operatorname{Tr}\left(\delta_{A}^{+}J^{+}B\wedge
J^{-}B\wedge\eta\right)\ .$ (105)

In doing so, we assume that we evaluate $\chi$ for a Haydys-Witten solution,
which is the situation we are interested in for the Weitzenböck formula.

Since we eventually need to pullback $\chi$ to the boundary, it is important
to understand the interplay between the vector subbundles $\ker\eta$ and
$T(\partial_{i}M)$, with a particular focus on the properties of $J$. Thus,
let $U=[0,1)_{s_{i}}\times\partial_{i}M^{5}$ be a cylindrical end and let $u$
denote the inward-pointing unit normal vector. This means that near a compact
end $u=\partial_{s_{i}}$, while at a non-compact end
$u={s_{i}}\partial_{s_{i}}$. Assume that $g(u,v)=\cos\theta$ is constant on
all of $U$.

If $\theta=0$, $u$ and $v$ are parallel and $\ker\eta=T\partial_{i}M$. In this
situation $J$ simply corresponds to an almost Hermitian structure on
$\partial_{i}M$. If $\theta\neq 0$, $u$ and $v$ are linearly-independent, non-
vanishing vector fields on the tubular neighbourhood and span a regular
distribution $\Delta_{(u,v)}\subset TU$ of rank two. Since $\ker\eta$ has rank
four, the two distributions intersect in a line bundle $L\subset TU$. At each
point, $L$ specifies a direction in the $(u,v)$-plane that is perpendicular to
$v$. We denote the generating unit vector field by $v^{\perp}$ and fix its
orientation such that $g(u,v^{\perp})=-\sin\theta$, such that it points into
the interior. The vector fields are related by $v=\sin\theta w+\cos\theta u$
and $v^{\perp}=\cos\theta w-\sin\theta u$.

Starting from $v^{\perp}$, we can always find a local basis of $\ker\eta$ that
interacts nicely with the almost complex structure $J$. It is given by
$\\{v^{\perp},w_{1}=Jv^{\perp},w_{2},w_{3}=Jw_{2}\\}$, where $w_{2}$ is some
local section of $T\partial_{i}M$ that is orthogonal to $v$, $v^{\perp}$, and
$Jv^{\perp}$. This induces an associated basis of $\Omega^{2}_{v,+}(U)$, given
by

$\displaystyle e_{i}=\eta^{\perp}\wedge
w_{i}^{\flat}+\tfrac{1}{2}\epsilon_{ijk}w_{j}^{\flat}\wedge w_{k}^{\flat}$
(106)

and that satisfies $Je_{1}=+e_{1}$ and $Je_{2/3}=-e_{2/3}$.

We start our investigation of the asymptotics of $\chi$ with Nahm pole
boundaries $\partial_{\mathrm{NP}}M$ in subsection 5.1. Though we only
establish Theorem A in the context of pure Nahm pole boundary conditions, we
also include a short discussion of the expected behaviour of $\chi$ near knot
boundaries $\partial_{K}M$ in subsection 5.2. Subsequently, we discuss
Kapustin-Witten ends with either finite energy or Nahm poles in subsection
5.3.

###  5.1 Nahm Pole Boundaries

Consider a Nahm pole boundary component $\partial_{\mathrm{NP}}M^{5}$. Let
$W^{4}\times[0,1)_{y}$ denote a tubular neighbourhood, where $W^{4}$ is a
complete Riemannian manifold without boundary, and write $\mu_{W^{4}}$ for the
induced volume form. Write $u=\partial_{y}$ for the inward-pointing unit
normal vector field and assume $g(u,v)=\cos\beta$ is constant, with
$\beta\in[0,\pi/2)$. Furthermore, denote by $i_{y}:W^{4}\hookrightarrow
W^{4}\times[0,1)_{y}$ inclusion of $W^{4}$ at $y$.

Recall from section 2 that there is a local basis
$\\{v^{\perp},w_{i}\\}_{i=1,2,3}$ of $\ker\eta$, where each $w_{i}$ is
parallel to $TW^{4}$, and associated to it a basis $\\{e_{i}\\}_{i=1,2,3}$ of
$\Omega^{2}_{v,+}(W^{4}\times[0,1)_{y})$ for which
$J^{-}B=B_{2}e_{2}+B_{3}e_{3}$. Let $(s,x^{i},y)_{i=1,2,3}$ be coordinates
that make these into coordinate vector fields. With that choice we have
$v=\sin\beta\partial_{s}+\cos\beta\partial_{y}$,
$v^{\perp}=\cos\beta\partial_{s}-\sin\beta\partial_{y}$, $w=\partial_{s}$, and
$w_{i}=dx^{i}$.

######  Proposition .

Assume $(A,B)$ is a $\beta$-twisted regular Nahm pole solution of the Haydys-
Witten equations. If $\beta$ is generic, $F_{\omega}=0$ and
$\nabla^{\omega}_{s}(C^{1,+})_{2}+\nabla^{\omega}_{1}(C^{1,+})_{3}=\nabla^{\omega}_{s}(C^{1,+})_{3}-\nabla^{\omega}_{1}(C^{1,+})_{2}=0$,
then there is a $\delta>0$ and a constant $C$ such that

$\displaystyle i^{\ast}_{y}\chi\sim Cy^{\delta}\mu_{W^{4}}\quad(y\to 0)$ (107)

###### Proof.

Consider first the term $\chi_{1}=-2\operatorname{Tr}(F_{A}\wedge J^{-}B)$.
Postponing the discussion of the $ds$-components for the moment, let’s
concentrate on the contributions from $(F_{A})_{ij}dx^{i}\wedge dx^{j}$.
Observe that $e_{i}$ pulls back to $\cos\beta ds\wedge
dx^{i}+\frac{1}{2}\epsilon_{ijk}dx^{j}\wedge dx^{k}$. Fixing an orientation in
which $\mu_{W^{4}}=\sqrt{g}ds\wedge dx^{1}\wedge dx^{2}\wedge dx^{3}$, we find
that these contributions are given by
$\big{(}(F_{A})_{12}B_{3}-(F_{A})_{13}B_{2}\big{)}\cos\beta\mu_{W^{4}}$.

Plugging the expansions of section 4 and section 4 into this formula, we
encounter the following terms at orders $y^{-3}$ and $y^{-1}$

$\displaystyle
y^{-3}i_{y}^{\ast}\;\operatorname{Tr}([\phi_{\rho}^{\tau}\wedge\phi_{\rho}^{\tau}]\wedge
J^{-}\phi_{\rho})\propto\operatorname{Tr}([\mathfrak{t}_{1}^{\tau},\mathfrak{t}_{2}^{\tau}]\mathfrak{t}_{3})-\operatorname{Tr}([\mathfrak{t}_{1}^{\tau},\mathfrak{t}_{3}^{\tau}]\mathfrak{t}_{2})=-2\operatorname{Tr}(\mathfrak{t}_{2}\mathfrak{t}_{3})=0$
(108)
$\displaystyle\begin{multlined}y^{-1}i_{y}^{\ast}\;\operatorname{Tr}([\phi_{\rho}^{\tau}\wedge\phi_{\rho}^{\tau}]\wedge
J^{-}C^{1,+})+y^{-1}i_{y}^{\ast}\;\operatorname{Tr}([\phi_{\rho}^{\tau}\wedge(C^{1,+})^{\tau}]\wedge
J^{-}\phi_{\rho})\\\
\propto\operatorname{Tr}\left(-\mathfrak{t}_{2}(C^{1,+})_{3}-\mathfrak{t}_{3}(C^{1,+})_{2}\right)+\operatorname{Tr}\left(\mathfrak{t}_{2}(C^{1,+})_{3}+\mathfrak{t}_{3}(C^{1,+})_{2}]\right)=0\end{multlined}y^{-1}i_{y}^{\ast}\;\operatorname{Tr}([\phi_{\rho}^{\tau}\wedge\phi_{\rho}^{\tau}]\wedge
J^{-}C^{1,+})+y^{-1}i_{y}^{\ast}\;\operatorname{Tr}([\phi_{\rho}^{\tau}\wedge(C^{1,+})^{\tau}]\wedge
J^{-}\phi_{\rho})\\\
\propto\operatorname{Tr}\left(-\mathfrak{t}_{2}(C^{1,+})_{3}-\mathfrak{t}_{3}(C^{1,+})_{2}\right)+\operatorname{Tr}\left(\mathfrak{t}_{2}(C^{1,+})_{3}+\mathfrak{t}_{3}(C^{1,+})_{2}]\right)=0$
(111)

Meanwhile, at order $\mathcal{O}(1)$, we have the following contributions

$\displaystyle
i_{y}^{\ast}\;\operatorname{Tr}([\phi_{\rho}^{\tau}\wedge\phi_{\rho}^{\tau}]\wedge
J^{-}C^{2,+})$ $\displaystyle=0$ $\displaystyle
i_{y}^{\ast}\;\operatorname{Tr}([\phi_{\rho}^{\tau}\wedge\phi_{\rho}^{\tau}]\wedge
J^{-}D^{1,-})$ $\displaystyle=0$ (112) $\displaystyle
i_{y}^{\ast}\;\operatorname{Tr}([\phi_{\rho}^{\tau}\wedge(C^{2,+})^{\tau}]\wedge
J^{-}\phi_{\rho})$ $\displaystyle=0$ $\displaystyle
i_{y}^{\ast}\;\operatorname{Tr}([\phi_{\rho}^{\tau}\wedge(D^{1,-})^{\tau}]\wedge
J^{-}\phi_{\rho})$ $\displaystyle=0$ (113)

On the left we used that the subspaces $V_{1},V_{2}\subset\mathfrak{g}$ are
orthogonal with respect to the trace (keeping in mind that the action of
$\phi_{\rho}\in V_{1}$ preserves $V_{2}$), while the equations on the right
similarly use $D^{1,-}\in V_{1}^{-}$ insofar that $D^{1,-}\propto\phi_{\rho}$,
such that the trace vanishes by the same calculation as at order
$\mathcal{O}(y^{-3})$.

According to section 4, the only $ds$-component of $F_{A}$ can arise from
$d_{\omega}C^{1,+}$, which appears at $\mathcal{O}(1)$. As we have seen, all
other contributions vanish, so we are left with

$\displaystyle i_{y}^{\ast}\chi_{1}=$
$\displaystyle-2i_{y}^{\ast}\,\operatorname{Tr}(\sin\beta\cos\beta\
d_{\omega}(C^{1,+})^{\tau}\wedge J^{-}\phi_{\rho})$ (114) $\displaystyle=$
$\displaystyle\begin{aligned} &-2\sin\beta\cos^{2}\beta\ \mu_{W^{4}}\
\operatorname{Tr}\left((\nabla^{\omega}_{s}(C^{1,+})_{2}+\nabla^{\omega}_{1}(C^{1,+})_{3})\mathfrak{t}_{3}+(\nabla^{\omega}_{s}(C^{1,+})_{3}-\nabla^{\omega}_{1}(C^{1,+})_{2})\mathfrak{t}_{2}\right)\\\
&-2\sin\beta\cos^{2}\beta\ \mu_{W^{4}}\
\operatorname{Tr}\left(\nabla^{\omega}_{3}(C^{1,+})_{1}\mathfrak{t}_{2}-\nabla^{\omega}_{2}(C^{1,+})_{1}\mathfrak{t}_{3}\right)+\mathcal{O}(y^{\delta})\end{aligned}$
(115)

The first line of the result vanishes by assumption, while the remaining terms
cancel in combination with $\chi_{2}$.

Hence, moving on to $\chi_{2}=-2\operatorname{Tr}(\delta_{A}^{+}J^{+}B\wedge
J^{-}B\wedge\eta)$. Note that $i_{y}^{\ast}\eta=\sin\beta ds$ and as a result
only the $dx^{2}$\- and $dx^{3}$-components of $\delta_{A}^{+}J^{+}B$ can
contribute to this expression. Specifically, we get
$i_{y}^{\ast}\chi_{2}=2\operatorname{Tr}\big{(}(\delta_{A}^{+}J^{+}B)_{2}B_{2}+(\delta_{A}^{+}J^{+}B)_{3}B_{3}\big{)}\sin\theta\mu_{W^{4}}$.
Upon plugging in section 4 and section 4 one encounters essentially the same
expressions as for $\chi_{1}$. The only non-zero terms are again those that
contain derivatives of $C^{1,+}$, which are now given by

$\displaystyle i_{y}^{\ast}\chi_{2}=2\sin\beta\cos^{2}\beta\mu_{W^{4}}\
\operatorname{Tr}\left(\nabla^{\omega}_{3}(C^{1,+})_{1}\mathfrak{t}_{2}-\nabla^{\omega}_{2}(C^{1,+})_{1}\mathfrak{t}_{3}\right)+\mathcal{O}(y^{\delta})$
(116)

These cancel the remnants of $i_{y}^{\ast}\chi_{1}$, which concludes the
proof. ∎

###  5.2 Knot boundaries

Consider a boundary face of type $\partial_{K}M$ and remember that it arises
by blowup of a knot surface $\Sigma_{K}$. Let $\Sigma_{K}\times
H^{2}\times[0,1)_{R}$ denote a tubular neighbourhood, where $H^{2}$ is the
two-dimensional hemisphere. Moreover, assume that the glancing angle $\theta$
between $v$ and $\Sigma_{K}$ is constant. If $\theta\neq\pi/2$, there is a
non-vanishing vector field $u$ parallel to $\Sigma_{K}$ such that
$g(u,v)=\sin\theta$.

In analyzing the behavior of the function $\chi$ near $\partial_{K}M$, we
encounter some difficulties. The polyhomogeneous expansion of $(a,b)$ in this
case starts at $y^{0}$. While the leading order of $\chi$ remains essentially
unchanged, given by straightforward analogues of the $y^{-3}$-terms from
earlier, the additional $y^{0}$-modes of $a$ and $b$ lead to new
contributions, beginning at $\mathcal{O}(y^{-2})$.

To determine these contributions, we need to repeat the analysis of the
polyhomogeneous expansion near $\partial_{K}M$, but now based on the knot
singularity model solutions. In this case, one considers model solutions
$(A^{\lambda,\theta},B^{\lambda,\theta})$ that are solutions of a
$\theta$-twisted version of the extended Bogomolny equations (TEBE).
Unfortunately, unlike the Nahm pole solutions, these knot singularity
solutions are to the most part known only implicitly. For $G=SU(N)$ and
$\theta=\pi/2$ the solutions have been described in [Mik12], while the
existence of solutions for $G=SU(2)$ and general $\theta\in(0,\pi/2]$ is
described by a continuation argument due to [Dim22].

For the specific case with $G=SU(2)$ and $\theta=\pi/2$, the solutions have
been described in [Wit11] and can be given explicitly. To do so, introduce
coordinates $(s,t,\psi,\vartheta,R)$ on $\Sigma_{K}\times
H^{2}\times[0,1)_{R}$, $(s,t)$ are local coordinates on the surface
$\Sigma_{K}$ that we will also collectively refer to as $z$, while
$R\in[0,1)$, $\psi\in[0,\pi/2]$ and $\vartheta\in[0,2\pi]$ are global
coordinates on the filled hemisphere. Let $(\mathfrak{t}_{i})_{i=1,2,3}$
denote a standard basis of $\mathfrak{su}(2)$ and view $\mathfrak{t}_{1}$ as
the generator of a fixed Cartan subalgebra. Introduce the
$\mathfrak{sl}(2,\mathbb{C})$-valued function $\varphi=\phi_{2}-i\phi_{3}$ as
a convenient combination of the components $\phi_{2}$ and $\phi_{3}$ of
$B=\sum_{1}^{3}\phi_{i}e_{i}$. Similarly, denote by
$E=\mathfrak{t}_{2}-i\mathfrak{t}_{3}$, $H=\mathfrak{t}_{1}$, and
$F=\mathfrak{t}_{2}+i\mathfrak{t}_{3}$ the elements of an
$\mathfrak{sl}(2,\mathbb{C})$-triple $(E,H,F)$. The knot singularity solutions
with charge $\lambda\in\mathbb{Z}$ are given by

$\displaystyle\begin{split}A_{\vartheta}&=-(\lambda+1)\cos^{2}\psi\frac{(1+\cos\psi)^{\lambda}-(1-\cos\psi)^{\lambda}}{(1+\cos\psi)^{\lambda+1}-(1-\cos\psi)^{\lambda+1}}\
H\\\\[4.30554pt]
\phi_{1}&=-\frac{\lambda+1}{R}\frac{(1+\cos\psi)^{\lambda+1}+(1-\cos\psi)^{\lambda+1}}{(1+\cos\psi)^{\lambda+1}-(1-\cos\psi)^{\lambda+1}}\
H\\\\[4.30554pt] \varphi&=\frac{(\lambda+1)}{R}\frac{\sin^{\lambda}\psi\
\exp(i\lambda\vartheta)}{(1+\cos\psi)^{\lambda+1}-(1-\cos\psi)^{\lambda+1}}\
E\\\\[4.30554pt] A_{s}&=A_{t}=A_{R}=A_{\psi}=0\end{split}$ (117)

We refer to these as the untwisted knot singularity model solutions.

The higher orders in a polyhomogeneous expansion of $(A,B)$ around these model
solutions was previously investigated by He [He18, HM19]. Originally this was
for Nahm pole solutions of the $\theta=\pi/2$ Kapustin-Witten equations, i.e.
in four dimensions. However, the $\theta=\pi/2$ Kapustin-Witten equations are
equivalent to a dimensional reduction of the Haydys-Witten equations along a
direction perpendicular to $v$, and along which $K$ extends to $\Sigma_{K}$.
After an inconsequential reinterpretation of field components, the
polyhomogeneous expansion carries over to the Haydys-Witten equations with
knot singularity. Indeed, in that context the glancing angle between $v$ and
$\Sigma_{K}$ is $\theta=\pi/2$, such that the untwisted knot singularity
solutions are natural boundary conditions.

######  Theorem  ([He18, HM19]).

Let $(A,B)$ be a solution of the Haydys-Witten equations that satisfies Nahm
pole boundary conditions with a knot singularity of weight $\lambda$ and with
glancing angle $\theta=\pi/2$. Correspondingly, write $A=A^{\lambda,\pi/2}+a$
and $B=B^{\lambda,\pi/2}+b$ with
$a,b\in\mathcal{O}(R^{-1+\epsilon}s^{-1+\epsilon})$. Then $a$ and $b$ are
polyhomogeneous in $R$ and $s$, with non-negative exponents. In particular
they satisfy the estimates

$\displaystyle\left|\nabla^{\ell}_{z}\nabla^{m}_{R}\nabla^{n}_{\cos\psi}\
a\right|_{\mathcal{C}^{0}}\leq
C_{\ell,m,n}R^{-\epsilon-m}(\cos\psi)^{2-\epsilon-n}\
,\quad\left|\nabla^{\ell}_{z}\nabla^{m}_{R}\nabla^{n}_{\cos\psi}\
b\right|_{\mathcal{C}^{0}}\leq
C_{\ell,m,n}R^{-\epsilon-m}(\cos\psi)^{1-\epsilon-n}$ (118)

for any $\epsilon>0$ and $\ell,m,n\in\mathbb{N}$.

We can use this to determine the leading order of $\chi$ in the case of
$\theta=\pi/2$ knot singularities along $\partial_{K}M$. For this, let
$i_{R}:\Sigma_{K}\times H^{2}\hookrightarrow\Sigma_{K}\times
H^{2}\times[0,1)_{R}$ denote inclusion at radius $R$ and write
$\mu_{\Sigma_{K}\times H^{2}_{R}}$ for the pullback of the volume form under
$i_{R}$, i.e. to the hemisphere (or cylinder) of radius $R$. First, since the
$dR$-components of $F_{A}$ drop out under pullback and the remaining
components contain at most one derivative $\nabla_{\cos\psi}$, only components
of $F_{A}$ of order $\mathcal{O}(R^{-\epsilon}(\cos\psi)^{1-\epsilon})$
contribute to $\chi_{1}$. Furthermore,
$J^{-}B=\mathcal{O}(R^{-1}(\cos\psi)^{-1})$, such that
$\chi_{1}=-2\operatorname{Tr}(F_{A}\wedge J^{-}B)\sim
CR^{-1-\epsilon}(\cos\psi)^{-\epsilon})\mu_{\Sigma_{K}\times H^{2}_{R}}$
Second, $\delta_{A}^{+}J^{+}B$ does not contain $\nabla_{R}$ and the leading
order contributions of the product $J^{+}B\wedge J^{-}B$ at
$R^{-2}(\cos\psi)^{-2}$ vanish by construction of the underlying Nahm pole
($\operatorname{Tr}(HE)=0$). We conclude that also
$\chi_{2}=-2\operatorname{Tr}(\delta_{A}^{+}J^{+}B\wedge J^{-}B\wedge\eta)\sim
CR^{-1-\epsilon}(\cos\psi)^{-\epsilon})\mu_{\Sigma_{K}\times H^{2}_{R}}$. In
fact, when the assumptions of subsection 5.1 are satisfied, the exponent of
$\cos\psi$ can be improved to $(\cos\psi)^{\delta}$, for some $\delta>0$. This
follows from the fact that $R\cos\psi=y$ and for fixed $R\neq 0$, the
expansion in $\cos\psi$ must be consistent with the expansion in $y^{\delta}$
given in subsection 5.1.

All in all, if $(A,B)$ are Haydys-Witten solutions that satisfy (untwisted)
Nahm pole boundary conditions near $\partial_{K}M$, then $\chi$ is of order
$R^{-1-\epsilon}(\cos\psi)^{\delta}$, for some $\delta>0$. It is natural to
expect that for general $\theta\in(0,\pi/2]$ the behavior of the
polyhomogeneous expansions of $A$ and $B$ near $\partial_{K}M$ follows a
similar pattern to the one we observed for the pure $\beta$-twisted Nahm pole
solutions at $\partial_{\mathrm{NP}}M$. Specifically, if
$A=A^{\lambda,\theta}+a$ and $B=B^{\lambda,\theta}+b$ with respect to the
twisted model solutions, it is still true that $a$ and $b$ admit
polyhomogeneous expansions of order
$\mathcal{O}(R^{-\epsilon}(\cos\psi)^{-\epsilon})$. In analogy to the
cancellations in the case of Nahm pole boundary conditions, we then expect
that $a$ and $b$ are restricted in a way that makes any contributions to
$\chi$ below $R^{-1-\epsilon}(\cos\psi)^{\delta}$ vanish.

In any case, as we will see below, the following asymptotic behaviour of
$\chi$ near a knot boundary $\partial_{K}M$ is sufficient to extend Theorem A
to situations with knot singularities:

$\displaystyle\chi\sim CR^{-2+\delta}(\cos\psi)^{\delta}\mu_{\Sigma_{K}\times
H^{2}_{R}}\quad(R\to 0)$ (119)

It’s likely that such a result only holds under certain conditions on the
topology and geometry of $E\to M^{5}$ and
$\Sigma_{K}\subset\partial_{\mathrm{NP}}M$.

###  5.3 Kapustin-Witten ends

We now consider Kapustin-Witten ends $\partial_{\mathrm{KW}}M^{5}$ and
$\partial_{\mathrm{NP-KW}}M^{5}$. As explained in section 3, we take this to
mean non-compact ends at which fields converge to a Kapustin-Witten solution
with either finite energy or with Nahm pole at a corner of $M^{5}$,
respectively. We let $[0,1)_{s}\times W^{4}$ be a tubular neighbourhood and
denote by $i_{s}:W^{4}\hookrightarrow[0,1)_{s}\times W^{4}$ inclusion of
$W^{4}$ at $s$. Write $u=s\partial_{s}$ for the inward-pointing unit normal
vector field and assume $g(u,v)=\cos\theta$ is constant. As usual, if
$\theta\neq 0$, let $w$ be the non-vanishing unit vector field on $W^{4}$ with
respect to which $v=\cos\theta u+\sin\theta w$.

We begin with a general result about the pullback of $\chi$ in dependence of
the limiting field configuration. Hence, assume $(A,B)$ approaches a solution
of the $\theta$-Kapustin-Witten equations $(\tilde{A},\phi)$ on $W^{4}$. In a
local basis $(s,t,x^{i})_{i=1,2,3}$ with $w=\partial_{t}$, the pullback of
$\chi=2\operatorname{Tr}(F_{A}\wedge J^{-}B+\delta_{A}^{+}J^{+}B\wedge
J^{-}B\wedge\eta)$ is then given by

$\displaystyle\tfrac{1}{2}\lim_{s\to
0}i_{s}^{\ast}\chi=\cos\theta\operatorname{Tr}\big{(}(F_{\tilde{A}}^{+})_{t2}\phi_{2}+(F_{\tilde{A}}^{+})_{t3}\phi_{3}\big{)}\mu_{W^{4}}+\sin\theta\operatorname{Tr}\big{(}\nabla^{\tilde{A}}_{2}\phi_{1}\
\phi_{3}-\nabla^{\tilde{A}}_{3}\phi_{1}\ \phi_{2}\big{)}\mu_{W^{4}}$ (120)

Here, $F_{\tilde{A}}^{+}$ denotes the self-dual part with respect to the four-
dimensional Hodge operator $\star_{W^{4}}$.

#### Finite Energy Solutions.

Kapustin and Witten observed that Kapustin-Witten solutions on closed
manifolds are highly restricted [KW07]. A similar result was established by
Nagy and Oliveira for finite energy solutions on ALE and ALF gravitational
instantons [NO21]. The energy in question, usually called Kapustin-Witten
energy, is defined by the functional

$\displaystyle E_{\mathrm{KW}}=\int_{W^{4}}\left\lVert
F_{\tilde{A}}\right\rVert^{2}+\left\lVert\nabla^{\tilde{A}}\phi\right\rVert^{2}+\left\lVert[\phi\wedge\phi]\right\rVert^{2}$
(121)

Nagy and Oliveira observed that for finite energy solutions the norm of the
Higgs field is bounded, which yields surprisingly strong restrictions for
solutions on $\mathbb{R}^{4}$ and $S^{1}\times\mathbb{R}^{3}$, when combined
with a result by Taubes [Tau17]. This has recently been generalized to
arbitrary ALE and ALF gravitational instantons, as was already conjectured by
Nagy and Oliveira. The situation is summarized by the following two theorems.

######  Theorem ([KW07, Gagliardo2014]).

Let $E\to W^{4}$ be an $SU(2)$ principal bundle over a compact manifold
without boundary. Assume $(A,\phi)$ satisfies the $\theta$-Kapustin-Witten
equations with $\theta\in(0,\pi)$. If $E\to W^{4}$ has non-zero Pontryagin
number, then $A$ and $\phi$ are identically zero. Otherwise $A+i\phi$ is a
flat $PSL(2,\mathbb{C})$ connection; equivalently $F_{A}=[\phi\wedge\phi]$ and
$\nabla^{A}\phi=0$.

######  Theorem ([Ble23]).

Let $(A,\phi)$ be a finite energy solution of the $\theta$-Kapustin-Witten
equations with $\theta\neq 0\pmod{\pi}$ on an ALE or ALF gravitational
instanton and let $G=SU(2)$. Then $A$ is flat, $\phi$ is
$\nabla^{A}$-parallel, and $[\phi\wedge\phi]=0$.

In combination with equation (120), we immediately arrive at the following
result.

######  Proposition .

Let $G=SU(2)$ and $\theta\nequiv 0\pmod{\pi}$. Assume $(A,B)$ approaches a
finite energy solution of the $\theta$-Kapustin-Witten equations on $W^{4}$ as
$s\to 0$. If $W^{4}$ is an ALE or ALF gravitational instanton, or if $W^{4}$
is compact and either  (i) $E\to W^{4}$has non-zero Pontryagin number or  (ii)
$\theta=\pi/2$,  then $\lim_{s\to 0}i_{s}^{\ast}\chi=0$.

###### Proof.

If $W^{4}$ is an ALE or ALF gravitational instanton, both terms in (120)
vanish directly. The first since $\tilde{A}$ is flat and the second since
$\phi$ is $\nabla^{\tilde{A}}$ parallel.

If $W^{4}$ is compact, $A$ and $\phi$ can only be non-zero if the Pontryagin
number of $E\to W^{4}$ is zero and, furthermore, in that case
$F_{\tilde{A}}=[\phi\wedge\phi]$ and $\nabla^{\tilde{A}}\phi=0$. It follows
that $\lim_{s\to
0}i_{s}^{\ast}\chi=2\cos\theta\operatorname{Tr}([\phi_{1},\phi_{2}]\phi_{3})\mu_{W^{4}}$,
which vanishes if $\theta=\pi/2$. ∎

We make two observations. First, if $\theta\equiv 0\pmod{\pi}$, the natural
boundary condition for $(A,B)$ are finite energy solutions $(\tilde{A},B,C)$
of the Vafa-Witten equations on $W^{4}$. On $\mathbb{R}^{4}$ the Vafa-Witten
and $\theta=0$ Kapustin-Witten equations are equivalent by an identification
of $\phi=Cdx^{0}+B_{0i}dx^{i}$. According to the result by Taubes mentioned
earlier, any solution with bounded $\left\lVert\phi\right\rVert$ satisfies
$\sigma(B,B)=[B,C]=0$ and $\nabla^{A}B=\nabla^{A}C=0$ [Tau17].

It is not currently known if Vafa-Witten solutions on ALX spaces have a
similar property. However, the Vafa-Witten equations are still closely related
to the $\theta=0$ version of the Kapustin-Witten equations, and Taubes’ result
has been established for the latter in [Ble23]. One might thus expect that, at
least in certain situations, the vanishing results carry over to Vafa-Witten
solutions. Whenever this is the case, the Vafa-Witten equations
$F_{\tilde{A}}^{+}=\sigma(B,B)+[B,C]$ reduce to anti-self-dual equations for
$\tilde{A}$, in which case $\lim_{s\to
0}i_{s}^{\ast}\chi=F_{\tilde{A}}^{+}\wedge i_{s}^{\ast}J^{-}B=0$, as we found
above.

Second, the fact that $\chi$ converges to zero holds at two ends of a spectrum
of asymptotic volume growth of $W^{4}$. On the one hand, in the case of
compact manifolds with bounded volume or equivalently asymptotic volume growth
of order $r^{0}$, and on the other hand, on ALF and ALE manifolds with
asymptotic volume growth of order $r^{3}$ and $r^{4}$, respectively. For ALG
and ALH gravitational instantons, the proof strategy of Nagy and Oliveira
result doesn’t work, because they rely on the existence of a positive Green’s
function for the Laplacian. The approach to Taubes’ dichotomy fails for ALH
manifolds for much the same reason.

To the best of the authors knowledge it is currently not known if an analogue
of the results of Nagy and Oliveira should be expected to be true or false on
ALG and ALH gravitational instantons. However, from the physics perspective
there is no obvious reason to single out intermediate volume growth like that.
On the contrary, in Witten’s approach to Khovanov homology, where one
considers $M^{5}=\mathbb{R}_{s}\times X^{3}\times\mathbb{R}^{+}_{y}$, it is
natural to expect simplifications whenever $X^{3}$ has additional
substructure.

For example, one expects that Khovanov homology arises for $X^{3}=S^{3}$, in
which case $M^{5}$ has an ALH end $W^{4}=\mathbb{R}_{s}\times S^{3}$ at
$y\to\infty$. Also, Gaiotto and Witten recovered the Jones polynomial purely
by adiabatically braiding solutions of the EBE under the assumption that
$X^{3}=\mathbb{R}_{t}\times\Sigma^{2}$, in which case one encounters an ALG
manifold at $y\to\infty$. It thus seems plausible to postulate that the
results of this section hold, more generally, whenever $W^{4}$ is a complete
Ricci-flat Riemannian manifold with sectional curvature bounded from below and
$(A,B)$ approaches a finite energy solution of the $\theta$-Kapustin-Witten
equations ($\theta\neq 0,\pi$) or the Vafa-Witten equations ($\theta=0$).

#### Nahm Poles at Corners.

We now discuss the behaviour of $\chi$ at non-compact ends of the class
$\partial_{\mathrm{NP-KW}}M^{5}$. As before, we denote by $[0,1)_{s}\times
W^{4}$ a tubular neighbourhood. In this situation we still demand that $(A,B)$
converges to a $\theta$-Kapustin-Witten solution $(\tilde{A},\phi)$ on $W^{4}$
as $s\to 0$. But in contrast to earlier, we now assume that $(\tilde{A},\phi)$
satisfies $\beta$-twisted Nahm pole boundary conditions at an adjacent corner
of $M^{5}$ – or equivalently at a boundary of $W^{4}$. Observe that
$(\tilde{A},\phi)$ cannot have finite energy if it exhibits a Nahm pole, such
that subsection 5.3 doesn’t apply.

In the absence of a finite energy condition, we now also have to specify
boundary conditions at non-compact ends $[0,1)_{s^{\prime}}\times X^{3}$ of
$W^{4}$. We demand that $\tilde{A}+i\phi$ converges to a flat $G_{\mathbb{C}}$
connection on $X^{3}$ as $s^{\prime}\to 0$. From now on we refer to these
configurations simply as Nahm pole solutions of the $\theta$-Kapustin-Witten
equations.

######  Remark.

Note that a non-compact end $[0,1)_{s^{\prime}}\times X^{3}$ of $W^{4}$
corresponds to a “corner at infinity” $[0,1)_{s}\times[0,1)_{s^{\prime}}\times
X^{3}$ of $M^{5}$. This corner is adjacent to two non-compact ends of $M^{5}$,
at which $(A,B)$ converges to a corresponding solution of the $\theta$\- or
$\theta^{\prime}$-Kapustin-Witten equation, respectively. The two associated
asymptotic boundary conditions, which demand that both $\tilde{A}+i\phi$ and
$\tilde{A}^{\prime}+i\phi^{\prime}$ are flat connections on $X^{3}$, have to
be consistent with the fact that both Kapustin-Witten solutions arise from the
common five-dimensional fields $(A,B)$. Put differently: If we view $B$ as a
one-form by the usual isomorphism, the pullback of $A+iB$ converges to a flat
connection on $X^{3}$ as $s,s^{\prime}\to 0$.

We similarly need to ensure that the boundary conditions are compatible at
corners $[0,1)_{s}\times X^{3}\times[0,1)_{y}$ that are adjacent to a
$\theta$-Kapustin-Witten end as $s\to 0$ and a $\beta$-Nahm pole as $y\to 0$.
This is only the case if $\beta=\pi/2-\theta$, since otherwise the
$\beta$-twisted Nahm pole model solutions are not solutions of the
$\theta$-Kapustin-Witten equations.

######  Proposition .

Assume $(A,B)$ approaches a solution of the $\theta$-Kapustin-Witten equations
on $W^{4}$. Then

$\displaystyle\lim_{s\to 0}i_{s}^{\ast}\chi$
$\displaystyle=\tfrac{2}{3}i_{0}^{\ast}\operatorname{Tr}\left(\sigma(B,B)\wedge
B\right)+2\tfrac{\sin\theta}{\cos^{2}\theta}\
i_{0}^{\ast}d\operatorname{Tr}\left(\imath_{w}(J^{+}B)\wedge J^{-}B\right)\ .$
(122)

and the expression is exact if $\theta=\pi/2$.

###### Proof.

This is a rewriting of equation (120). Since $(\tilde{A},\phi)$ satisfy the
$\theta$-Kapustin-Witten equations, we can replace the field strength by
$F_{\tilde{A}}=\frac{1}{2}[\phi\wedge\phi]-\cot\theta
d_{\tilde{A}}\phi+\csc\theta\star_{4}d_{\tilde{A}}\phi$. After a short
calculation (and slightly miraculous cancellations), we find

$\displaystyle\lim_{s\to
0}i_{s}^{\ast}\chi=2\cos\theta\operatorname{Tr}([\phi_{1},\phi_{2}]\phi_{3})\mu_{W^{4}}+\sin\theta\;\operatorname{Tr}(\nabla^{A}_{2}(\phi_{1}\phi_{3})-\nabla^{A}_{3}(\phi_{1}\phi_{2}))\mu_{W^{4}}\
,$ (123)

which is a local representation of the expression above and shows that the
right hand side is exact if $\cos\theta=0$ ∎

##  6 Vanishing of the Boundary Term

We can now show that the contributions from the exact term in the Weitzenböck
formula of Equation 17 vanishes when the various conditions we have
encountered in section 4 and section 5 are satisfied. In summary we make the
following assumptions:

  1. (A1)

At $\partial_{\mathrm{NP}}M^{5}$ the fields satisfy regular $\beta$-twisted
Nahm pole boundary conditions for some generic $\beta$. Writing
$A=y^{-1}\sin\beta\phi_{\rho}^{\tau}+\omega+a$ and
$B=y^{-1}\cos\beta\phi_{\rho}+b$, assume that $F_{\omega}=0$ and that
$\nabla^{\omega}_{s}b_{2}+\nabla^{\omega}_{1}b_{3}=\mathcal{O}(y^{2})$ and
$\nabla^{\omega}_{s}b_{3}-\nabla^{\omega}_{1}b_{2}=\mathcal{O}(y^{2})$ (cf.
subsection 5.1).

  2. (A2)

At $\partial_{K}M^{5}$ the fields are asymptotic to knot singularity models
and there is some $\delta>0$ such that $i^{\ast}_{R}\chi\sim
CR^{-2+\delta}(\cos\psi)^{\delta}\mu_{\Sigma_{K}\times H^{2}_{R}}$ as $R\to 0$
(cf. subsection 5.2).

  3. (A3)

At $\partial_{\mathrm{KW}}M^{5}$ the fields approach a finite energy solution
of the $\theta$-Kapustin-Witten equations. The boundary face
$\partial_{\mathrm{KW}}M^{5}$ is either  (a) an ALE or ALF gravitational
instanton,  (b) a compact manifold on which the bundle
$E\to\partial_{\mathrm{KW}}M^{5}$ has non-zero Pontryagin number, or  (c) a
compact manifold with incidence angle $\theta=\pi/2$ (cf. subsection 5.3).

  4. (A4)

At $\partial_{\mathrm{NP-KW}}M^{5}$ the incidence angle is $\theta=\pi/2$ and
the fields approach Kapustin-Witten solutions with Nahm poles at boundaries.
Moreover, at non-compact cylindrical ends of $\partial_{\mathrm{NP-KW}}M^{5}$,
the combination $A+iB$ converges to a flat $G_{\mathbb{C}}$ connection that
satisfies $J^{-}\sigma(B,B)=0$. (cf. subsection 5.3).

######  Theorem .

Let $G=SU(2)$, $M^{5}$ a manifold with poly-cylindrical ends, $v$ a non-
vanishing vector field that approaches ends at a constant angle, and $J$ an
almost Hermitian structure on $\ker\eta$. Assume
${\mathop{\mathbf{HW}}{}_{v\,\/}}(A,B)=0$ and that (A1) \- (A4) are satisfied,
then ${\mathop{\mathbf{dHW}}{}_{v,J\,\/}}(A,B)=0$.

###### Proof.

Our starting point is a regularized version of the Weitzenböck formula of
section 2, obtained by restricting the domain of integration to the compact
submanifold with corner $M^{5}_{\epsilon}$ introduced in section 3 and taking
$\epsilon\to 0$.

$\displaystyle\int_{M^{5}}\left\lVert{\mathop{\mathbf{HW}}{}_{v\,\/}}(A,B)\right\rVert^{2}=\int_{M^{5}}\left\lVert{\mathop{\mathbf{dHW}}{}_{v,J\,\/}}(A,B)\right\rVert^{2}+\lim_{\epsilon\to
0}\int_{M^{5}_{\epsilon}}d\chi$ (124)

According to Stokes’ theorem, the contributions of the exact term are now
determined by

$\displaystyle\lim_{\epsilon\to 0}\int_{M_{\epsilon}}d\chi=\sum_{i\in
I}\lim_{\epsilon\to 0}\int_{\partial_{i}M_{\epsilon}}\chi\ .$ (125)

We address the integrals for each of the four boundary classes independently.

#### Nahm Pole Boundaries:

Let $W^{4}\times[0,1)_{y}$ be a tubular neighbourhood of a Nahm pole boundary
$\partial_{\text{NP}}M^{5}$. The boundary face $\partial_{NP}M_{\epsilon}$ is
a subset of the $\epsilon$-displacement $\\{\epsilon\\}\times
W^{4}\hookrightarrow[0,1)_{s}\times W^{4}$. We know from subsection 5.1 that
there is some $\delta>0$ such that $i^{\ast}_{y}\chi\sim
Cy^{\delta}\mu_{W^{4}}$ as $y\to 0$. This provides the estimates

$\displaystyle\lim_{\epsilon\to
0}\left|\int_{\partial_{\text{NP}}M_{\epsilon}}i_{\epsilon}^{\ast}\chi\right|\leq\lim_{\epsilon\to
0}\int_{\partial_{\text{NP}}M_{\epsilon}}\left|i_{\epsilon}^{\ast}\chi\right|\leq\lim_{\epsilon\to
0}\int_{W^{4}\times\\{\epsilon\\}}\left|C\right|\epsilon^{\delta}\mu_{W^{4}}=0$
(126)

We have used that the leading order of the pullback of $\chi$ extends
unchanged to all of $W^{4}$, due to compatibility of boundary conditions at
corners.

#### Knot Boundaries:

Let $\Sigma_{K}\times H^{2}_{\psi,\vartheta}\times[0,1)_{R}$ be a tubular
neighbourhood of a knot boundary $\partial_{K}M^{5}$, where
$H^{2}_{\psi,\vartheta}$ denotes the two-dimensional hemisphere, parametrized
by $\psi\in[0,\pi/2]$ and $\vartheta\in[0,2\pi]$. The associated boundary of
the regularized manifold $\partial_{K}M^{5}_{\epsilon}$ is contained in the
$\epsilon$-displacement $\Sigma_{K}\times
H^{2}\times\\{\epsilon\\}=\Sigma_{K}\times H^{2}_{\epsilon}$. The pullback of
the volume form to the hemisphere of radius $R=\epsilon$ is given by
$\mu_{\Sigma_{K}\times H^{2}_{\epsilon}}=\epsilon^{2}d\psi
d\vartheta\mu_{\Sigma_{K}}$. Assuming explicitly that $i_{R}^{\ast}\chi\sim
cR^{-2+\delta}(\cos\psi)^{\delta}\mu_{\Sigma_{K}\times H^{2}_{R}}$ as $R\to
0$, we find

$\displaystyle\lim_{\epsilon\to
0}\left|\int_{\partial_{K}M^{5}_{\epsilon}}i_{\epsilon}^{\ast}\chi\right|\leq\lim_{\epsilon\to
0}\int_{\Sigma_{K}\times
H^{2}\times\\{\epsilon\\}}\left|i_{\epsilon}^{\ast}\chi\right|\leq\lim_{\epsilon\to
0}\int_{\Sigma_{K}}\int_{H^{2}}\left|C\right|\epsilon^{-2+\delta}\epsilon^{2}d\psi
d\vartheta\mu_{\Sigma_{K}}=0$ (127)

where we have used the Fubini-Tonelli theorem to split off integration along
$\Sigma_{K}$. In extending the integral from $\partial_{K}M_{\epsilon}$ to all
of $\Sigma_{K}\times H\times\\{\epsilon\\}$, we have used the compatibility of
knot singularities with the pure Nahm pole boundary conditions (which are the
only boundary conditions that are adjacent to knot singularities) at the
corner $\cos\psi\to 0$.

#### Kapustin-Witten Ends:

Let $[0,1)_{s}\times W^{4}$ be a tubular neighbourhood of a Kapustin-Witten
end $\partial_{\text{KW}}M^{5}$. subsection 5.3 states that $\lim_{s\to
0}i_{s}^{\ast}\chi=0$. Since $\partial_{\mathrm{KW}}M_{\epsilon}\subset W^{4}$
and $W^{4}$ is ALE or ALF, its volume grows asymptotically at most with
$\epsilon^{-4}$. Looking back at (120), the rate of decay of $\chi$ is
determined by how fast $F_{A}$ and $\nabla^{A}\phi$ approach zero as $s\to 0$.
Since the Haydys-Witten equations represent flow equations of the Kapustin-
Witten equations, a typical solution is expected to decay exponentially fast
towards the stationary solution. We conclude that in the limit $\epsilon\to
0$:

$\displaystyle\lim_{\epsilon\to
0}\left|\int_{\partial_{\text{KW}}M^{5}_{\epsilon}}\chi\right|\leq\lim_{\epsilon\to
0}\int_{\partial_{\text{KW}}M^{5}_{\epsilon}}\left|i_{\epsilon}^{\ast}\chi\right|=0$
(128)

#### Kapustin-Witten Ends with Nahm Poles:

Let $[0,1)_{s}\times W^{4}$ be a non-compact end of $M^{5}$ and assume that
$g(\partial_{s},v)=0$, i.e. $\theta=\pi/2$. As before, denote by
$i_{s}:W^{4}\hookrightarrow[0,1)_{s}\times W^{4}$ inclusion of $W^{4}$ at $s$.

Assume $(A,B)$ converges to a Kapustin-Witten solution $(\hat{A},\phi)$ that
exhibits a Nahm pole at a boundary of $W^{4}$. Since $\theta=\pi/2$,
subsection 5.3 states that $\lim_{s\to 0}i_{s}^{\ast}\chi=d\omega$, where
$\omega=\operatorname{Tr}\left(\imath_{w}(J^{+}B)\wedge J^{-}B\right)$. It
follows that

$\displaystyle\lim_{s\to 0}\int_{\partial_{\text{NP-
KW}}M_{\epsilon}}i_{s}^{\ast}\chi=\int_{W^{4}}d\omega$ (129)

and it remains to determine the integral of $d\omega$ over $W^{4}$.

For this, let $B_{r}(p)$ denote the (four-dimensional) ball of radius $r$
centered at some point $p\in W^{4}$ and $\mu_{B_{r}(p)}$ its volume form. A
classic result by Yau states [Yau76, Theorem 3 & Appendix (ii)]: If
$\liminf_{r\to\infty}r^{-1}\int_{B_{r}(p)}\left|\omega\right|\mu_{B_{r}(p)}=0$,
then $\int_{W^{4}}d\omega=0$.

Near the boundary we can rely on section 4. Applied to the Kapustin-Witten
solution $(\tilde{A},\phi)$ with $\beta=\pi/2-\theta=0$, this yields

$\displaystyle\omega=y^{-2}\operatorname{Tr}(\mathfrak{t}_{1}\mathfrak{t}_{2}+\mathfrak{t}_{1}\mathfrak{t}_{3})+y^{0}\operatorname{Tr}(\mathfrak{t}_{1}(C^{1,+})_{2}+\mathfrak{t}_{1}(C^{1,+})_{3}+(C^{1,+})_{1}\mathfrak{t}_{2}+(C^{1,+})_{1}\mathfrak{t}_{3})+\mathcal{O}(y^{\delta})\qquad(y\to
0)$ (130)

The term proportional to $y^{-2}$ vanishes and the term at constant order is
assumed to be integrable on $X^{3}$, such that contributions from the Nahm
pole to the integral $\int_{B_{r}(p)}\left|\omega\right|$ are harmless.

At a non-compact end $[0,1)_{s}\times X^{3}$ of $W^{4}$, the asymptotic
boundary condition states that $A+iB$ approaches a flat
$SL(2,\mathbb{C})$-connection on $X^{3}$ that satisfies $J^{-}\sigma(B,B)=0$.
Equivalently, the component $\phi_{1}$ of $J^{+}B$ commutes with the
components $\phi_{2/3}$ of $J^{-}B$. This is for example the case if $X^{3}$
is a product $S^{1}\times\Sigma$, where $\Sigma$ is a Riemann surface and the
almost complex structure $J$ is the direct sum of complex structures on the
cylinder $[0,1)_{s}\times S^{1}$ and $\Sigma$. In any case, since
$[\phi_{1},\phi_{2}]=[\phi_{1},\phi_{3}]=0$ we find that
$\omega=\operatorname{Tr}(\phi_{1}\phi_{2})+\operatorname{Tr}(\phi_{1}\phi_{3})\to
0$. Assuming $A+iB$ converges to the flat connection faster than the volume of
the geodesic balls $B_{r}(p)$ grows, we conclude that $\int_{W^{4}}d\omega=0$.

#### Conclusion:

Since all boundary contributions to the exact term vanish in the limit
$\epsilon\to 0$, we arrive at

$\displaystyle\int_{M^{5}}\left\lVert{\mathop{\mathbf{HW}}{}_{v\,\/}}(A,B)\right\rVert^{2}=\int_{M^{5}}\left\lVert{\mathop{\mathbf{dHW}}{}_{v,J\,\/}}(A,B)\right\rVert^{2}+\lim_{\epsilon\to
0}\int_{M^{5}_{\epsilon}}d\chi=\int_{M^{5}}\left\lVert{\mathop{\mathbf{dHW}}{}_{v,J\,\/}}(A,B)\right\rVert^{2}$
(131)

Seeing that the integrands on both sides are non-negative, we find that
whenever ${\mathop{\mathbf{HW}}{}_{\,\/}}(A,B)=0$ also
${\mathop{\mathbf{dHW}}{}_{\,\/}}(A,B)=0$, which concludes the proof.

∎

## References

  * [Whi57] Hassler Whitney  “Geometric Integration Theory”  Princeton University Press, 1957  DOI: 10.1515/9781400877577
  * [Yau76] Shing-Tung Yau  “Some Function-Theoretic Properties of Complete Riemannian Manifold and Their Applications to Geometry”  In _Indiana University Mathematics Journal_ 25.7, 1976, pp. 659–670
  * [Mel95] Richard B. Melrose  “Geometric Scattering Theory”, Stanford Lectures  Cambridge University Press, 1995
  * [Mel96] Richard B Melrose  “Differential Analysis on Manifolds with Corners”, 1996  URL: http://math.mit.edu/~rbm/book.html
  * [KW07] Anton Kapustin and Edward Witten  “Electric-Magnetic Duality and the Geometric Langlands Program”  In _Commun. Number Theory Phys._ 1.1, 2007, pp. 1–236
  * [Wit11] Edward Witten  “Fivebranes and Knots”  In _Quantum Topol._ 3.1, 2011, pp. 1–137
  * [GU12] Michael Gagliardo and Karen Uhlenbeck  “Geometric Aspects of the Kapustin–Witten Equations”  In _J. Fixed Point Theory Appl._ 11.2, 2012, pp. 185–198
  * [Hen12] Måns Henningson  “Boundary Conditions for Geometric-Langlands Twisted N = 4 Supersymmetric Yang-Mills Theory”  In _Phys. Rev. D_ 86.8, 2012, pp. 085003
  * [Mik12] Victor Mikhaylov  “On the Solutions of Generalized Bogomolny Equations”  In _J. High Energy Phys._ 2012.5, 2012, pp. 112
  * [MW14] Rafe Mazzeo and Edward Witten  “The Nahm Pole Boundary Condition”  In _Contemporary Mathematics_ The influence of Solomon Lefschetz in geometry and topology.621, 2014
  * [Hay15] Andriy Haydys  “Fukaya-Seidel Category and Gauge Theory”  In _Journal of Symplectic Geometry_ 13.1, 2015, pp. 151–207
  * [MW17] Rafe Mazzeo and Edward Witten  “The KW Equations and the Nahm Pole Boundary Condition with Knots”, 2017  arXiv:1712.00835 [hep-th]
  * [Tau17] Clifford Henry Taubes  “Growth of the Higgs Field for Solutions to the Kapustin-Witten Equations on R4̂”, 2017  arXiv:1701.03072 [math-ph]
  * [He18] Siqi He  “The Expansions of the Nahm Pole Solutions to the Kapustin-Witten Equations”, 2018  arXiv:1808.03886 [math]
  * [HM19] Siqi He and Rafe Mazzeo  “Classification of Nahm Pole Solutions of the Kapustin-Witten Equations on $S1̂\times \Sigma\times \mathbb{}R{}+̂$”, 2019  arXiv:1901.00274
  * [NO21] Ákos Nagy and Gonçalo Oliveira  “The Kapustin–Witten Equations on ALE and ALF Gravitational Instantons”  In _Lett Math Phys_ 111.4, 2021, pp. 87
  * [Dim22] Panagiotis Dimakis  “Model Knot Solutions for the Twisted Bogomolny Equations”, 2022  arXiv:2204.06617 [math-ph]
  * [Ble23] Michael Bleher  “Growth of the Higgs Field for Kapustin-Witten Solutions on ALE and ALF Gravitational Instantons”, 2023  arXiv:2306.17017 [math-ph]

Gagliardo2014Gagliardo2012 Haydys2010Haydys2015 Mazzeo2013aMazzeo2014