# The moduli space of solutions to the Extended Bogomolny equations on
$\Sigma\times\mathbb{R}_{+}$

Panagiotis Dimakis  450 Serra Mall, Bldg 380, Stanford, CA 94305
<EMAIL_ADDRESS>
###### Abstract.

We study moduli spaces of solutions to the extended Bogomolny equations on
$\Sigma\times\mathbb{R}_{+,y}$ with gauge group
$\operatorname{SL}(2,\mathbb{C})$ satisfying the generalized Nahm pole
boundary condition as $y\to 0$ and limiting to complex flat connections as
$y\to\infty$. Refining the Kobayashi-Hitchin correspondence of [HM20] we
identify these moduli spaces with certain holomorphic lagrangian sub-manifolds
inside the moduli space of Higgs bundles.

##  1\. Introduction

The aim of this paper is to study the moduli structure of solutions to the
dimensional reduction of the Kapustin-Witten equations on manifolds of the
form $\Sigma\times\mathbb{R}_{+,y}$ where $\Sigma$ is a closed Riemann surface
of genus $g\geq 2$ satisfying the generalized Nahm pole boundary condition as
$y\to 0$ and converging to a flat $\operatorname{SL}(2,\mathbb{C})$ connection
as $y\to\infty$.

In order to motivate the question studied in this paper we briefly introduce
the Kapustin-Witten equations (KW) and Witten’s suggestion [KW06, Wit11, GW12,
MW14, MW17] that solutions to the KW equations satisfying specific
geometrically motivated asymptotic conditions should contain information about
the topology of the underlying space. Given a four-manifold $(M^{4},g)$, a
$G$-bundle $E$ over $M$ a connection $A$ on $E$ and an ad$(E)$-valued $1$-form
$\phi$, the KW equations are given by

$\begin{split}F_{A}-\phi\wedge\phi+\star d_{A}\phi&=0\\\
d_{A}\star\phi&=0\end{split}$ (1.1)

When $M=W\times\mathbb{R}_{+,y}$ and $L\hookrightarrow W\times\\{0\\}$ is an
embedded link, Witten conjectured that the counting solutions to the KW
equations satisfying the generalized Nahm pole boundary condition as $y\to 0$
[MW17] and converging to a complex flat connection as $y\to\infty$ should
recover the coefficients of the Jones polynomial of the link $L$ when
$W=S^{3}$ and define a generalization of the Jones polynomial in general.

Constructing solutions to the KW equations satisfying the required asymptotic
conditions when the link $L$ is non-empty is a major obstacle in the above
proposal. As an intermediate step towards constructing solutions Gaiotto and
Witten [GW12] suggested an adiabatic approach to this problem. Consider a
Heegard splitting $W=H_{1}\cup_{\Sigma}H_{2}$ and stretch the metric so that
$W$ contains a long neck $\Sigma\times[-L,L]$. Choosing the splitting
appropriately and taking the limit $L\to\infty$, one would like to understand
solutions to the KW equations on
$M=\Sigma\times\mathbb{R}_{x_{1}}\times\mathbb{R}_{+,y}$ and the link being
$n$ straight lines paralle to the $x_{1}$-axis. If we require that the
solutions are $x_{1}$-invariant, then we obtain the extended Bogomolny
equations (EB).

In [HM20] the authors proved that there exists a bijective correspondence
between solutions to the EB equations and triples
$(\overline{\partial}_{E},\Phi,\mathcal{L})$, where
$(\overline{\partial}_{E},\Phi)$ is a stable Higgs pair on the restriction of
$E$ on $\Sigma$ and $\mathcal{L}\hookrightarrow E$ is a holomorphically
embedded line bundle. This Kobayashi-Hitchin type correspondence is set
theoretic.

In order to state the main theorems of the present paper, we need certain
structural results for the moduli space of Higgs bundles $\mathcal{M}_{H}$. In
[CW19] the authors prove the existence of a Byalinicki-Birula stratification
of $M_{H}$. Each stratum is a holomorphic fibration by holomorphic lagrangian
sub-manifolds $W^{0}(\overline{\partial}_{0},\Phi_{0})$ over a connected
component of the set of complex variations of Hodge structure.When the gauge
group is $\operatorname{SL}(2,\mathbb{C})$ the different strata are indexed by
an even integer in the interval $\\{0,...,2g-2\\}$.

In the adiabatic problem described above, to each parallel straight line
$l_{j}$ we attach a positive integer $k_{j}$ called the magnetic charge. The
points where the $n$ parallel straight lines intersect $\Sigma\times\\{0\\}$
with the associated magnetic charges determine an effective divisor
$\mathcal{D}:=\\{(p_{1},k_{1}),...,(p_{n},k_{n})\\}$. Denote by
$|\mathcal{D}|:=\sum\limits_{j=1}^{n}k_{j}$ the total magnetic charge
associated to the divisor. We call $\mathcal{D}$ even if the total magnetic
charge is even.

######  Theorem 1.1.

Given an even effective divisor $\mathcal{D}$ denote the moduli space of
solutions to the EB equations with divisor $\mathcal{D}$ as
$\underline{\operatorname{EBE}}_{\mathcal{D}}$. We define
$\mathcal{L}:=K(-\mathcal{D})^{1/2}$ where $K$ is the canonical bundle of
$\Sigma$.

  * •

When $|\mathcal{D}|<2g-2$ the moduli space
$\underline{\operatorname{EBE}}_{\mathcal{D}}$ is diffeomorphic to
$W^{0}(\overline{\partial}_{0},\Phi_{0})$ with
$E\cong\mathcal{L}\oplus\mathcal{L}^{\star}$ holomorphically and $\Phi_{0}$
the unique compatible nilpotent Higgs field.

  * •

When $|\mathcal{D}|\geq 2g-2$ let $\mathcal{M}_{\mathcal{L}}$ be the
subvariety of stable bundles inside $\mathcal{M}_{H}$ which are extensions of
the form

$0\rightarrow\mathcal{L}\rightarrow E\rightarrow\mathcal{L}^{\star}\rightarrow
0.$

The moduli space $\underline{\operatorname{EBE}}_{\mathcal{D}}$ is
diffeomorphic to a holomorphic Lagrangian inside $\mathcal{M}_{H}$ which is
topologically the total space of a fiber bundle with affine fibers over
$\mathcal{M}_{\mathcal{L}}$ minus the zero section.

The proof of theorem (1.1) is comprised of two steps. The first step is the
analysis of the kernel of the linearization of the EB equations modulo gauge
transformations. In order to do this we analyse how the generalized Nahm pole
boundary condition [MW17, HM20] affects the elliptic complex construction of
Hitchin’s equations. The second step is the proof that
$\underline{\operatorname{EBE}}_{\mathcal{D}}$ is non-empty for any even
effective divisor $\mathcal{D}$. When $|\mathcal{D}|<2g-2$ this can be proven
directly. In order to treat the general case, we introduce a coordinate-free
definition of the small section condition.

Section 2 contains background material on the moduli space of Higgs bundles.
Section 3 introduces the extended Bogomolny equations. Section 4 introduces
the asymptotic conditions we require our solutions to satisfy. We discuss the
generalized Nahm pole boundary condition and the small section in detail.
Section 5 contains the proof of theorem (1.1). Section 6 discusses
generalizations of the result in this paper.

###  1.1. Acknowledgements

The author would like to thank Rafe Mazzeo and Richard Wentworth for many
illuminating conversations.

##  2\. The geometry of Hitchin’s moduli space

###  2.1. The Higgs bundle moduli space

Let $E\rightarrow\Sigma$ be a rank $2$ complex vector bundle over a Riemann
surface $\Sigma$ of genus $g\geq 2$.

######  Definition 2.1.

An $\operatorname{SL}(2,\mathbb{C})$-Higgs bundle on $\Sigma$ is a pair
$(\overline{\partial}_{E},\Phi)$ such that

  * •

$\overline{\partial}_{E}:\Omega^{0}(E)\rightarrow\Omega^{0,1}(E)$ is a
holomorphic structure on $E$,

  * •

$\Phi:E\rightarrow E\otimes K$ is holomorphic:
$\overline{\partial}_{E}\Phi=0$, and traceless: $\operatorname{Tr}(\Phi)=0$.

The holomorphic section $\Phi$ is called the Higgs field. We fix the notation
$\mathcal{E}:=(E,\overline{\partial}_{E})$ to mean the complex vector bundle
together with a given holomorphic structure. Denote by $\mathcal{H}$ the set
of $\operatorname{SL}(2,\mathbb{C})$-Higgs bundles on $E$. $\mathcal{H}$
admits an action of the gauge group $\operatorname{SL}(2,E)$ given by

$g\cdot(\overline{\partial}_{E},\Phi)=(g^{-1}\circ\overline{\partial}_{E}\circ
g,g^{-1}\circ\Phi\circ g).$

Since the space of holomorphic structures on $E$ is an affine space modelled
on $\Omega^{0,1}(\mathfrak{sl}(E))$, the tangent space to $\mathcal{H}$ at
$(\overline{\partial}_{E},\Phi)$ is given by pairs
$(\beta,\phi)\in\Omega^{0,1}(\mathfrak{sl}(E))\oplus\Omega^{1,0}(\mathfrak{sl}(E))$
such that $\Phi+\phi$ is holomorphic to first order with respect to
$\overline{\partial}_{E}+\beta$. Concretely,

$T_{(\overline{\partial}_{E},\Phi)}\mathcal{H}:=\\{(\beta,\phi)\in\Omega^{0,1}(\mathfrak{sl}(E))\oplus\Omega^{1,0}(\mathfrak{sl}(E))~{}|~{}\overline{\partial}_{E}\phi+[\Phi,\beta]=0\\}.$

The space $\mathcal{H}$ carries a natural complex structure $I$ given by

$I(\beta,\phi)=(i\beta,i\phi),$ (2.1)

and

$\omega_{I}^{\mathbb{C}}((\beta_{1},\phi_{1}),(\beta_{2},\phi_{2}))=i\int\limits_{\Sigma}Tr(\phi_{2}\wedge\beta_{1}-\phi_{1}\wedge\beta_{2})$
(2.2)

defines a holomorphic symplectic form on $\mathcal{H}$ with respect to $I$.

In order to define the Higgs bundle moduli space we want to consider the
quotient of $\mathcal{H}$ by the action of the gauge group
$\operatorname{SL}(2,E)$. In order to obtain a well behaved moduli space we
need to restrict the possible Higgs bundles considered. This leads to the
notion of stability.

######  Definition 2.2.

A Higgs bundle $(\overline{\partial}_{E},\Phi)$ is called

  * •

_semistable_ if for all $\Phi$-invariant line sub-bundles
$\mathcal{L}\subseteq\mathcal{E}$, it holds that $\deg(\mathcal{L})\leq 0$,

  * •

_stable_ if for all $\Phi$-invariant line sub-bundles
$\mathcal{L}\subseteq\mathcal{E}$, it holds that $\deg(\mathcal{L})<0$, and

  * •

_polystable_ if it is semistable and a direct sum of stable Higgs bundles.

Let $\mathcal{H}^{ps}\subseteq\mathcal{H}$ denote the set of polystable Higgs
bundles. We define the Higgs bundle moduli space

$\mathcal{M}_{H}:=\mathcal{H}^{ps}/\operatorname{SL}(2,E).$

It is proven by Hitchin [Hit87] that
$\mathcal{M}^{st}_{H}\subseteq\mathcal{M}_{H}$, the open subset of equivalence
classes of stable Higgs bundles is a smooth manifold of complex dimension
$6g-6$. The complex structure $I$ and symplectic form
$\omega_{I}^{\mathbb{C}}$ are invariant under the action of the gauge group
and therefore give $\mathcal{M}^{st}_{H}$ the structure of a holomorphic
symplectic manifold.

######  Theorem 2.3 (Nonabelian Hodge correspondence).

A Higgs bundle $(\overline{\partial}_{E},\Phi)$ is polystable if and only if
there exists a hermitian metric $h$ on $E$ such that

$F_{(\overline{\partial}_{E},h)}+[\Phi,\Phi^{\star_{h}}]=0,$ (2.3)

where $F_{(\overline{\partial}_{E},h)}$ is the curvature of the Chern
connection $\overline{\partial}_{E}+\partial_{E}^{h}$ associated to the pair
$(\overline{\partial}_{E},h)$ and $\Phi^{\star_{h}}$ is the adjoint of $\Phi$
with respect to $h$. The equation (2.3) is called Hitchin’s equation and the
metric $h$ solving (2.3) is reffered to as the _harmonic_ metric.

Using the harmonic metric $h$ associated to $(\overline{\partial}_{E},\Phi)$
we can define a different complex structure on
$T_{(\overline{\partial}_{E},\Phi)}$ which in terms of $(\beta,\phi)$ can be
written as

$J(\beta,\phi)=(i\phi^{\star_{h}},-i\beta^{\star_{h}}).$

The composition $K=IJ$ defines a third complex structure and it is a theorem
of Hitchin [Hit87] that the three different complex structures give $M_{H}$
the structure of a hyperkähler manifold.

###  2.2. The $\mathbb{C}^{\star}$-action on $\mathcal{M}_{H}$

The Higgs bundle moduli space carries a holomorphic
$\mathbb{C}^{\star}$-action given by

$\xi\cdot[(\overline{\partial}_{E},\Phi)]:=[(\overline{\partial}_{E},\xi\Phi)].$

It follows from the properness of the Hitchin fibration that the limit
$\lim\limits_{\xi\to 0}[(\overline{\partial}_{E},\xi\Phi)]$ always exists in
$\mathcal{M}_{H}$ and is a fixed point of the $\mathbb{C}^{\star}$-action. The
following theorem of Hitchin [Hit87] characterizes fixed points of the action:

######  Theorem 2.4.

A point $[(\overline{\partial}_{E},\Phi)]\in\mathcal{M}_{H}$ is a fixed point
of the $\mathbb{C}^{\star}$-action if either of the following holds:

  * •

$\Phi=0$ and $(E,\overline{\partial}_{E})$ is a stable bundle,

  * •

there is a $C^{\infty}$-splitting $E=\mathcal{L}\oplus\mathcal{L}^{\star}$
with deg$(\mathcal{L})>0$ and with respect to which

$\overline{\partial}_{E}=\begin{pmatrix}\overline{\partial}_{\mathcal{L}}&0\\\
0&\overline{\partial}_{\mathcal{L}^{\star}}\end{pmatrix}~{}\text{and}~{}\Phi=\begin{pmatrix}0&0\\\
\Phi_{1}&0\end{pmatrix},$ (2.4)

where $\Phi_{1}:\mathcal{L}^{\star}\rightarrow\mathcal{L}\otimes K$ is
holomorphic.

######  Definition 2.5.

Given a fixed point of the $\mathbb{C}^{\star}$-action
$[(\overline{\partial}_{0},\Phi_{0})]\in\mathcal{M}_{H}$, the _stable
manifold_ associated to it is defined as

$W^{0}(\overline{\partial}_{0},\Phi_{0}):=\Bigl{\\{}[(\overline{\partial}_{E},\Phi)]\in\mathcal{M}_{H}~{}|~{}\lim\limits_{\xi\to
0}[(\overline{\partial}_{E},\xi\Phi)]=[(\overline{\partial}_{0},\Phi_{0})]\Bigr{\\}}.$

The following theorem on the structure of
$W^{0}(\overline{\partial}_{0},\Phi_{0})$ is proved in [CW19] :

######  Theorem 2.6.

The stable manifold $W^{0}(\overline{\partial}_{0},\Phi_{0})$ is an
$\omega_{I}^{\mathbb{C}}$ holomorphic Lagrangian of $\mathcal{M}_{H}$.

Before we state the next proposition [CW19, Proposition 4.2], let us introduce
some notation. Let

$N_{+}:=Hom(\mathcal{L},\mathcal{L}^{\star})~{},~{}L:=\left(Hom(\mathcal{L},\mathcal{L})\oplus
Hom(\mathcal{L}^{\star},\mathcal{L}^{\star})\right)\cap\mathfrak{sl}(E).$
(2.5)

######  Proposition 2.7.

$[(\overline{\partial}_{E},\Phi)]\in W^{0}(\overline{\partial}_{0},\Phi_{0})$
if and only if after a gauge transformation

$\Phi-\Phi_{0}\in\Omega^{1,0}(L\oplus
N_{+})~{}\text{and}~{}\overline{\partial}_{E}-\overline{\partial}_{0}\in\Omega^{0,1}(N_{+}).$

##  3\. The extended Bogomolny equations

###  3.1. Hermitian geometry

Let $\Sigma$ denote a Riemann surface with local holomorphic coordinate
$z=x_{2}+ix_{3}$ and $E$ an $SU(2)$-bundle over
$\Sigma\times\mathbb{R}_{y}^{+}$. If $A$ is a connection on $E$, $\phi$ an
ad$(E)$-valued $1$-form and $\phi_{1}$ a scalar field, then the extended
Bogomolny equations take the form

$\begin{split}F_{A}-\phi\wedge\phi&=\star\,d_{A}\phi_{1}\\\
\,d_{A}\phi+\star[\phi,\phi_{1}]&=0\\\ \,d_{A}^{\star}\phi&=0.\end{split}$
(3.1)

Here $\star$ is the extension of the Hodge star to
$\Omega^{\star}(\text{ad}(E))$ on $\Sigma\times\mathbb{R}_{+,y}$ equipped with
a fixed product metric $g=g_{0}^{2}|\,dz|^{2}+\,dy^{2}$. In order to do this
we need to choose a Hermitian metric $H$ on $E$ representing the
$\operatorname{SU}(2)$-structure.

It was observed in [Wit11, GW12] that under the assumption that $\phi_{y}=0$,
the extended Bogomolny equations have a Hermitian Yang-Mills structure. The
boundary and asymptotic conditions that we will impose at $y=0$ and
$y\to\infty$ when we study solutions to the EBE force $\phi_{y}=0$ as shown in
[He19]. In order to make the Hermitian Yang-Mills structure apparent, we re-
write the equations (3.1) in terms of the operators

$\begin{split}\mathcal{D}_{1}&=(D_{2}+iD_{3})\,d\overline{z}=(\partial_{x_{2}}+i\partial_{x_{3}}+[A_{2}+iA_{3},\cdot])\,d\overline{z}\\\
\mathcal{D}_{2}&=[\phi_{2}-i\phi_{3},\cdot]\,dz=[\phi_{z},\cdot]\,dz\\\
\mathcal{D}_{3}&=D_{y}-i[\phi_{1},\cdot]=\partial_{y}+[A_{y}-i\phi_{1},\cdot]\\\
\end{split}$ (3.2)

The equations (3.1) take the form

$\begin{split}[\mathcal{D}_{i},\mathcal{D}_{j}]=0,\,i,j=1,\,2,\,&3,\\\
\frac{i}{2}\Lambda\left([\mathcal{D}_{1},\mathcal{D}_{1}^{\dagger_{H}}]+[\mathcal{D}_{2},\mathcal{D}_{2}^{\dagger_{H}}]\right)&+[\mathcal{D}_{3},\mathcal{D}_{3}^{\dagger_{H}}]=0,\end{split}$
(3.3)

where $\Lambda:\Omega^{1,1}\to\Omega^{0}$ is the inner product with the Kähler
form (normalized as $\frac{i}{2}\,dz\wedge\,d\overline{z}$ when the metric on
$\Sigma$ is flat). The adjoints are taken with respect to $H$ and the pairing

$(\alpha,\beta)\rightarrow\int\limits_{\Sigma\times\mathbb{R}^{+}}\alpha\wedge\star\overline{\beta}.$

######  Lemma 3.1.

Assume that the data $(E,\mathcal{D}_{1},\mathcal{D}_{2},\mathcal{D}_{3},H)$
satisfy (3.3). Then so do the data $(E,g^{-1}\circ\mathcal{D}_{1}\circ
g,g^{-1}\circ\mathcal{D}_{2}\circ g,g^{-1}\circ\mathcal{D}_{3}\circ
g,g^{\dagger}Hg)$ .

The above lemma from [Dim22a] shows how to move between different gauges. We
study equations (3.3) under two different gauge fixing conditions, each one
having its own benefits.

The first is called holomorphic gauge. Notice that assuming a solution to
(3.3) is smooth on $\Sigma\times(0,\infty)_{y}$, there exists a gauge
transformation such that $g^{-1}\circ\mathcal{D}_{3}\circ g=\partial_{y}$.
Then the equations
$[\mathcal{D}_{1},\mathcal{D}_{3}]=[\mathcal{D}_{2},\mathcal{D}_{3}]=0$ imply
that in this gauge $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ do not depend on
$y$. Let $E_{y}$ be the restriction of $E$ to the slice $\Sigma\times\\{y\\}$.
$\mathcal{D}_{1}$ in the above gauge is a
$\overline{\partial}_{E_{y}}$-operator satisfying $\mathcal{D}_{1}^{2}=0$ and
therefore by the Newlander-Nirenberg theorem, it defines a holomorphic
structure $\mathcal{E}_{y}$ on $E_{y}$. Now, if we denote the
$K_{\Sigma}$-valued endomorphism $\mathcal{D}_{2}$ by $\Phi_{y}$ on $E_{y}$,
the equation $[\mathcal{D}_{1},\mathcal{D}_{2}]=0$ implies that in this gauge
$(\mathcal{D}_{1},\mathcal{D}_{2},\mathcal{D}_{3})=(\overline{\partial}_{E},\Phi,\partial_{y})$
where the first two entries are the data of a Higgs bundle on $E_{y}$. In this
gauge the last equation in (3.3) takes the form

$F_{(\overline{\partial}_{E},H)}-g_{0}^{2}\partial_{y}(H^{-1}\partial_{y}H)+[\Phi,\Phi^{\star_{H}}]=0.$
(3.4)

######  Remark 3.2.

Notice that Hitchin’s equation (2.3) can be obtained from (3.4) by requiring
that the solution be $y$-independent.

The second gauge that we use is called unitary gauge. Assuming that we have a
solution $(E,\overline{\partial}_{E},\Phi,\partial_{y},H)$ to equations (3.3)
in holomorphic gauge, one can express the metric $H=g^{\dagger}g$ for a
complex gauge transformation. Then $(E,g\circ\overline{\partial}_{E}\circ
g^{-1},g\circ\Phi\circ g^{-1},g\circ\partial_{y}\circ
g^{-1},(g^{\dagger})^{-1}Hg^{-1})$ is also a solution. In this gauge the
connection and fields $(A,\phi,\phi_{1})$ take the following form as functions
of $g$:

$\begin{split}A_{\overline{z}}&=-(\overline{\partial}g)g^{-1},~{}A_{z}=(g^{\dagger})^{-1}\partial
g^{\dagger}\\\ \phi_{z}&=g\varphi
g^{-1},~{}\phi_{\overline{z}}=(g^{\dagger})^{-1}\varphi^{\dagger}g^{\dagger}\\\
A_{y}&=\frac{1}{2}((\partial_{y}g)g^{-1}-(g^{\dagger})^{-1}\partial_{y}g^{\dagger})\\\
\phi_{1}&=\frac{i}{2}((\partial_{y}g)g^{-1}+(g^{\dagger})^{-1}\partial_{y}g^{\dagger}).\end{split}$
(3.5)

It is straightforward that in this gauge the connection and fields are
unitary.

##  4\. Boundary conditions

As explained in the introduction, we want to study solutions to the EBE on
$\Sigma\times\mathbb{R}_{+}$ satisfying certain geometrically motivated
asymptotic conditions at $y=0$ and as $y\to\infty$. We require that the triple
$(A,\phi,\phi_{1})$ converges to a flat irreducible
$\operatorname{SL}(2,\mathbb{C})$-connection as $y\to\infty$ and satisfies the
generalized Nahm pole boundary condition [MW17] as $y\to 0$ . The first
condition is straightforward to analyze. It follows from (2.3) that it is
equivalent to the metric $H$ converging to a solution of Hitchin’s equation
(2.3) as $y\to\infty$ and equivalently to the Higgs bundle
$(\overline{\partial}_{E},\Phi)$ appearing in the holomorphic gauge picture to
be stable. This is also the reason we require the genus of $\Sigma$ to satisfy
$g\geq 2$.

###  4.1. The generalized Nahm pole boundary condition

In order to explain the asymptotic condition we impose at $y=0$, we need to
introduce model solutions to equation (3.4). We start with analyzing the field
$\varphi=\varphi_{z}\,dz$. Working in parallel holomorphic gauge, the
equations
$[\mathcal{D}_{1},\mathcal{D}_{2}]=[\mathcal{D}_{3},\mathcal{D}_{2}]=0$ imply
that the matrix $\varphi_{z}$ has holomorphic entries. We make the simplifying
assumption that this matrix is nilpotent. This implies that there exists a
holomorphic gauge transformation $g$ that takes this matrix to

$\varphi_{z}=\begin{pmatrix}0&P(z)\\\ 0&0\end{pmatrix},$ (4.1)

where $P(z)$ is holomorphic. We restrict to holomorphic functions that have at
most polynomial growth as $r\to\infty$, which by Liouville’s theorem must then
be polynomials. For the model solutions we let $P(z)=z^{k}$ for
$k\in\mathbb{N}$.

To solve (3.4) we also need to choose an ansatz for $H$. The simplest possible
ansatz is given by

$H_{k}=\begin{pmatrix}e^{-u_{k}}&0\\\ 0&e^{u_{k}}\end{pmatrix}.$ (4.2)

Since it is natural to expect that a solution with $P(z)=z^{k}$ be
rotationally symmetric around the axis $z=0$, we assume that $u_{k}$ is a
function of $r:=|z|$ and $y$. The unitary triple corresponding to this data is
given by

$\begin{split}A_{k}&=r\partial_{r}u_{k}(r,y)\begin{pmatrix}\frac{i}{2}&0\\\
0&-\frac{i}{2}\end{pmatrix}\,d\theta\\\
\phi_{z,k}&=e^{-u_{k}}\varphi_{z}=e^{-u_{k}}z^{k}\begin{pmatrix}0&1\\\
0&0\end{pmatrix}\\\
\phi_{1,k}&=\partial_{y}u_{k}(r,y)\begin{pmatrix}\frac{i}{2}&0\\\
0&-\frac{i}{2}\end{pmatrix}.\end{split}$ (4.3)

Inserting these $\varphi_{z}$ and $H$ into (3.4), we obtain the equation

$\left(\frac{\partial^{2}}{\partial x_{2}^{2}}+\frac{\partial^{2}}{\partial
x_{3}^{2}}+\frac{\partial^{2}}{\partial
y^{2}}\right)u_{k}+r^{2k}e^{-2u_{k}}=0.$ (4.4)

We look for solutions $u_{k}$ which are regular on
$\mathbb{C}\times(0,\infty)_{y}$ and which blow up in a specific way as $y\to
0$ away from $r=0$. Let us elaborate on this. Consider the special case where
$k=0$. In this case we consider solutions which only depend on $y$ and hence
need to solve the equation

$u_{0}^{\prime\prime}+e^{-2u_{0}}=0.$ (4.5)

There is a two-parameter family of solutions:

$\begin{split}u_{0}(y)&=\text{log}\left(\frac{\sinh(b(y+c))}{b}\right)\text{~{}when~{}}b\neq
0,\\\ u_{0}(y)&=\text{log}(y+c)\text{~{}when~{}}b=0.\end{split}$ (4.6)

We are looking for solutions which are singular at $y\to 0$ so we set $c=0$.
Regardless of whether $b=0$ or not, the solution $u_{0}(y)\sim\text{log}(y)$
as $y\to 0$.

######  Remark 4.1.

The solutions corresponding to $b\neq 0$ force $\phi_{1}$ to tend to a non-
zero traceless matrix as $y\to\infty$. Solutions with this asymptotic
condition are called real symmetry breaking solutions. Unless $k=0$, the
existence of model solutions of this form has not been rigorously proven. For
interesting predictions and the possible geometric significance of these
solutions see section $2.4$ in [GW12].

To treat the case $k>0$, we must specify the asymptotic conditions for the
model solution. Away from $z=0$, the field

$\varphi_{z}=\begin{pmatrix}0&z^{k}\\\ 0&0\end{pmatrix}$ (4.7)

is gauge equivalent to the field corresponding to $k=0$ through the
holomorphic gauge transformation

$g=\begin{pmatrix}e^{-ik\theta/2}&0\\\ 0&e^{ik\theta/2}\end{pmatrix}.$ (4.8)

Thus, as $y\to 0$ away from $z=0$, we want $u_{k}$ to be gauge equivalent to
$u_{0}$ and hence require the asymptotic condition that
$u_{k}\sim\text{log}(y)$ as $y\to 0$. We also require that $u_{k}$ is smooth
on $\mathbb{C}\times(0,\infty)$. To find an explicit solution, define

$v_{k}=u_{k}-(k+1)\text{log}(r).$ (4.9)

The point of this transformation is to make the differential equation (4.4)
homogeneous of order $-2$. Indeed equation (4.4) takes the form

$\Delta v_{k}+r^{-2}e^{-2v_{k}}=0.$ (4.10)

This equation is scale invariant so it is reasonable to look for solutions
which respect this symmetry. We thus consider $v_{k}$ as a function of
$\sigma=y/r$. Equation (4.4) can be further reduced to

$(\sqrt{\sigma^{2}+1}\,\partial_{\sigma})^{2}v_{k}+e^{-2v_{k}}=0.$ (4.11)

Under the change of variables $\sigma=\sinh(\tau)$ the equation becomes

$v_{k}^{\prime\prime}+e^{-2v_{k}}=0.$ (4.12)

Solutions to this equation which are singular along $y=0$ are given by (4.6)
with $y$ being replaced by $\tau$. However, since we require $u_{k}$ to be
regular away from the boundary, $v_{k}$ must be asymptotic to
$-(k+1)\text{log}(r)$ along the positive $z$-axis. This condition is satisfied
iff $b=k+1$ in which case the solution takes the form

$v_{k}=\text{log}\left(\frac{\sinh((k+1)\tau)}{k+1}\right).$ (4.13)

Going back to $u_{k}$ and the variable $\sigma$, we get the model solution

$e^{u_{k}}=\frac{(\sqrt{r^{2}+y^{2}}+y)^{k+1}-(\sqrt{r^{2}+y^{2}}-y)^{k+1}}{2(k+1)}.$
(4.14)

We are in a position to describe the boundary conditions for a general
solution to (3.3). We give definitions both for the unitary triple
$(A,\phi,\phi_{1})$ and for the metric $H$. In the following, we use the
variable $\psi=\tan^{-1}(r/y)$.

######  Definition 4.2.

The unitary triple $(A,\phi,\phi_{1})$ satisfies the Nahm pole boundary
condition with knot singularity of charge $k$ at
$(p,0)\in\mathbb{C}\times\mathbb{R}^{+}$ if, in some gauge,

$(A,\phi_{z},\phi_{1})=(A_{k},\phi_{z,k},\phi_{1,k})+\mathcal{O}(\rho^{-1+\epsilon}\psi^{-1+\epsilon})$
(4.15)

for some $\epsilon>0$.

We say that a hermitian metric $H$ satisfies the Nahm pole boundary condition
with knot singularity of charge $k$ at
$(p,0)\in\mathbb{C}\times\mathbb{R}^{+}$ if there exists a section $s\in
i\mathfrak{su}(E,H_{k})$ such that $H=H_{k}e^{s}$ and $|s|+|y\,ds|\leq
C\rho^{\epsilon}\psi^{\epsilon}$ for some $\epsilon>0$.

###  4.2. The holomorphic line bundle

Suppose that the triple $(A,\phi_{z},\phi_{1})$ satisfies the extended
Bogomolny equations (3.3) and the asymptotic conditions described above. We
further require the set of points $(p,0)\in\Sigma\times\mathbb{R}^{+}$ at
which the solution is up to local holomorphic gauge asymptotic to a positive
charge model solution to be finite. We denote the collection of positively
charged points together with their multiplicities as $\mathcal{D}$. Near each
point on the boundary not in $\mathcal{D}$ there exists a local homolorphic
frame in which

$\phi_{1,0}=\frac{1}{2y}\begin{pmatrix}i&0\\\
0&-i\end{pmatrix}+\mathcal{O}(\rho^{-1+\epsilon}\psi^{-1+\epsilon}).$ (4.16)

Let $s_{1}$ and $s_{2}$ be local holomorphic coordinates for the bundle $E$ so
that $s_{1}$ corresponds to $(1,0)^{\dagger}$ and $s_{2}$ to
$(0,1)^{\dagger}$. Consider a generic section $s=a_{1}s_{1}+a_{2}s_{2}$ of the
bundle $E|_{y=1}$ and parallel transport it using the operator
$\mathcal{D}_{3}$. Then the equation
$\mathcal{D}_{3}s=\partial_{y}s-i\phi_{1}s=0$ implies that as $y\to 0$

$s(y)=(a_{1}y^{-1/2}+\mathcal{O}(y^{-1/2+\epsilon}))s_{1}+(a_{2}y^{1/2}+\mathcal{O}(y^{1/2+\epsilon}))s_{2}.$
(4.17)

Notice that the section $s_{2}$ is special since its parallel transport
vanishes like $y^{1/2}$ as $y\to 0$, whereas the parallel transport of a
generic section actually blows up like $y^{-1/2}$. This is called the small
section and we obtain an invariant description of the vanishing line bundle

$\mathcal{L}:=\\{s\in\Gamma(E)|_{(\Sigma-\mathcal{D})\times\mathbb{R}_{+}}:\mathcal{D}_{3}s=0,\lim\limits_{y\to
0}|y^{-1/2+a}s|=0\\}$ (4.18)

for all $0<a<1$. Since $\mathcal{L}$ is locally spanned by the holomorphic
section $s_{2}$ it is holomorphic. Notice that this line bundle is defined
away from the divisor $\mathcal{D}$. In order to show that it does in fact
extend over these points, repeat the same argument as above with $\phi_{1,k}$
in position of $\phi_{1,0}$. Let $(r_{j},y)$ be local cylindrical coordinates
around each of the positively charged points and
$\psi_{j}:=\tan^{-1}(r_{j}/y)$.

It turns out that around each positively charged point $p_{j}$, while a
generic section blows up like $\psi_{j}^{-1/2}$ when parallel transported
using $\mathcal{D}_{3}$, there is a distinguished section which in fact
vanishes like $\psi_{j}^{1/2}$. Here $\psi_{j}$ is just the polar angle at
$p_{j}$. Thus, we can obtain a globally defined holomorphic line sub-bundle in
the following way. Let $U_{\mathcal{D}}$ be a union of open half balls
containing $\mathcal{D}$. Then

$\begin{split}\mathcal{L}:=\\{s\in\Gamma(E):\mathcal{D}_{3}s=0,&\lim\limits_{y\to
0}|y^{-1/2+a}s|=0~{}\text{on}~{}\Sigma\times\mathbb{R}_{+}-U_{\mathcal{D}}\\\
\text{and}~{}&\lim\limits_{\psi_{j}\to
0}|\psi_{j}^{-1/2+a}s|=0~{}\text{on}~{}U_{\mathcal{D}}\\}.\end{split}$ (4.19)

##  5\. Moduli structure of solutions to EBE on $\Sigma\times\mathbb{R}_{+}$

###  5.1. The tangent space

We start by briefly recalling the work of [HM20]. In this paper, the authors
classify solutions to the extended Bogomolny equations on
$\Sigma\times\mathbb{R}_{+}$ with the asymptotic conditions of the previous
section, namely

  * •

As $y\to 0$ the solution should satisfy the GNPB condition for a given divisor
$\mathcal{D}\subseteq\Sigma\times\\{0\\}$,

  * •

As $y\to\infty$ the solution should converge to a $y$-independent solution of
(3.3).

Working in parallel holomorphic gauge, requiring that the solution converge to
a $y$-independent solution as $y\to\infty$ implies that the metric $H$ should
converge to a solution of the equation

$F_{(\overline{\partial}_{E},H)}+[\Phi,\Phi^{\star_{H}}]=0$ (5.1)

which is Hitchin’s equation (2.3). Denoting the _set_ of solutions satisfying
the above asymptotics $\underline{\operatorname{EBE}}$, the main theorem in
[HM20] can be stated as follows:

######  Theorem 5.1.

There exists a bijective correspondence between
$\underline{\operatorname{EBE}}$ and holomorphic triples
$(\overline{\partial}_{E},\Phi,\mathcal{L})$ where
$(\overline{\partial}_{E},\Phi)$ is a stable Higgs pair and
$\mathcal{L}\hookrightarrow E$ is a holomorphically embedded line bundle.

######  Remark 5.2.

The divisor $\mathcal{D}$ can be extracted from the holomorphic data as
follows. Consider the holomorphic map

$1\wedge\Phi:\mathcal{L}\oplus\mathcal{L}\otimes
K^{-1}\xrightarrow{1\oplus\Phi}\mathcal{E}\xrightarrow{\text{det}}\mathcal{O}.$
(5.2)

This map descends to a map

$F:\mathcal{L}^{2}\otimes K^{-1}\rightarrow\mathcal{O},$ (5.3)

and thus gives rise to a holomorphic section of $K\otimes\mathcal{L}^{-2}$.
The zeroes and multiplicities of this section specify the divisor
$\mathcal{D}$.

######  Remark 5.3.

The bijective correspondence proven in [HM20] is a _set_ correspondence.
Although the question is implicit in the paper’s introduction, the authors did
not investigate the _moduli_ behavior of solutions to the extended Bogomolny
equations.

Motivated by the adiabatic approach to the Kapustin-Witten equations explained
in the introduction of [HM20] we restrict ourselves to
$\underline{\operatorname{EBE}}_{\mathcal{D}}$, the set of solutions to EBE
with the asymptotic behavior described above and with fixed divisor
$\mathcal{D}$. We want to linearize (3.3) around a solution
$(\mathcal{D}_{1},\mathcal{D}_{2},\mathcal{D}_{3})$ and understand the kernel
of the linearization modulo gauge transformations. Let

$\begin{split}\widetilde{\mathcal{D}_{1}}&=\mathcal{D}_{1}+\epsilon b,\\\
\widetilde{\mathcal{D}_{2}}&=\mathcal{D}_{2}+\epsilon\phi,\\\
\widetilde{\mathcal{D}_{3}}&=\mathcal{D}_{3}+\epsilon\phi_{1},\\\ \end{split}$
(5.4)

where $b\in\Omega^{0,1}(\mathfrak{g}_{E})$,
$\phi\in\Omega^{1,0}(\mathfrak{g}_{E})$ and
$\phi_{1}\in\Omega^{0}(\mathfrak{g}_{E})$. Since we are considering the
linearization up to gauge transformations, we may work in the gauge where
$\phi_{1}=0$. Assuming this condition, we have removed some of the gauge
freedom but not all of it. In order to see this, let $H$ be the solution
metric corresponding to the solution
$(\mathcal{D}_{1},\mathcal{D}_{2},\mathcal{D}_{3})$ of (3.3). Write
$H=g^{\dagger}g$. Then, we know that we can express

$\begin{split}\mathcal{D}_{1}&=g\circ\overline{\partial}_{E}\circ g^{-1}\\\
\mathcal{D}_{2}&=g\circ\Phi\circ g^{-1}\\\
\mathcal{D}_{3}&=g\circ\partial_{y}\circ g^{-1},\end{split}$ (5.5)

where $(\overline{\partial}_{E},\Phi)$ is the stable Higgs pair associated to
the solution $(\mathcal{D}_{1},\mathcal{D}_{2},\mathcal{D}_{3})$ of (3.3)
coming from the $y\to\infty$ asymptotic condition. For any $y$-independent
gauge transformation $g_{1}$, the gauge transformation $gg_{1}g^{-1}$
preserves $\mathcal{D}_{3}=\widetilde{\mathcal{D}_{3}}$ and therefore we also
need to consider quotients by these gauge transfomrations. Notice that the
gauge condition $\phi_{1}=0$ implies that the linearizations of equations
$[\mathcal{D}_{1},\mathcal{D}_{3}]=0$ and
$[\mathcal{D}_{2},\mathcal{D}_{3}]=0$ are given by

$\begin{split}\mathcal{D}_{3}b&=0\\\ \mathcal{D}_{3}\phi&=0,\end{split}$ (5.6)

while the linearization of the other two equations is gauge equivalent to the
linearization of Hitchin’s self duality equations [Hit87]. The extra gauge
freedom is precisely the gauge freedom that appears in Hitchin’s paper
[Hit87]. The equations (5.6) imply that $\beta:=g\circ b\circ g^{-1}$ and
$\varphi:=g\circ\phi\circ g^{-1}$ are $y$-independent and the previous
observation implies that the kernel of the last two equations after
quotienting by gauge transformations can be identified with the tangent space
of the Hitchin moduli space at the stable pair
$(\overline{\partial}_{E},\Phi)$.

We are not done yet. We have not investigated the conditions that we need to
impose so that these deformations preserve the divisor $\mathcal{D}$. This is
the essential step. In order for this to happen, it must hold that $b,\phi$
blow up at most like $\mathcal{O}(y^{-1+\epsilon})$ as $y\to 0$.

We know that fixing $\mathcal{D}$ implies that $\mathcal{L}\cong
K(-\mathcal{D})^{1/2}$ and that $E\cong\mathcal{L}\oplus\mathcal{L}^{\star}$
smoothly but not necessarily holomorphically. In a neighborhood of a point at
the boundary we can always choose a local holomorphic frame so that
$(1,0)^{\dagger}$ represents a section of $\mathcal{L}$ and $(0,1)^{\dagger}$
represents a section of $\mathcal{L}^{\star}$. In these local holomorphic
coordinates

$\overline{\partial}_{E}=\begin{pmatrix}\overline{\partial}_{\mathcal{L}}&\beta_{1}\\\
0&\overline{\partial}_{\mathcal{L}^{\star}}\end{pmatrix}$ (5.7)

and the $y$-independent field is given by

$\varphi=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}.$ (5.8)

We also know for example from [HM20] that in this coordinate system

$g=\begin{pmatrix}e^{-\frac{u_{k}}{2}}&0\\\
0&e^{\frac{u_{k}}{2}}\end{pmatrix}+\mathcal{O}(y^{-1/2+\epsilon})$ (5.9)

where $k=0$ away from points of $\mathcal{D}$ and $k=k_{j}$ at
$p_{j}\in\mathcal{D}$. From the formula $\phi=g^{-1}\circ\varphi\circ g$ we
see that the only way in which $\phi=\mathcal{O}(y^{-1+\epsilon})$ as $y\to 0$
is if $c=0$ identically in (5.8). Since the holomorphic embedding
$\mathcal{L}\hookrightarrow E$ is fixed when we fix $\mathcal{D}$,

$\beta=\begin{pmatrix}0&\beta_{2}\\\ 0&0\end{pmatrix}$ (5.10)

in the local holomorphic coordinates we have introduced. Therefore we have
proven that the tangent space of $\underline{EBE}_{\mathcal{D}}$ at the point

$(\mathcal{D}_{1},\mathcal{D}_{2},\mathcal{D}_{3})=(g\circ\overline{\partial}_{E}\circ
g^{-1},g\circ\Phi\circ g^{-1},g\circ\partial_{y}\circ g^{-1})$

is given by the subspace of the tangent space to
$(\overline{\partial}_{E},\Phi)$ inside the Hitchin space which is defined as
those pairs $(\beta,\varphi)$ inside the tangent space which up to gauge
belong to

$\Omega^{0,1}(N_{+})\oplus\Omega^{1,0}(L\oplus N_{+}).$

We show that this subspace is a holomorphic Lagrangian subspace of the tangent
space. To do this, start with a stable Higgs pair given by

$\overline{\partial}_{E}=\begin{pmatrix}\overline{\partial}_{\mathcal{L}}&b\\\
0&\overline{\partial}_{\mathcal{L}^{\star}}\end{pmatrix}~{}\text{and}~{}\Phi_{0}=\begin{pmatrix}\Phi_{2}&\Phi_{3}\\\
\Phi_{1}&\Phi_{4}\end{pmatrix}.$ (5.11)

Define the operators

$D^{\prime\prime}:=\overline{\partial}_{E}+\Phi,~{}D^{\prime}:=\partial^{H}_{E}+\Phi^{\star_{H}}.$
(5.12)

Consider the cochain complex $C_{+}(\overline{\partial}_{E},\Phi)$

$\Omega^{0}(N_{+})\xrightarrow{D^{\prime\prime}}\Omega^{0,1}(N_{+})\oplus\Omega^{1,0}(L\oplus
N_{+})\xrightarrow{D^{\prime\prime}}\Omega^{1,1}(L\oplus N_{+}).$ (5.13)

We claim that $H^{1}(C_{+}(\overline{\partial}_{E},\Phi))$ of this complex,
which we have shown above to be the tangent space to a solution of
$\underline{EBE}_{\mathcal{D}}$, is of dimension $6g-6$. The proof is
essentially lemma $3.6$ from [CW19]. In that paper the authors only use this
in the case where $\deg\mathcal{L}>0$ but the proof works in the general case.
Re-write the complex in the form

$\not{D}:=(D^{\prime\prime})^{\star}+D^{\prime\prime}:\Omega^{0,1}(N_{+})\oplus\Omega^{1,0}(L\oplus
N_{+})\rightarrow\Omega^{0}(N_{+})\oplus\Omega^{1,1}(L\oplus N_{+}).$ (5.14)

Since $(\overline{\partial}_{E},\Phi)$ is stable,
$H^{0}(C_{+}(\overline{\partial}_{E},\Phi))=H^{2}(C_{+}(\overline{\partial}_{E},\Phi))=0$.
Therefore, dim$H^{1}(C_{+}(\overline{\partial}_{E},\Phi))=$ index $\not{D}$.
Deforming the Higgs field to zero does not change the index but it decouples
the operators. Thus

$\text{index }\not{D}=\text{index }\overline{\partial}_{0}-\text{index
}\overline{\partial}_{1},$

where $\overline{\partial}_{0}$ is the $\overline{\partial}$-operator on
sections of $(L\oplus N_{+})\otimes K$ induced by $\overline{\partial}_{E}$
and $\overline{\partial}_{1}$ is similarly defined for sections of $N_{+}$.
Since deg$L=0$, by Riemann-Roch:

$\begin{split}\text{index }\overline{\partial}_{0}=\deg N_{+}+(\text{rank
}L+\text{rank }N_{+})(g-1)\\\ \text{index }\overline{\partial}_{1}=\deg
N_{+}-(\text{rank }N_{+})(g-1).\end{split}$

Since

$\text{rank }L+2\text{rank }N_{+}=\text{rank }\mathfrak{sl}(2,E)=6,$

index $\not{D}=6g-6$ as wanted. Finally the fact that this subspace is
holomorphic and isotropic follow immediately from plugging $(\beta,\varphi)$
of the above form into the expressions (2.1) and (2.2).

###  5.2. Holomorphic Lagrangians

In the previous subsection we showed that the tangent space at a solution
inside $\underline{\operatorname{EBE}}_{\mathcal{D}}$ is in a natural way a
holomorphic Lagrangian subspace of the tangent space to the stable Higgs pair
associated to the solution. The aim of this section is to prove two
statements. First, when the degree of the line bundle $\mathcal{L}$ is
positive, we show that the stable Higgs pairs associated to the elements of
$\underline{\operatorname{EBE}}_{\mathcal{D}}$ are exactly the points of the
stable manifold $W^{0}(\overline{\partial}_{0},\Phi_{0})$ for a uniquely
determined $\mathbb{C}^{\star}$-fixed point
$(\overline{\partial}_{0},\Phi_{0})$. Furthermore, the tangent space at each
point $[(\overline{\partial}_{E},\Phi)]\in
W^{0}(\overline{\partial}_{0},\Phi_{0})$ is exactly the tangent space to the
solution in $\underline{\operatorname{EBE}}_{\mathcal{D}}$ at the point
determined by $[(\overline{\partial}_{E},\Phi)]$. This proves that as long as
$|\mathcal{D}|<2g-2$, the moduli space
$\underline{\operatorname{EBE}}_{\mathcal{D}}$ is diffeomorphic to one of the
stable manifolds $W^{0}(\overline{\partial}_{0},\Phi_{0})$.

When the degree of $\mathcal{L}$ is non-positive, we show that the moduli
spaces are diffeomorphic to certain holomorphic Lagrangians inside the Hitchin
moduli space which however are no longer stable submanifolds. From the
previous section we know that, assuming the existence of a point
$[(\overline{\partial}_{E},\Phi)]$ inside the Hitchin moduli space together
with a holomorphic embedding $\mathcal{L}\hookrightarrow E$ such that the
zeroes and multiplicities of the section $1\wedge\Phi$ of
$K\otimes\mathcal{L}^{-2}$ are given by $\mathcal{D}$, the set of points with
this property must be a holomorphic Lagrangian submanifold of the Hitchin
moduli space.

We start with the $\deg\mathcal{L}>0$ case.

######  Theorem 5.4.

Assume that the degree of the holomorphic line subbundle in the effective
triple $(\overline{\partial}_{E},\Phi,\mathcal{L})$ is strictly positive. Then
the set of stable Higgs pairs $(\overline{\partial}_{E},\Phi)$ which preserve
the map $F$ are precisely those in the stable manifold
$W^{0}(\overline{\partial}_{0},\Phi_{0})$ where

$\overline{\partial}_{0}=\begin{pmatrix}\overline{\partial}_{\mathcal{L}}&0\\\
0&\overline{\partial}_{\mathcal{L}^{\star}}\end{pmatrix}~{}\text{and}~{}\Phi_{0}=\begin{pmatrix}0&0\\\
\Phi_{1}&0\end{pmatrix}.$ (5.15)

Here $\Phi_{1}$ is specified by the map $F$ completely.

######  Remark 5.5.

The latter statement follows easily by writing the map $F$ in local
holomorphic coordinates. Indeed $\mathcal{L}$ is locally spanned by
$(1,0)^{\dagger}$ and thus $1\wedge\Phi_{0}$ is locally written as

$\begin{pmatrix}1\\\ 0\end{pmatrix}\wedge\begin{pmatrix}0&0\\\
\Phi_{1}&0\end{pmatrix}\begin{pmatrix}1\\\ 0\end{pmatrix}=\Phi_{1}.$

###### Proof.

The first step is to notice that $\mathcal{L}$ is uniquely determined by the
map $F$. Indeed, the map $F$ is equivalent to a holomorphic section of the
line bundle $K\otimes\mathcal{L}^{-2}$. This identifies
$K\otimes\mathcal{L}^{-2}\cong\mathcal{O}(D)$ where
$D=\sum\limits_{j=1}^{n}k_{j}p_{j}$ is the divisor given by the knot points
$p_{j}$ and their multiplicities $k_{j}$. Thus $\mathcal{L}\cong K(-D)^{1/2}$.

The next observation is that if $(\overline{\partial}_{E},\Phi,\mathcal{L})$
is an effective triple corresponding to the divisor $D$, then so is the whole
$\mathbb{C}^{\star}$-family $(\overline{\partial}_{E},\xi\Phi,\mathcal{L})$.
Indeed multiplying the field by $\xi$ does not affect the zeroes and
multiplicities. Now, it is clear that the $\mathbb{C}^{\star}$-fixed point
(5.16) is an effective triple with divisor $D$. From proposition (2.7) it
holds that any stable Higgs pair in the stable manifold
$W^{0}(\overline{\partial}_{0},\Phi_{0})$ up to gauge transformation is of the
form

$\overline{\partial}_{0}=\begin{pmatrix}\overline{\partial}_{\mathcal{L}}&\alpha\\\
0&\overline{\partial}_{\mathcal{L}^{\star}}\end{pmatrix}~{}\text{and}~{}\Phi_{0}=\begin{pmatrix}\Phi_{2}&\Phi_{3}\\\
\Phi_{1}&-\Phi_{2}\end{pmatrix}.$ (5.16)

This in particular implies that $\mathcal{L}$ is still a holomorphic line
subbundle of $\mathcal{E}$ and that in local holomorphic coordinates $F$ is
still given by $\Phi_{1}$. Therefore it is indeed true that any element
$[(\overline{\partial}_{E},\Phi)]\in W^{0}(\overline{\partial}_{0},\Phi_{0})$
satisfies that $(\overline{\partial}_{E},\Phi,\mathcal{L})$ is an effective
triple corresponding to the divisor $D$. Clearly, the above proof also implies
that these are all the effective triples corresponding to $D$.

∎

Next, we focus on the case when $\deg\mathcal{L}\leq 0$. In this case, we
cannot apply the structure theory from [CW19] to produce an explicit point
inside the Hitchin moduli space giving rise to a solution in
$\underline{\operatorname{EBE}}_{\mathcal{D}}$. What we will do is derive an
invariant characterisation of the section $1\wedge\Phi$ of
$K\otimes\mathcal{L}^{-2}$ which will ensure the existence of such points. In
fact, we prove more:

######  Theorem 5.6.

The moduli spaces $\underline{\operatorname{EBE}}_{\mathcal{D}}$ are always
non-empty. The holomorphic Lagrangians inside the Hitchin space diffeomorphic
to these spaces have the following structure. Let $M_{\mathcal{L}}$ denote the
subvariety of stable bundles inside $M_{H}$ which are extensions of the form

$0\rightarrow\mathcal{L}\rightarrow E\rightarrow\mathcal{L}^{\star}\rightarrow
0.$ (5.17)

The holomorphic lagrangians diffeomorphic to $\underline{\operatorname{EBE}}$
are total spaces of fiber bundles with affine fibers over $M_{\mathcal{L}}$
minus the zero section.

######  Remark 5.7.

The reason we do not include the zero section is because when the Higgs field
$\Phi=0$, the section $1\wedge\Phi$ of $K\otimes\mathcal{L}^{-2}$ does not
make sense. It is not clear if the holomorphic lagrangian stays smooth if one
adds the zero section or not.

######  Remark 5.8.

The above characterization of the moduli spaces
$\underline{\operatorname{EBE}}_{\mathcal{D}}$ seems to suggest that when
$\deg\mathcal{L}\leq 0$ the moduli spaces are no longer topologically trivial.

######  Conjecture 5.9.

Let $M_{st}$ denote the moduli space of stable bundles. In [Hit87], Hitchin
showed that the co-tangent bundle $T^{\star}M_{st}$ can be embedded into
$M_{H}$ and $M_{H}$ can be thought of as a partial compactification of
$T^{\star}M_{st}$. Let $i:T^{\star}M_{st}\hookrightarrow M_{H}$ denote the
embedding. We conjecture that the pullback of the holomorphic lagrangians when
$\deg\mathcal{L}\leq 0$ is given by the co-normal bundle of $M_{\mathcal{L}}$
inside $T^{\star}M_{st}$ endowed with the natural symplectic structure.

###### Proof.

The non-trivial part of the proof is to show that the moduli spaces are non-
empty. We start with the following exact diagram

---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\text{End}\,\mathcal{E})_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}^{\star}\otimes\mathcal{E}^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}\otimes\mathcal{E}^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{E}\otimes\mathcal{E}^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$tr$\textstyle{\mathcal{L}^{\star}\otimes\mathcal{E}^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}\otimes\mathcal{L}^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{\Sigma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$
(5.18)

Here $\mathcal{E}$ denotes the vector bundle $E$ together with its complex
structure. The second line of the diagram is given by the short exact sequence
(5.17) tensored with $\mathcal{E}^{\star}$. The proof of the exactness of this
diagram is standard. Notice that since $\deg\,\mathcal{E}=0$,
$\mathcal{E}\cong\mathcal{E}^{\star}$. Tensoring the short exact sequence
(5.17) with $K\otimes\mathcal{L}^{-1}$, using the aforementioned isomorphism
and the exactness of the above commutative diagram we have

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}^{2}\otimes
K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\text{End}\,\mathcal{E})_{0}\otimes
K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}^{\star}\otimes\mathcal{E}^{\star}\otimes
K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}^{\star}\otimes\mathcal{E}\otimes
K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\otimes\mathcal{L}^{-2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$
(5.19)

Therefore, we have constructed a holomorphic surjective map from
$(\text{End}\,\mathcal{E})_{0}\otimes K$ to $K\otimes\mathcal{L}^{-2}$.
Writing everything in local holomorphic coordinates it is easy to see that
this is the map sending the Higgs field $\Phi$ to the holomorphic section
$1\wedge\Phi$. The surjectivity of the map therefore implies that for any
holomorphic section of $K\otimes\mathcal{L}^{-2}$ there exists holomorphic
data producing this section and therefore the moduli spaces are non-empty as
wanted.

∎

##  6\. Further directions

The result of this paper is for the gauge group
$\operatorname{SL}(2,\mathbb{C})$. Both [CW19] and [HM20] strongly suggest
that similar results should be true when the gauge group is
$\operatorname{SL}(n,\mathbb{C})$. One significant difficulty in this more
general setting is that the fixed points of the $\mathbb{C}^{\star}$-action
have more complicated structure. At a $\mathbb{C}^{\star}$-fixed point $E\cong
E_{1}\oplus...\oplus E_{k}$ holomorphically where each $E_{i}$ is a stable
bundle. Unless each $E_{i}$ is a line bundle, one needs to modify the
generalized Nahm pole boundary condition in order to construct solutions to
the EB equations from such holomorphic data. In order to do this one needs to
obtain model solutions like the ones described in section (4) where the Higgs
field (4.1) can now be an arbitrary strictly upper triangular $n\times n$
matrix with holomorphic entries which grow at most polynomially. Unlike the
case studied in [Mik12], it is not clear whether in this more general setting
one can find a solution using a diagonal ansatz for the hermitian metric as in
(4.2). Even in the situation when each $E_{i}$ is a line bundle, it is not
completely clear how to generalize theorem (1.1) unless there exists a Higgs
field $\Phi$ such that $E\cong E_{1}\oplus...\oplus E_{k}$ with its induced
holomorphic structure from the splitting together with $\Phi$ form a fixed
point of the $\mathbb{C}^{\star}$-action. In this case it is straightforward
to generalize theorem (5.4).

One can ask the above question for arbitrary gauge group. In [Mik12] certain
conjectures are made as to the existence and structure of model solutions in
this general setting. The equivalent of [CW19] does not exist for more general
groups.

A natural generalization of this paper is to allow the divisor $\mathcal{D}$
to vary and try to describe the moduli structure of this larger space of
solutions. As long as the number of points and respective magnetic charges do
not change, so that no points collide or separate, one can pull back by a
diffeomorphism of $\Sigma$ so that the divisor stays fixed. One then wants to
study the moduli space of the EB equations with fixed divisor but with varying
metric on $\Sigma$. In this paper we have proven that given a metric on
$\Sigma$, we can associate to the triple $(\Sigma,g,\mathcal{D})$ a moduli
space which is diffeomorphic to a holomorphic lagrangian inside the moduli of
Higgs bundles $\mathcal{M}_{H}$ with complex structure (2.1) coming from the
complex structure of $\Sigma$. Following the steps of the proof of theorem
(1.1) it is true that the moduli space
$\underline{\operatorname{EBE}}_{\mathcal{D}}$ only depends on the metric up
to conformal equivalence. Therefore, it is natural to expect that the moduli
spaces $\underline{\operatorname{EBE}}_{\mathcal{D}}$ form a fibre bundle over
the Teichmüller space associated to $\Sigma$. In fact this fibration should
descend to a fibration over the Riemann moduli space $\mathcal{M}_{g,0}$. In
[BDD20] such a fibration is constructed when $|\mathcal{D}|=0$ and it is shown
to extend to the Deligne-Mumford compactification of $\mathcal{M}_{g,0}$. We
conjecture that the same properties should hold for any divisor with
$|\mathcal{D}|<2g-2$. This could have applications to both knot theory and to
the asymptotic geometry of $\mathcal{M}_{H}$.

A final way in which one can generalize the results of this paper is to work
with the twisted Bogomolny equations (TBE) [HM19, Dim22b]. Model solutions
have been constructed in [Dim22b] and the methods of [HM20] should apply to
this setting. It is expected that when $|\mathcal{D}|<2g-2$ the associated
holomorphic lagrangians are diffeomorphic to the spaces $W_{\alpha}^{\lambda}$
described in [CW19]. It follows from [CW19] that $W_{\alpha}^{\lambda}$ is
diffeomorphic to $W_{\alpha}^{0}$ described above. When $|\mathcal{D}|\geq
2g-2$ it seems to be the case that the moduli space
$\underline{\operatorname{TBE}}_{\mathcal{D}}$ is no longer diffeomorphic to
$\underline{\operatorname{EBE}}_{\mathcal{D}}$. This is because while the
$\mathbb{C}^{\star}$ limit of a stable Higgs pair with stable underlying
bundle has zero Higgs field and therefore the small section is not defined,
this is no longer the case in the setting of the twisted Bogomolny equations
[GW12, Section 3.3].

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