# The SO(3) Vortex Equations over Orbifold Riemann Surfaces

Mariano Echeverria

###### Abstract.

We study the general properties of the moduli spaces of $SO(3)$ vortices over
orbifold Riemann surfaces and use these to characterize the solutions of the
$SO(3)$ monopole equations on Seifert manifolds following in the footsteps of
Mrowka, Ozsv th and Yu.

We also study the solutions to the $SO(3)$ monopole equations on
$S^{1}\times\varSigma$ in order to motivate the construction of a version of
monopole Floer homology, which we call framed monopole Floer homology, in
analogy with the construction given by Kronheimer and Mrowka for the case of
instanton Floer homology.

Finally, the $SO(3)$ vortex moduli spaces provide a nice toy model for recent
work due to Feehan and Leness regarding the study of a natural Morse-Bott
function on the moduli space of $SO(3)$ monopoles over Kahler manifolds. In
particular, we compute the Morse-Bott indices of this function.

##  1\. Introduction

###  1.1. $SO(3)$ vortices and Morse-Bott Theory

The study of vortices on Riemann surfaces has been so fruitful since the work
of Jaffe and Taubes [49] that it would be entirely reasonable to assume that
on this topic there is nothing new under the Sun. Indeed, the vortex equations
have been studied and generalized in multiple directions (see for example [20,
19, 18, 44, 90, 77, 21, 12, 93, 17]), and they have proved incredibly useful
in 3- and 4-manifold topology due to its interaction with the Seiberg-Witten
equations [94, 85, 86, 87]. However, by scraping the bottom of the barrel one
can still find some new things as we now explain.

First of all, an $SO(3)$ vortex consists of a pair $(C,\varUpsilon)$, where
$C$ is a unitary connection on a $U(2)$ bundle $E$ over a Riemann surface
$\varSigma$, and $\varUpsilon$ is a section of $E$. The connection $C$ must
induce a predetermined connection $C^{\det}$ on $\det E$, which is why we are
really studying the $SO(3)$ vortex equations instead of the $U(2)$ vortex
equations. The equations $(C,\varUpsilon)$ must satisfy are the equations
(13), that is,

$\displaystyle*F_{C}^{0}-i\left[\varUpsilon\varUpsilon^{*}-\frac{1}{2}|\varUpsilon|^{2}I_{E}\right]=0$
$\displaystyle\bar{\partial}_{C}\varUpsilon=0$

The meaning of these equations is explained in Section (3), but they probably
already look familiar to most readers. By studying the solutions to the
$SO(3)$ vortex equations modulo gauge transformations one obtains the $SO(3)$
vortex moduli space $\mathcal{M}(\varSigma,E)$.

The study of these equations has a long history. For example, in his PhD
thesis (see also [43, Section 5]), Garc a-Prada studied the moduli space of
non-abelian vortices from a gauge theoretic and algebraic geometric point of
view. From the latter perspective, these moduli spaces can be identified with
the moduli spaces of stable pairs, and Garc a-Prada interpreted stable pairs
of arbitrary rank on any compact Kahler manifold $X$ as stable bundles on
$X\times\mathbb{CP}^{1}$, which allowed him to encode the stability parameter
of the moduli space of stable pairs in terms of the stability condition in the
sense of Mumford of the corresponding stable bundle.

Around the same time, Bradlow and Daskalopoulous [19, 16] studied the moduli
spaces of stable pairs for the case of a Riemann surface from a more analytic
point of view, and a bit later the moduli spaces for the rank two case were
studied by Thaddeus [90] and Bertram [11] from the algebraic geometric point
of view.

The case of the $U(2)$ vortex equations was also studied extensively from a
gauge-theoretic perspective by Bradlow and Daskalopoulous [19, 16], as well as
Bradlow, Daskalopoulous and Wentworth [19]. More importantly, the moduli space
of $SO(3)$ vortices for Riemann surfaces without marked points is a particular
case of the general framework developed in the paper by Garc a-Prada, Gothen
and Mundet i Riera [45].

For our purposes it was important to work with the more general setup of
orbifold Riemann surfaces, using the ideas of Furuta and Steer [42] for the
case of the Yang-Mills equations (which is the orbifold version of Atiyah and
Bott’s seminal work [6]). We use gauge-theoretic techniques instead of
geometric invariant theory (GIT), which is also used to study these moduli
spaces from the algebraic geometric point of view. In fact, Bertram describes
briefly the construction of the moduli space of stable parabolic pairs [11,
Section 3] (using GIT techniques).

One reason for studying the moduli space of $SO(3)$ vortices on Riemann
surfaces is that they provide useful toy models complementing recent work of
Feehan and Leness, who studied the $SO(3)$ monopole moduli spaces over Kahler
surfaces from a Morse-theoretic point of view. Following the work of Hitchin
[48], Feehan and Leness [34] studied the $L^{2}$ norm of the spinor of an
$SO(3)$ monopole as a Morse-Bott function on the moduli space of $SO(3)$
monopoles over Kahler manifolds, which in particular means they had to compute
the Morse-Bott indices of the critical sets of this function. We do the
analogous computations for $SO(3)$ vortices over a Riemann surfaces. Since we
are also finding these formulas for the case of orbifolds, the work [70] by
Nasatyr and Steer also served as an important model. We now explain how the
computation of these indices is done.

As mentioned earlier, $\mathcal{M}(\varSigma,E)$ is obtained by studying the
solutions to the $SO(3)$ vortex equations modulo gauge. In this case, the
gauge group used is the determinant-one gauge group $\mathcal{G}^{\det}(E)$,
which is not the gauge group of all unitary automorphisms of $E$. The upshot
of doing this is that there is a residual symmetry on the moduli space of
$SO(3)$ vortices modulo $\mathcal{G}^{\det}(E)$.

More precisely, there is a circle action on the moduli space obtained by
rescaling the section $\varUpsilon$, in other words, $\varUpsilon\rightarrow
e^{i\theta}\varUpsilon$. Associated to this circle action, there is a moment
map $\mu$ which is essentially the $L^{2}$ norm of the section, that is,
$\mu(C,\varUpsilon)=\frac{1}{2}\|\varUpsilon\|_{L^{2}(\varSigma)}^{2}$.
Following Bradlow, Daskalopoulous and Wentworth [19], the idea is to study
$\mu$ as a Morse-Bott function on the $SO(3)$ vortex moduli space (they
studied this for the $U(2)$ vortex moduli space for the case of a smooth
Riemann surface, i.e, without marked points).

The critical sets of $\mu$ can be identified with the fixed points of the
circle action, which consists of the moduli space of projectively flat
connections on $\varSigma$, and certain moduli spaces of abelian $U(1)$
vortices on $\varSigma$. The abelian vortices satisfy equations (20), which
read

$\displaystyle*F_{C_{L}}-\frac{i}{2}|\alpha|^{2}=*\frac{1}{2}F_{C^{\det}}$
$\displaystyle\bar{\partial}_{C_{L}}\alpha=0$

Here $E$ has reduced as $E=L\oplus(L^{*}\otimes\det E)$, and $C_{L}$ is a
connection on $L$, while $\alpha$ is a section of $L$. Via an easy Chern-Weil
argument (still valid in the orbifold case, see lemma (20)) we must have
$c_{1}(L)\leq\frac{1}{2}c_{1}(E)$, where in the orbifold case these Chern
classes are in general rational cohomology classes, and Poincar duality is
used to regard these as rational numbers. Any $SO(3)$ vortex which is not a
projectively flat connection nor an abelian vortex will be called an
_irreducible_ _$SO(3)$ vortex_, following the standard terminology in gauge
theory.

In any case, a theorem due to Frankel [40] shows that the indices of these
critical sets equals the dimension of the subspace on which the circle action
acts with negative weight. Feehan and Leness recently [34] computed these
indices for the analogue of $\mu$ in the case of $SO(3)$ monopoles over a
Kahler surface. In section Theorem (35) we compute these indices for the case
of an orbifold Riemann surface.

Before giving the general formula for the index, we remark that if our Riemann
surface is regarded as an orbifold Riemann surface we then write
$\check{\varSigma}=(\varSigma,(p_{1},a_{1}),\cdots,(p_{n},a_{n}))$, where the
marked points $p_{1},\cdots,p_{n}$ have been assigned positive integers
$a_{1},\cdots,a_{n}$. Moreover, an orbifold $U(2)$ bundle $\check{E}$ over
$\check{\varSigma}$ carries some _isotropy data,_ which consists of integers
$b_{i}^{\pm}$ at each marked points satisfying certain conditions which we
recall in the next section.

######  Theorem 1.

General Properties of the $SO(3)$ vortex moduli spaces and indices of the
Morse-Bott function $\mu$:

a) Suppose that $\check{E}$ is an orbifold $U(2)$ bundle over
$\check{\varSigma}=(\varSigma,(p_{1},a_{1}),\cdots,(p_{n},a_{n}))$ with
isotropy data $(b_{1}^{\pm},\cdots,b_{n}^{\pm})$. Then if
$[C,\varUpsilon]\in\mathcal{M}(\check{\varSigma},\check{E})$ is an irreducible
$SO(3)$ vortex, the moduli space is smooth at this point, and of dimension

$\dim_{\mathbb{R}}\mathcal{M}(\check{\varSigma},\check{E})=2\left(g-1+c_{1}(\det\check{E})+n-n_{0}-\sum_{i=1}^{n}\frac{b_{i}^{-}+b_{i}^{+}}{a_{i}}\right)$

where $n_{0}=\\#\\{i\mid b_{i}^{-}=b_{i}^{+}\\}$.

b) There is an $S^{1}$ action on $\mathcal{M}(\check{\varSigma},\check{E})$
whose fixed point set consists of the moduli space of projectively flat
connections on $\text{ad}\check{E}$ and the moduli spaces of abelian orbifold
vortices associated to the reductions
$\check{E}=\check{L}\oplus(\check{L}^{*}\otimes\det\check{E})$, with
$c_{1}(\check{L})\leq\frac{1}{2}c_{1}(\check{E})$.

If one assumes that $E$ is of odd degree in the smooth case or $\check{E}$ is
an odd power of the fundamental orbifold line bundle $\check{L}_{0}$ in the
case where the $a_{i}$ are mutually coprime, and also that

$c_{1}(\check{E})>2c_{1}(K_{\check{\varSigma}})=2\left(2g-2+n-\sum_{i=1}^{n}\frac{1}{a_{i}}\right)$

where $K_{\check{\varSigma}}$ represents the canonical orbifold line bundle,
then $\mathcal{M}(\check{\varSigma},\check{E})$ is also smooth at the
reducible $SO(3)$ vortices. In fact,
$\mathcal{M}(\check{\varSigma},\check{E})$ is a smooth closed Kahler manifold.

In particular, the function $\mu$ is a Morse-Bott function on
$\mathcal{M}(\check{\varSigma},\check{E})$ and if the orbifold line bundle
$\check{L}$ has isotropy $b_{i}$, the index of abelian vortex associated to
the reduction $\check{E}=\check{L}\oplus(\check{L}^{*}\otimes\det\check{E})$
is

$\text{ind}(\check{E},\check{L})=2\left[g-1+c_{1}(\det\check{E})-2c_{1}(\check{L})+\sum_{i\mid\epsilon_{i}=1}\frac{b_{i}^{+}-b_{i}^{-}}{a_{i}}+n_{-}+\sum_{i\mid\epsilon_{i}=-1}\frac{b_{i}^{-}-b_{i}^{+}}{a_{i}}\right]$

where $\epsilon_{i}=1$ if $b_{i}=b_{i}^{+}$, $\epsilon_{i}=-1$ if
$b_{i}=b_{i}^{-}$. Here $n_{-}=\\#\\{i\mid b_{i}=b_{i}^{-}<b_{i}^{+}\\}$.

######  Remark 2.

a) The proof of the previous theorem is the main content of Sections 3, 4 and
5.

b) We point out that in general some assumption on the degree of $\check{E}$
seems necessary. For example, the cokernel of the linearized $SO(3)$ vortex
map at a projectively flat connection can basically be identified with
$\text{coker}\bar{\partial}_{C}\simeq H^{1}(\check{E})$ and the only way to
guarantee the vanishing of $H^{1}(\check{E})$ for a semistable bundle
$\check{E}$ is by taking $c_{1}(\check{E})$ sufficiently large.

c) Also, $c_{1}(\check{L})\leq\frac{1}{2}c_{1}(\check{E})$ is a necessary but
not sufficient condition for the appearance of a moduli space of abelian
vortices. One also needs for $\check{L}$ to have the correct isotropy data and
for the background degree of $\check{L}$ to be non-negative. We explain this
terminology in Section (2) and illustrate this with the examples we do at the
end of the paper.

d) An easy way to guarantee that the moduli space
$\mathcal{M}^{*}(\check{\varSigma},\check{E})$ of irreducible $SO(3)$ vortices
is empty is by knowing that there are no orbifold line bundles $\check{L}$
compatible with the isotropy data of $\check{E}$ (this is more common that one
might expect as the examples in Section (8) illustrate). The reason for this
is that the function $\mu$ must achieve an absolute maximum on
$\mathcal{M}(\check{\varSigma},\check{E})$, but if
$\mathcal{M}^{*}(\check{\varSigma},\check{E})\subset\mathcal{M}(\check{\varSigma},\check{E})$
is non-empty then the maximum must occur at an abelian vortex.

e) Finally, $c_{1}(\check{E})>2c_{1}(K_{\check{\varSigma}})$ does guarantee in
the smooth case that the dimension of the moduli space of $SO(3)$ vortices is
positive [except when $g=0$]. Likewise, we will see in Section (8) examples
where this condition is not enough to guarantee positive dimensional moduli
spaces.

In the smooth case (no marked points on the Riemann surface), the formula is
quite clean and simply reads

$\text{ind}(E,L)=2(g-1+\deg E-2\deg L)$

As an example of what one can do with the formula for the index, choosing
$\deg L=0$ we see that the index is $2(g-1+\deg E)$, which is the dimension
formula for the moduli space of $SO(3)$ vortices in the smooth case. This
means that this moduli space of abelian vortices (which in fact ends up being
a single point), corresponds to the maximum for the function $\mu$.

Strictly speaking, these indices can be computed using the orbifold version of
Riemann-Roch even in the case where we are not imposing conditions on
$\check{E}$ which guarantee that the moduli spaces are smooth at the reducible
points, so they are best interpreted (in the terminology of Feehan and Leness)
as _virtual_ Morse-Bott indices.

###  1.2. Framed Monopole Floer Homology and $SO(3)$ monopoles on
$S^{1}\times\varSigma$

A second motivation for studying the moduli space of $SO(3)$ vortices over
Riemann surfaces which could be of more interest to low-dimensional
topologists is the conjecture due to Kronheimer and Mrowka relating the framed
instanton homology $HI^{\\#}(Y)$ of a 3 manifold and the tilde version
$\widetilde{HM}(Y)$ of monopole Floer homology (equivalently, the hat version
of Heegaard Floer homology $\widehat{HF}(Y)$). More precisely, a special case
of [56, Conjecture 7.24] states the following:

######  Conjecture 3.

[56, Conjecture 7.24] Let $Y$ denote a closed oriented 3-manifold. Then

$HI^{\\#}(Y)\simeq\widehat{HF}(Y)\otimes\mathbb{C}\simeq\widetilde{HM}(Y)\otimes\mathbb{C}$

A definition of
$\widetilde{HM}(Y)=\bigoplus_{\mathfrak{s}\in\text{Spin}^{c}(Y)}\widetilde{HM}(Y,\mathfrak{s})$
can be found in [15, 88] and by the work of Kutluhan, Lee and Taubes [57], it
is isomorphic to
$\widehat{HF}(Y)=\bigoplus_{\mathfrak{s}\in\text{Spin}^{c}(Y)}\widehat{HF}(Y,\mathfrak{s})$.
The Euler characteristic of these groups are

$\chi(\widetilde{HM}(Y))=\begin{cases}|H_{1}(Y,\mathbb{Z})|&b_{1}(Y)=0\\\
0&b_{1}(Y)\neq 0\end{cases}$

That $HI^{\\#}(Y)$ has the same Euler characteristic as $\widetilde{HM}(Y)$
was shown by Scaduto in [80, Corollary 1.4], and the isomorphism between the
groups has been verified for certain cases in [80, 9, 8].

The equality up to a sign between the Euler characteristics in the more
general case of the sutured versions of monopole, Heegaard and instanton Floer
homologies has been shown recently by Zhenkun Li and Fan Ye in their paper
[59, Theorem 1.2].

At this point it is useful to recall how $HI^{\\#}(Y)$ is defined. One first
considers the ordinary instanton Floer homology $HI(Y\\#T^{3},\gamma)$, where
$\gamma$ represents the Poincar dual for an admissible $U(2)$ bundle $E$ over
$Y\\#T^{3}$ (more precisely, $\gamma$ is a loop which represents one of the
circle factors of $T^{3}$).

The group $HI(Y\\#T^{3},\gamma)$ is $\mathbb{Z}/8\mathbb{Z}$ graded, and
$HI^{\\#}(Y)$ is isomorphic to four consecutive summands of
$HI(Y\\#T^{3},\gamma)$. In the special case of $Y=S^{3}$, it then follows that
$HI^{\\#}(S^{3})$ corresponds to four summands of $HI(T^{3},\gamma)$, which is
two dimensional since it has two generators and no differential, so
$HI^{\\#}(S^{3})$ ends up being one-dimensional [54, Section 4.1].

Notice that the Euler characteristic of these groups satisfies

(1)
$\chi(HI^{\\#}(Y))=\frac{1}{2}\chi(HI(Y\\#T^{3},\gamma))=\frac{1}{2}\chi(CI(Y\\#T^{3},\gamma))$

where $CI(Y\\#T^{3},\gamma)$ denotes the chain complex used to define the
Floer groups.

On the other hand, $\widetilde{HM}(Y)$ can be defined in terms of a mapping
cone construction which uses the monopole Floer chain complexes on $Y$ [15,
57]. For us, the important fact is that $\widetilde{HM}(Y)$ is defined in
terms of data on $Y$, without using additional 3-manifolds like $T^{3}$ as in
the case of $HI^{\\#}(Y)$.

Thus, in order to try to compare $HI^{\\#}(Y)$ and $\widetilde{HM}(Y)$ it may
be convenient to use an alternative construction of $\widetilde{HM}(Y)$ so
that it is also defined on $Y\\#T^{3}$. As communicated to the author by Tom
Mrowka, this alternative construction of $\widetilde{HM}(Y)$ was known to him
and Peter Kronheimer, and probably to the experts at large as well.

We motivate the construction in Section (7), but basically we consider
$HM(Y\\#T^{3},\omega,\varGamma)$, that is, the monopole Floer homology on
$Y\\#T^{3}$ associated to a suitable non-balanced perturbation and a local
coefficient system $\varGamma$, which is needed in order to obtain a well
defined differential. We call this version of monopole Floer homology _framed
monopole Floer homology_ , and denote it as $HM^{\\#}(Y)$ in analogy to the
notation for the case of instantons.

It may not be clear why study $HM(Y\\#T^{3},\omega,\varGamma)$ in a paper that
is about $SO(3)$ vortices over Riemann surfaces, but the main idea is that if
we think of $T^{3}$ as $S^{1}\times T^{2}$, then the perturbation term
$\omega$ appears naturally from how the abelian vortices embed in the moduli
space of $SO(3)$ vortices on $T^{2}$.

To be more specific, we first need to know how the solutions to the $SO(3)$
monopole equations on $S^{1}\times\varSigma$ compare to the $SO(3)$ vortices
on $\varSigma$. In Section (7) we prove (for the Seiberg-Witten case see [66,
68]):

######  Theorem 4.

Suppose that $(B,\varPsi)$ is an irreducible $SO(3)$ monopole for the spin-u
structure

$V=(\mathbb{C}\oplus K_{\varSigma}^{-1})\otimes E$

where $E$ is the pullback of a $U(2)$ bundle on $\varSigma$ under the obvious
projection map $S^{1}\times\varSigma\rightarrow\varSigma$. Write

$\varPsi=\alpha\oplus\beta\in\varGamma(E)\oplus\varGamma(K_{\varSigma}^{-1}\otimes
E)$

Then either $\beta$ vanishes identically or $\alpha$ vanishes identically.

a) The solutions with $\beta\equiv 0$ can be identified with the irreducible
$SO(3)$ vortices for $(\varSigma,E)$.

b) The solutions with $\alpha\equiv 0$ can be identified (via Serre duality)
with the irreducible $SO(3)$ vortices for
$(\varSigma,K_{\varSigma}^{-1}\otimes E)$.

Moreover, if we assume that $c_{1}(E)>2c_{1}(K_{\varSigma})$, then only
solutions of type a) can occur.

Returning to our discussion of framed monopole homology, choose the $U(2)$
bundle $E$ over $T^{2}$ with $c_{1}(E)=1$. The expected dimension of the
moduli space of $SO(3)$ vortices is two, so after dividing by the circle
action we get a manifold which is topologically an interval, the maximum of
this interval corresponding to the unique abelian vortex inside this moduli
space, and the minimum of this interval corresponding to the unique
projectively flat connection on $\text{ad}E$ 111Recall that irreducible
projectively flat connections on Riemann surfaces correspond to stable
bundles, and in the case of an elliptic curve these were classified by Atiyah
[5]. For the point of view of algebraic geometry, we are looking at the moduli
space of stable bundles with a fixed determinant, which is why we get a single
element..

This forces $HM(T^{3},\omega)$ to be one dimensional (since Seiberg-Witten
monopoles on $S^{1}\times\varSigma$ are in bijection with abelian vortices on
$\varSigma$). Therefore, we can think of the moduli space of $SO(3)$ vortices
on $T^{2}$ as providing a _canonical_ cobordism between the unique abelian
vortex (which generates $HM^{\\#}(S^{3})=HM(T^{3},\omega)$), and the unique
projectively flat connection on $T^{2}$ (which can be taken as a generator for
$HI^{\\#}(S^{3})$).

Thus, from this point of view, the isomorphism between $HM^{\\#}(S^{3})$ and
$HI^{\\#}(S^{3})$ can be understood as a consequence of the fact that there is
a _canonical_ cobordism between the unique generator for $HM^{\\#}(S^{3})$ and
the unique generator for $HI^{\\#}(S^{3})$. We should also point out that here
we haven’t perturbed the $SO(3)$ monopole equations on $S^{1}\times\varSigma$,
since otherwise the moduli space would be zero dimensional, instead of an
interval as we have in our situation.

We should also note that the chain complex $CM^{\\#}(Y)=CM(Y\\#T^{3},\omega)$
which yields $HM(Y\\#T^{3},\omega,\varGamma)$ is finitely generated, thus the
Euler characteristic of $HM^{\\#}(Y)$ is well defined with respect to the
Novikov field implicit in our choice of $\varGamma$. In particular,

(2) $\chi(HM^{\\#}(Y))=\chi(CM^{\\#}(Y))$

This immediately begs the questions:

1) How is $\chi(HM^{\\#}(Y))$ related to $\chi(\widetilde{HM}(Y))$?

2) How is $\chi(HM^{\\#}(Y))$ related to $\chi(HI^{\\#}(Y))$?

Regarding the first question, a K nneth formula proven by Kutluhan, Lee and
Taubes in [57] relates $CM(Y\\#T^{3},\omega,\varGamma)$ to a mapping cone
$S_{U}(CM(Y\sqcup T^{3}),\omega,\varGamma))$. Since the right hand side still
involves a local coefficient system, this is still not quite isomorphic to
$\widetilde{HM}(Y)$ (which is defined over integer coefficients for example),
but rather a twisted version $\widetilde{HM}(Y,\varGamma)$ (see the discussion
preceding our theorem (48) for more details). But the K nneth formula implies
that

(3) $\chi(HM^{\\#}(Y))=\chi(\widetilde{HM}(Y))$

Regarding the second question, thanks to work in progress by Aleksander Doan
and Chris Gerig [25], which is based on ideas of Kronheimer and Mrowka, after
appropriate choices of orientations

$\chi(CM(Y\\#T^{3},\omega))=\frac{1}{2}\chi(CI(Y\\#T^{3},\gamma))$

so combining (1), (2) and (3) we obtain

$\chi(\widetilde{HM}(Y))=\chi(HM^{\\#}(Y))=\chi(HI^{\\#}(Y))$

The interesting feature of the idea of Kronheimer and Mrowka used by Doan and
Gerig is that the equality between both Euler characteristics is obtained by
studying the $SO(3)$ monopole equations on the $3$-manifold, so in a sense is
a low dimensional version of Witten’s conjecture. One could then try to see if
this can also be used for relating the Floer homologies $HM^{\\#}(Y)$ and
$HI^{\\#}(Y)$, but that is outside the scope of this work.

Alternatively, in order to relate the Euler characteristics one can use the
fact that for 3-manifolds with first Betti number the Euler characteristic of
the instanton Floer homology groups is (up to a sign) twice the Casson-Walker-
Lescop invariant $\lambda_{CWL}(Y)$ [79, Theorem 1.1], and again up to a sign
$\lambda_{CWL}(Y)$ equals the sum of the Turaev torsion functions on the
3-manifold [92], which are known to agree with the Seiberg-Witten invariant
[65].

###  1.3. $SO(3)$ monopoles on Seifert manifolds

Finally, the fact that we studied the $SO(3)$ vortices over orbifold Riemann
surfaces allows us to describe the solutions to the $SO(3)$ monopole equations
on Seifert manifolds in terms of the $SO(3)$ vortex moduli spaces, which is
the analogue of the computations done by Mrowka, Ozsv th and Yu [67] for the
case of the Seiberg-Witten equations.

Just as in their paper, one must work with connections which are spinorial
with respect to a metric connection compatible with the Seifert structure
(which is not the Levi-Civita connection). This connection can be thought of
as an adiabatic connection [74]. In Section (8) we find:

######  Theorem 5.

Suppose that $\pi:Y=S(\check{L})\rightarrow\check{\varSigma}$ is a Seifert
manifold arising as the unit circle bundle of an orbifold line bundle
$\check{L}\rightarrow\check{\varSigma}$. Let $(B,\varPsi)$ denote an
irreducible $SO(3)$ monopole for the spin-u structure

$V=(\mathbb{C}\oplus\pi^{*}(K_{\check{\varSigma}}^{-1}))\otimes\pi^{*}(\check{E})$

where $\check{E}$ is an $U(2)$ bundle over $\varSigma$. Write

$\varPsi=\alpha\oplus\beta\in\varGamma(\pi^{*}(\check{E}))\oplus\varGamma(\pi^{*}(K_{\check{\varSigma}}^{-1})\otimes\pi^{*}(\check{E}))$

Then either $\beta$ vanishes identically or $\alpha$ vanishes identically.

a) The solutions with $\beta\equiv 0$ can be identified with the irreducible
$SO(3)$ vortices for $(\check{\varSigma},\check{E}^{\prime})$. Here
$\check{E}^{\prime}$ is any $U(2)$ bundle over $\check{\varSigma}$ satisfying
$\det\check{E}^{\prime}\simeq\det\check{E}$.

b) The solutions with $\alpha\equiv 0$ can be identified (via Serre duality)
with the irreducible $SO(3)$ vortices for
$(\varSigma,K_{\varSigma}^{-1}\otimes\check{E}^{\prime})$, where
$\check{E}^{\prime}$ has the same meaning as in part a).

Moreover, if we assume that $c_{1}(\check{E})>2c_{1}(K_{\check{\varSigma}})$,
then only solutions of type a) can occur.

######  Remark 6.

a) We should point out that if one were interested in relating the flat
$SU(2)$ or $SO(3)$ connections on $Y=S(\check{L})$ with solutions to the
Seiberg-Witten equations on $Y$, then the right $U(2)$ bundles to look at are
those which satisfy $\det\check{E}\simeq\check{L}^{k}$ for some integer $k$,
since these are the ones Furuta and Steer used to relate the projectively flat
connections on $Y$ with solutions to the Yang-Mills equations on
$\check{\varSigma}$ [42, Theorem 3.7]

b) At the end of Section (8) we work out the example of the Poincar homology
sphere. In this case one can choose $\det\check{E}\simeq\check{L}_{0}$ where
$\check{L}_{0}$ satisfies $c_{1}(\check{L}_{0})=\frac{1}{2\cdot 3\cdot
5}=\frac{1}{30}$ and generates the topological Picard group for orbifold lines
bundles over $S^{2}(2,3,5)$.

There are six $U(2)$ bundles $\check{E}^{\prime}$ such that
$\det\check{E}^{\prime}\simeq\check{L}_{0}$, but only two end up being
interesting: call these $\check{E}_{1}$ and $\check{E}_{2}$. Each of these
contain one projectively flat connection.

The expected dimension of the moduli space of $SO(3)$ vortices for
$\check{E}_{1}$ and $\check{E}_{2}$ is $2$ and $0$ respectively, so after
taking the quotient by the $S^{1}$ action, it is clear that the second one
must be empty (in the sense that there are no irreducible $SO(3)$ vortices),
while the first one becomes one dimensional, thus providing a cobordism
between the unique abelian vortex and the unique projectively flat connection
on the bundle $\check{E}_{1}$.

So on the Poincar homology sphere, the moduli space of $SO(3)$ vortices
consists of:

i) The trivial $SU(2)$ connection,

ii) Two irreducible $SU(2)$ flat connections

iii) One Seiberg-Witten monopole,

iv) A one-dimensional moduli space of irreducible $SO(3)$ monopoles which
serves as a cobordism between the Seiberg-Witten monopole and one of the
irreducible flat connections.

It is interesting to compare the case of the Poincar homology sphere with that
of $T^{3}$, since in both situations we have a one dimensional moduli of
$SO(3)$ vortices connecting one of the flat connections which the unique
Seiberg-Witten monopole.

In fact, thanks to the work of Taubes [84], it is known that the Casson
invariant $\lambda_{C}(Y)$ [4] is morally $(1/2)$ the count of irreducible
flat $SU(2)$ connections on $Y$.

On the other hand, as Lim showed [60] (verifying a conjecture due to
Kronheimer), on an integer homology sphere $Y$, the count of irreducible
Seiberg-Witten monopoles, while not a topological invariant, can be made to
agree with $\lambda_{C}(Y)$ after adding a suitable correction term.

So morally there are twice as many flat connections as there are Seiberg-
Witten monopoles, and we find it is amusing to give examples where this
“principle” holds on the nose.

This also suggests studying the monopole and instanton Floer homologies of the
connected sum $Y\\#\varSigma(2,3,5)$, in analogy to studying the monopole and
instanton Floer homologies of $Y\\#T^{3}$.

We also describe the case of the Brieskorn homology sphere $\varSigma(2,3,7)$
near the end of the paper. The computations are slightly more complicated, but
one finds a similar picture to that of $\varSigma(2,3,5)$.

We finish this introduction by mentioning that in follow-up work we plan to
study the flowlines for the $SO(3)$ Chern-Simons-Dirac functional on Seifert
manifolds and $S^{1}\times\varSigma$ in terms of the geometry of the moduli
space of stable pairs on natural (orbifold) ruled surfaces associated to these
3-manifolds [47, 67, 78].

_Outline of the paper:_

In Section 2 we give a basic review of gauge theory on orbifolds. In many ways
this material is already common knowledge, so this section is basically
intended to fix some notation. If one is willing to invest some time in the
algebraic geometry literature, it is not difficult to find very general
formulas for the Hirzebruch-Grothendieck-Riemann-Roch index theorem on
orbifolds (or even stacks!). However, usually these formulas are not fleshed
out explicitly for the situations we have in mind, so we also work out some of
them in this section.

Section 3 introduces the $SO(3)$ vortex equations over an orbifold Riemann
surface. The discussion is pretty standard, and include things like the
deformation theory of the moduli spaces, the conditions under which abelian
vortices appear in the moduli space, and the Kahler structure on the moduli
spaces. The fact that we are working over an orbifold only means we need to
keep track of more topological data, but besides that it is almost
indistinguishable from the smooth case. We are also quite explicit when
describing the different structures on the moduli spaces, since many
simplifications occur from the fact that we are in dimension two.

Section 4 discusses the well-known correspondence between $SO(3)$ vortices and
stable pairs. Our use of the stable pairs point of view is very limited in
this work, and we mostly wrote it to clarify certain things in case the reader
is more acquainted with the $U(2)$ vortex equations.

In Section 5 we study the Morse-Bott function $\mu$ on the moduli space of
$SO(3)$ vortices. This is essentially the version for Riemann surfaces of the
work of Feehan and Leness [34], or the classical paper by Hitchin but now for
the case of $SO(3)$ vortices [48].

Section 6 describes some basic facts about the $SO(3)$ monopole equations on
3-manifolds. This material is an adaptation of [37], but we found it useful to
work it out explicitly because the reader may not be familiarized with certain
features of the $SO(3)$ monopole equations. We also describe briefly the
$U(2)$ monopole equations on 3-manifolds, which could be of interest for
technical reasons which we discuss in this section.

In Section 7 we describe the $SO(3)$ monopole equations on 3-manifolds of the
form $S^{1}\times\varSigma$, where $\varSigma$ is a Riemann surface. Our main
interest is the case when $\varSigma$ is the 2-torus $T^{2}$, since this
example is what motivated our definition of the _framed monopole Floer
homology_ of $Y$. We give its definition in this section, and discuss some of
the properties mentioned in the introduction.

Finally, in Section 8 we describe the behavior of the $SO(3)$ monopole
equations for Seifert manifolds and discuss the examples of $\varSigma(2,3,5)$
and $\varSigma(2,3,7)$.

######  Acknowledgement.

The idea to study the $SO(3)$ vortex equations on Riemann surfaces arose from
many helpful conversations with Paul Feehan and Tom Leness surrounding their
project [34] so the author would like to thank them for their encouragement
and many valuable suggestions. The author would also like to thank Yi-Jen Lee
for pointing out the K nneth formula in her paper [57] with Kutluhan and
Taubes, as well as Chris Woodward, Andrei Teleman, Oscar Garc a-Prada, Michael
Thaddeus, Dinesh Valluri, Aleksander Doan, Tom Mrowka and Matthew Stoffregen
for useful conversations and correspondence.

##  2\. A CrashCourse on Orbifolds

We now give a brief summary of gauge theory on orbifolds, which in the modern
literature (e.g [1, 3]) falls into the theory of analytic Deligne-Mumford
stacks.

For our purposes, the main arena will be that of a Riemann surface $\varSigma$
with finitely many marked points $p_{1},\cdots,p_{n}$. Since this material is
standard, we refer to [72, 67, 42, 71, 70, 23, 82, 83, 51, 53, 74, 75, 73, 14,
13, 2, 1, 58, 50] for more details and proofs of the main results we need.

We will follow the conventions of Kronheimer and Mrowka [54], and use a
$\check{}$ whenever we want to emphasize the orbifold structure of a geometric
object. For example, $\check{\varSigma}$ will denote an orbifold Riemann
surface, while $\varSigma$ will denote the underlying topological space (i.e,
we forget the marked points). So we can think of $\check{\varSigma}$ as a
shorthand notation for the data $(\varSigma,p_{1},\cdots,p_{n})$, with the
isotropy at each of the marked points (which will be recalled soon), being
implicit.

In general, a complex orbifold $\check{X}$ consists of a connected paracompact
complex space such that every point $p$ has an open neighborhood $U_{p}$ which
is of the form $U_{p}\simeq V_{p}/G_{p}$, where $V_{p}$ is a suitable complex
manifold and $G_{p}$ a finite group acting biholomorphically on $V_{p}$.

The germ $(\check{X},p)$ of an orbifold at $p$ is called the quotient germ
[14, Definition 1.1]. If $\dim_{\mathbb{C}}\check{X}=m$, we can assume that
$G_{p}$ is a finite subgroup of $GL_{m}(\mathbb{C})$, unique up to
conjugation, and $V_{p}$ is an open neighborhood of $0\in\mathbb{C}^{m}$ such
that $g(V_{p})=V_{p}$ for all $g\in G_{p}$. We can think of the quotient germ
$(\check{X},p)$ as $(\check{X},p)=(\mathbb{C}^{m},0)/G_{p}$ and the map
$\pi:(\mathbb{\mathbb{C}}^{m},0)\rightarrow(\check{X},p)$ is called the local
smoothing covering of $\check{X}$ at $p$.

We will assume that there are finitely many points $p_{1},\cdots,p_{n}$ of
$\check{X}$ where $G_{p}$ is non-trivial. In fact, we can assume that
$G_{p_{i}}\simeq\mathbb{Z}/a_{i}\mathbb{Z}$, where the action is
$z\ni\mathbb{C}^{m}\rightarrow e^{2\pi i/a_{i}}z$. In this case the natural
orbifold metric $\check{g}$ on $\check{X}$ is one with a conical singularity
at each $p_{i}$ of cone angle $2\pi/a_{i}$. Integration on orbifolds can also
be defined using a partition of unity, and the orbifold version of the de Rham
complex yields cohomology groups isomorphic to those of the underlying
topological space [58, Section 2.1]. In other words,

$H^{k}(\check{X};\mathbb{R})\simeq H^{k}(X;\mathbb{R})\simeq
H_{DR}^{k}(X;\mathbb{R}),\;\;\;k=0,1,\cdots,\dim_{\mathbb{R}}X$

with a similar statement for $\mathbb{C}$ instead of $\mathbb{R}$. Moreover,
the cup product of orbifold de Rham classes is induced by the wedge product.
By the same token, any element of $H^{k}(\check{X};\mathbb{R})$ can be
represented by a unique orbifold harmonic $k$ form.

Orbi-bundles can be defined in an analogous way to how we defined orbifolds.
Namely, we have the data $\pi:\check{E}\rightarrow\check{X}$ , where both
$\check{E},\check{X}$ are orbifolds and $\pi$ is a holomorphic surjective map.
If $F$ is a complex manifold, $\pi$ has fibre $F$ if for every $p\in\check{X}$
there is a local smoothing covering $U_{p}=V_{p}/G_{p}$ such that there is a
$G_{p}$ action on $V_{p}\times G_{p}$ with
$\check{E}\mid_{U_{p}}\simeq(V_{p}\times F)/G_{p}$ over $U_{p}$.

In general the fibres of $\pi$ are orbifolds, namely, $\pi^{-1}(p)=F/G_{p}$
for some action of $G_{p}$ on $F$. In particular, notice that the projection
$\pi$ is not required to be locally trivial, as opposed to the usual vector
bundle situation. Regardless, it still makes sense to talk about connections
on vector orbi-bundles [58, Section 2.2].

For the particular case of an orbifold line bundle
$\check{L}\rightarrow\check{X}$, its tensor powers $\check{L}^{\otimes q}$
will continue to be orbifold line bundles. In fact, one can take $q$
sufficiently large so that $\check{L}^{\otimes q}$ becomes locally trivial, in
which case one can define the orbifold Chern class of $\check{L}$ as

$c_{1}(\check{L})\equiv\frac{1}{q}c_{1}(\check{L}^{\otimes q})\in
H^{2}(\check{X};\mathbb{Q})$

Notice that in general this will be a class in the second cohomology of the
orbifold with _rational coefficients._ Alternatively, one can also use Chern-
Weil theory in order to define $c_{1}(\check{L})$ as

$c_{1}(\check{L})=\left[\frac{i}{2\pi}F_{\check{\nabla}}\right]$

where $[\cdot]$ denotes the orbifold de Rham class defined by the curvature of
an orbifold connection $\check{\nabla}$ on $\check{L}$.

For higher rank orbi-bundles $\check{E}$, the Chern classes $c_{i}(\check{E})$
can be defined in a similar way [14, Section 2]. Moreover, there is a natural
notion of an orbifold Euler characteristic of $\check{X}$, given by the
formula

$e(\check{X})=e(X)-\sum_{i=1}^{n}\left(1-\frac{1}{a_{i}}\right)$

In particular, this definition guarantees that the relation

$e(\check{X})=(-1)^{m}c_{m}(\varOmega_{\check{X}}^{1})$

continues to hold, where $c_{m}(\varOmega_{\check{X}}^{1})$ is the orbifold
Chern number of the orbifold holomorphic cotangent bundle.

When $\check{\mathscr{F}}$ is a locally free orbifold sheaf of rank $r$ on
$\check{X}$, the version of Hirzebruch-Grothendieck-Riemann-Roch reads in this
context [14, Section 3]:

(4)
$\chi(\check{X},\check{E})=\int_{\check{X}}\text{ch($\check{E}$})\text{td}(\check{X})+\sum_{i=1}^{n}\frac{1}{\\#G_{p_{i}}}\left(\sum_{g\in
G_{p_{i}}-\\{1_{m}\\}}\frac{\text{tr}(\rho(g))}{\det(1_{m}-g)}\right)$

where on the left hand side we mean the alternating sum of the dimensions of
the cohomology groups $H^{i}(\check{X},\check{E})$.

Here we are regarding $g\in G_{p}\simeq\mathbb{Z}/a_{i}\mathbb{Z}$ as an
$m\times m$ matrix, and $\rho(g)$ denotes the action of $g$ on the orbifold
bundle $\check{E}$.

In order to compare this formula with the one which appears in the paper by
Furuta and Steer [42, Theorem 1.5], we take $\check{X}=\check{\varSigma}$ and
assume $\check{E}=\check{L}$ is a line bundle. In this case,

$\begin{cases}\text{ch}(\check{L})=\text{rk}(\check{L})+c_{1}(\check{L})=1+c_{1}(\check{L})\\\
\text{td}(\check{L})=1+\frac{c_{1}(\check{L})}{2}\end{cases}$

Therefore we have

$\displaystyle\int_{\check{\varSigma}}\text{ch($\check{L}$})\text{td}(\check{\varSigma})$
$\displaystyle=$
$\displaystyle\int_{\check{\varSigma}}\left(1+c_{1}(\check{L})\right)\left(1+\frac{c_{1}(\check{\varSigma})}{2}\right)$
$\displaystyle=$
$\displaystyle\int_{\check{\varSigma}}\left(\frac{1}{2}\right)c_{1}(\check{\varSigma})+c_{1}(\check{L})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(2-2g-\sum_{i=1}^{n}\left(1-\frac{1}{a_{i}}\right)\right)+\int_{\check{\varSigma}}c_{1}(\check{L})$
$\displaystyle=$
$\displaystyle(1-g)-\sum_{i=1}^{n}\left(\frac{a_{i}-1}{2a_{i}}\right)+\int_{\check{\varSigma}}c_{1}(\check{L})$

Now we want to analyze the term $\sum_{g\in
G_{p_{i}}-\\{1\\}}\frac{\text{tr}(\rho(g))}{\det(1-g)}$.To understand how to
compute $\rho(g)$, it is useful to work with orbifold connections in a
slightly more concrete way [51, section 1.(ii)].

For the case of an orbifold line bundle $\check{L}$, near each marked point
$p$ we can choose polar coordinates $(r,\theta)$ on a disk neighborhood
$D_{p}$ of $p$ so that we have the connection form

$A_{\lambda}=i\lambda d\theta$

where $\lambda$ is a constant known the _holonomy parameter_. The reason for
this name is that holonomy of a connection $A$ on $D_{p}\backslash\\{p\\}$
whose matrix connection coincides with $A_{\lambda}$ near $p$ on the
positively-oriented small circles of constant $r$ will have a limiting
holonomy approximately equal to

$e^{-2\pi i\lambda}$

Strictly speaking, we want to study the previous situation _up to
conjugacy_[81], but since $U(1)$ is abelian this is not important, so it
suffices to take $0\leq\lambda<1$ in order to guarantee that $e^{-2\pi
i\lambda}$ exhausts all possible elements in $U(1)$. We are interested in
connections that differ from $A_{\lambda}$ by a smooth one form on
$\varSigma$, and for the orbifold interpretation to hold, we need to take
$\lambda\in\mathbb{Q}\cap[0,1)$. In fact, we will write $\lambda$ as
$\lambda=\frac{b}{a}$ for $0<b<a$. The parameter $b$ can then be identified
with what Furuta and Steer call the _isotropy data_ of the line bundle [42].
It is now clear that we can write

$\sum_{g\in
G_{p_{i}}-\\{1\\}}\frac{\text{tr}(\rho(g))}{\det(1-g)}=\sum_{k=1}^{a_{i}-1}\frac{e^{-2\pi
ikb_{i}/a_{i}}}{1-e^{2\pi ik/a_{i}}}$

Now we need to find a more concrete version of the previous sum. A similar
expression appears in [24, p.7], and the author would like to thank Dinesh
Valluri for suggesting how to prove the following lemma.

######  Lemma 7.

Let $\zeta$ denote an a-th root of unity, which for simplicity we take as
$\zeta=e^{2\pi i/a}$. Then for any $0<b<a$,

$\sum_{k=1}^{a-1}\frac{\zeta^{kb}}{1-\zeta^{k}}=\sum_{k=1}^{a-1}\frac{e^{2\pi
ikb/a}}{1-e^{2\pi ik/a}}=b-\frac{a+1}{2}$

###### Proof.

Step $1$:

Since $1=\zeta^{aj}=(\zeta^{j})^{a}$ we have that

$0=(\zeta^{j}-1)(1+\zeta^{j}+\cdots+(\zeta^{j})^{a-1})$

So for $0<j<a$, we have

(5) $\sum_{k=1}^{a-1}\zeta^{kj}=-1$

Step 2:

(6) $\sum_{k=1}^{a-1}\frac{1}{\zeta^{k}-1}=-\left(\frac{a-1}{2}\right)$

Observe that the left hand side of (6) equals

$\displaystyle\sum_{k=1}^{a-1}\frac{1}{\zeta^{k}-1}$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{a-1}\frac{1-\zeta^{k}+\zeta^{k}}{\zeta^{k}-1}$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{a-1}\left[-1+\left(\frac{\zeta^{k}}{\zeta^{k}-1}\right)\right]$
$\displaystyle=$
$\displaystyle-(a-1)+\sum_{k=1}^{a-1}\frac{\zeta^{k}}{\zeta^{k}-1}$
$\displaystyle=$ $\displaystyle-(a-1)+\sum_{k=1}^{a-1}\frac{1}{1-\zeta^{-k}}$

Now, notice that

$\displaystyle\sum_{k=1}^{a-1}\frac{1}{1-\zeta^{-k}}$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{a-1}\frac{1}{1-\zeta^{a-k}}$ $\displaystyle=$
$\displaystyle\frac{1}{1-\zeta^{a-1}}+\frac{1}{1-\zeta^{a-2}}+\cdots+\frac{1}{1-\zeta}$
$\displaystyle=$ $\displaystyle\sum_{k=1}^{a-1}\frac{1}{1-\zeta^{k}}$

Therefore, the proof of (6) becomes

$\displaystyle\sum_{k=1}^{a-1}\frac{1}{\zeta^{k}-1}=-(a-1)+\sum_{k=1}^{a-1}\frac{1}{1-\zeta^{k}}$
$\displaystyle\implies$
$\displaystyle\sum_{k=1}^{a-1}\frac{1}{\zeta^{k}-1}=-\frac{(a-1)}{2}$

Step 3:

(7) $\frac{\zeta^{kb}-1}{\zeta^{k}-1}=1+\zeta^{k}+\cdots+(\zeta^{k})^{b-1}$

This is obtained by factoring the numerator $\zeta^{kb}-1=(\zeta^{k})^{b}-1$.

Proof of Lemma:

$\displaystyle\sum_{k=1}^{a-1}\frac{\zeta^{kb}-1}{\zeta^{k}-1}$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{a-1}\left(\sum_{j=0}^{b-1}\zeta^{kj}\right)$
$\displaystyle=$ $\displaystyle\sum_{j=0}^{b-1}\sum_{k=1}^{a-1}\zeta^{kj}$
$\displaystyle=^{\text{Step}(1)}$
$\displaystyle(a-1)+\sum_{j=1}^{b-1}\underbrace{\sum_{k=1}^{a-1}\zeta^{kj}}_{-1}$
$\displaystyle=$ $\displaystyle a-1-(b-1)$ $\displaystyle=$ $\displaystyle
a-b$

Therefore,

(8)
$\sum_{k=1}^{a-1}\frac{\zeta^{kb}}{\zeta^{k}-1}-\sum_{k=1}^{a-1}\frac{1}{\zeta^{k}-1}=a-b$

Using step (2) we obtain

$\sum_{k=1}^{a-1}\frac{\zeta^{kb}}{\zeta^{k}-1}=-\frac{(a-1)}{2}+a-b=\frac{a+1}{2}-b$

Therefore

$\sum_{k=1}^{a-1}\frac{\zeta^{kb}}{1-\zeta^{k}}=b-\frac{a+1}{2}$

∎

Therefore

$\displaystyle\sum_{i=1}^{n}\frac{1}{\\#G_{p_{i}}}\left(\sum_{g\in
G_{p_{i}}-\\{1\\}}\frac{\text{tr}(\rho(g))}{\det(1-g)}\right)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\frac{1}{a_{i}}\left(\sum_{k=1}^{a_{i}-1}\frac{e^{-2\pi
ikb_{i}/a_{i}}}{1-e^{2\pi ik/a_{i}}}\right)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\frac{1}{a_{i}}\left(\sum_{k=1}^{a_{i}-1}\frac{e^{2\pi
ik(a_{i}-b_{i})/a_{i}}}{1-e^{2\pi ik/a_{i}}}\right)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\frac{1}{a_{i}}\left(a_{i}-b_{i}-\frac{a+1}{2}\right)$
$\displaystyle=$
$\displaystyle-\sum_{i=1}^{n}\frac{b_{i}}{a_{i}}+\sum_{i=1}^{n}\frac{a_{i}-1}{2a_{i}}$

So the orbifold version of Riemann Roch for line bundles is

$\displaystyle\dim_{\mathbb{C}}H^{0}(\check{\varSigma},\check{L})-\dim_{\mathbb{C}}H^{1}(\check{\varSigma},\check{L})$
$\displaystyle=$
$\displaystyle(1-g)-\sum_{i=1}^{n}\left(\frac{a_{i}-1}{2a_{i}}\right)+\int_{\check{\varSigma}}c_{1}(\check{L})-\sum_{i=1}^{n}\frac{b_{i}}{a_{i}}+\sum_{i=1}^{n}\frac{a_{i}-1}{2a_{i}}$
$\displaystyle=$
$\displaystyle(1-g)+\int_{\check{\varSigma}}c_{1}(\check{L})-\sum_{i=1}^{n}\frac{b_{i}}{a_{i}}$
$\displaystyle=$
$\displaystyle(1-g)+c_{1}(\check{L})-\sum_{i=1}^{n}\frac{b_{i}}{a_{i}}$

which agrees with [42, Theorem 1.5], under the usual abuse of notation that
identifies $c_{1}(\check{L})$ with a number.

######  Remark 8.

The quantity $c_{1}(\check{L})-\sum_{i=1}^{n}\frac{b_{i}}{a_{i}}$ is denoted
the background degree in [67, Definition 2.7]. As explained in [42, p.44], if
one thinks of $H^{*}(\check{\varSigma},\check{L})$ as
$H^{*}(\check{\varSigma},\mathcal{O}(\check{L}))$, where
$\mathcal{O}(\check{L})$ is the sheaf of holomorphic sections, then
$\mathcal{O}(\check{L})$ is a locally free sheaf with degree
$c_{1}(\check{L})-\sum_{i=1}^{n}\frac{b_{i}}{a_{i}}$. We will write

(9) $\deg_{B}(\check{L})=c_{1}(\check{L})-\sum_{i=1}^{n}\frac{b_{i}}{a_{i}}$

for this formula of the background degree.

One can also relate orbifold line bundles with divisors [70, Section 1B].
Namely, a divisor $\check{D}$ on $\check{\varSigma}$ is a finite linear
combination

$\check{D}=\sum_{p\in\varSigma}\frac{n_{p}}{a_{p}}\cdot p$

where $n_{p}\in\mathbb{Z}$, and $a_{p}=1$ if $p$ is not one of the marked
points on $\check{\varSigma}$. The degree of a divisor is then

$\deg\check{D}=\sum_{p}\frac{n_{p}}{a_{p}}$

Proposition 1.3 in [70] then describes a bijection between equivalence classes
of divisors and holomorphic prbifold line bundles. If $\check{L}_{\check{D}}$
is the line bundle corresponding to $\check{D}$, then
$c_{1}(\check{L}_{\check{D}})=\deg\check{D}$.

For our purposes it is also necessary to have a formula of Riemann-Roch for
$U(2)$ orbifold bundles $\check{E}$. One option would be to apply formula (4)
one more time, but we find it more elegant to bootstrap from Riemann-Roch for
orbifold line bundles $\check{L}$. This requires a small detour into the
classification of $U(2)$ orbifold bundles, which can be found in [42,
Proposition 1.8], as well as [70, Section 1A].

The isotropy data for a rank two $U(2)$ orbifold bundle $\check{E}$ will be
specified by the pairs of integers

$((b_{1}^{-},b_{1}^{+}),\cdots,(b_{n}^{-},b_{n}^{+}))$

where $0\leq b_{i}^{-},b_{i}^{+}<a_{i}$, for $i=1,\cdots,n$. In particular, we
can find orbifold line bundles $\check{L}^{-},\check{L}^{+}$ with isotropy
data $(b_{i}^{-})$ and $(b_{i}^{+})$ respectively. Then

$\check{E}=\check{L}^{-}\oplus\check{L}^{+}$

is an $U(2)$ orbifold bundle with isotropy data
$((b_{1}^{-},b_{1}^{+}),\cdots,(b_{n}^{-},b_{n}^{+}))$. Notice that the
notation seems to suggest that $b_{i}^{-}\leq b_{i}^{+}$, in fact, this is
assumed in [70, p.599]. This convention is related to which group of
automorphisms we are allowed to use.

Up to conjugacy, any matrix in $U(2)$ takes the form

$\left(\begin{array}[]{cc}e^{-2\pi i\lambda_{1}}&0\\\ 0&e^{-2\pi
i\lambda_{2}}\end{array}\right)$

where we can assume $0\leq\lambda_{1}\leq\lambda_{2}<1$. However, this
classification requires conjugating by arbitrary matrices in $U(2)$. In our
case we are interested in using the _determinant one gauge group_ , which
means that the bundle $\check{L}^{-}\oplus\check{L}^{+}$ needs to be
distinguished from the bundle $\check{L}^{+}\oplus\check{L}^{-}$, since an
automorphism which permutes both factors would locally take the form
$\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right)$, and this matrix,
while an element of $U(2)$, is not an element of $SU(2)$, because it has
determinant $-1$.

Therefore, with respect to the determinant one gauge group, we need to
consider the isotropy data as ordered pairs, without any assumption regarding
whether $b_{i}^{-}\leq b_{i}^{+}$ or not. This point is briefly made on [52,
p. 602] . However, ultimately we are interested in studying the moduli spaces
after taking the quotient by the residual $S^{1}$ action mentioned in the
introduction. Since $S^{1}\times_{\\{\pm I_{2}\\}}SU(2)$ can be identified
with $U(2)$ via the map $(e^{i\theta},A)\rightarrow e^{i\theta}A$, this point
doesn’t end up making a big difference, so it is fine to assume $b_{i}^{-}\leq
b_{i}^{+}$ while reading this paper.

In any case, the important point is that the orbifold version of Hirzebruch-
Grothendieck-Riemann-Roch states that the Euler characteristic
$\chi(\check{\varSigma},\check{E})$ is topological, so it suffices to find the
formula in the case of
$\chi(\check{\varSigma},\check{L}^{-}\oplus\check{L}^{+})$. By this we mean
the following: the spaces $H^{0}(\check{\varSigma},\check{E})$ and
$H^{1}(\check{\varSigma},\check{E})$ can be identified with the dimensions of
the kernel and cokernel of an orbifold Dolbeault operator
$\bar{\partial}_{\check{C}}:\varOmega^{0}(\check{\varSigma},\check{E})\rightarrow\varOmega^{0,1}(\check{\varSigma},\check{E})$,
where $\check{C}$ is an arbitrary $U(2)$ orbifold connection on $\check{E}$.

In particular, we can assume that $\check{C}$ is a $U(2)$ connection
compatible with the topological splitting
$\check{E}=\check{L}^{-}\oplus\check{L}^{+}$, which we write as
$\check{C}=\check{C}^{-}\oplus\check{C}^{+}$. In this situation it is clear
that
$\chi(\check{\varSigma},\check{E})=\chi(\check{\varSigma},\check{L}^{-})+\chi(\check{\varSigma},\check{L}^{+})$
and thus we find that

$\displaystyle\dim_{\mathbb{C}}H^{0}(\check{\varSigma},\check{E})-\dim_{\mathbb{C}}H^{1}(\check{\varSigma},\check{E})$
$\displaystyle=$
$\displaystyle\chi(\check{\varSigma},\check{L}^{-})+\chi(\check{\varSigma},\check{L}^{+})$
$\displaystyle=$
$\displaystyle(1-g)+c_{1}(\check{L}^{-})-\sum_{i=1}^{n}\frac{b_{i}^{-}}{a_{i}}+(1-g)+c_{1}(\check{L}^{+})-\sum_{i=1}^{n}\frac{b_{i}^{+}}{a_{i}}$
$\displaystyle=$ $\displaystyle
2(1-g)+c_{1}(\det\check{E})-\sum_{i=1}^{n}\frac{b_{i}^{-}+b_{i}^{+}}{a_{i}}$

In the special case of a $SU(2)$ orbifold bundle $\check{E}$, we can assume
that $0\leq b_{i}^{-}\leq[a_{i}/2]$, and also take
$b_{i}^{+}=a_{i}-b_{i}^{-}$. In particular, the previous formula reduces to

$2(1-g)-n$

Summarizing our findings, we have found

######  Theorem 9.

Riemann-Roch for orbifold Riemann surfaces:

a) If $\check{L}$ is an orbifold line bundle with isotropy data
$(b_{i})_{i=1,\cdots,n}$, then

(10)
$\dim_{\mathbb{C}}H^{0}(\check{\varSigma},\check{L})-\dim_{\mathbb{C}}H^{1}(\check{\varSigma},\check{L})=(1-g)+c_{1}(\check{L})-\sum_{i=1}^{n}\frac{b_{i}}{a_{i}}$

b) If $\check{E}$ is an orbifold $SU(2)$ bundle with isotropy data
$((b_{i}^{-},b_{i}^{+}=a_{i}-b_{i}^{-}))_{i=1}^{n}$, then

$\dim_{\mathbb{C}}H^{0}(\check{\varSigma},\check{E})-\dim_{\mathbb{C}}H^{1}(\check{\varSigma},\check{E})=2(1-g)-n$

c) If $\check{E}$ is an orbifold $U(2)$ bundle with isotropy data
$((b_{i}^{-},b_{i}^{+}))_{i=1}^{n}$, and determinant line bundle
$\det\check{E}$, then

(11)
$\dim_{\mathbb{C}}H^{0}(\check{\varSigma},\check{E})-\dim_{\mathbb{C}}H^{1}(\check{\varSigma},\check{E})=2(1-g)+c_{1}(\det\check{E})-\sum_{i=1}^{n}\frac{b_{i}^{-}+b_{i}^{+}}{a_{i}}$

Riemann-Roch is specially powerful when combined with Serre duality. Recall
that Serre duality states that

$H^{i}(\check{\varSigma},\check{E})\simeq
H^{1-i}(\check{\varSigma},K_{\check{\varSigma}}\otimes\check{E}^{*})$

Here $K_{\check{\varSigma}}$ is the canonical bundle, which as an orbifold
line bundle has Seifert invariants

(12) $(2g-2,a_{1}-1,\cdots,a_{n}-1)$

in the sense that $\deg_{B}K_{\check{\varSigma}}=2g-2$ and

$c_{1}(K_{\check{\varSigma}})=2g-2+\sum_{i=1}^{n}\frac{a_{i}-1}{a_{i}}=2g-2+n-\sum_{i=1}^{n}\frac{1}{a_{i}}$

Thus, suppose we want to check which conditions on $\check{L}$ guarantee that
$H^{1}(\check{\varSigma},\check{L})$ vanishes. First of all, by Serre duality,

$\dim H^{1}(\check{\varSigma},\check{L})=\dim
H^{0}(\check{\varSigma},K_{\check{\varSigma}}\otimes\check{L}^{*})$

If $\check{L}$ has isotropy $b_{i}$ at the point $p_{i}$ we define

$\epsilon_{i}=\begin{cases}1&0<b_{i}\\\ 0&b_{i}=0\end{cases}$

then the isotropy of $\check{L}^{*}$ at the point $p_{i}$ is
$\epsilon_{i}(a_{i}-b_{i})$, and the isotropy of
$K_{\check{\varSigma}}\otimes\check{L}^{*}$ is

$\begin{cases}a_{i}-1&\text{if }b_{i}=0\\\ a_{i}-1-b_{i}&\text{if
}a_{i}-1>b_{i}\\\ 0&\text{if }b_{i}=a_{i}-1\end{cases}$

Notice that the first case ends up being contained in the second one.

Clearly if $\deg_{B}(K_{\check{\varSigma}}\otimes\check{L}^{*})<0$ , then
$H^{1}(\check{\varSigma},\check{L})$ vanishes. Looking at the formula for the
background degree, this requires

$c_{1}(K_{\check{\varSigma}}\otimes\check{L}^{*})-\sum_{0\leq
b_{i}<a_{i}-1}\frac{a_{i}-1-b_{i}}{a_{i}}<0$

This is the same as $2g-2+n-\sum_{i=1}^{n}\frac{1}{a_{i}}-\sum_{0\leq
b_{i}<a_{i}-1}\frac{a_{i}-1-b_{i}}{a_{i}}<c_{1}(\check{L})$, or more
succinctly

$2g-2+\sum_{b_{i}=a_{1}-1}\frac{a_{i}-1}{a_{i}}+\sum_{0\leq
b_{i}<a_{i}-1}\frac{b_{i}}{a_{i}}<c_{1}(\check{L})$

But this ends up being equal to

$2g-2<\deg_{B}(\check{L})$

Therefore, we conclude that

######  Lemma 10.

Suppose that $\check{L}$ is an orbifold line bundle such that
$2g-2<\deg_{B}(\check{L})$. Then $H^{1}(\check{\varSigma},\check{L})$
vanishes.

######  Remark 11.

Observe that
$c_{1}(K_{\check{\varSigma}}\otimes\check{L}^{*})=c_{1}(K_{\check{\varSigma}})-c_{1}(\check{L})$,
so by [70, Corollary 1.4] $H^{1}(\check{\varSigma},\check{L})$ will also
vanish whenever $c_{1}(K_{\check{\varSigma}})<c_{1}(\check{L})$.

It is also important to mention the conditions an orbifold line bundle
$\check{L}$ must satisfy in order to appear in a reduction of an orbifold
$U(2)$ bundle $\check{E}$ [70, 42]. First of all, we must have the topological
splitting

$\check{E}=\check{L}\oplus\left(\check{L}^{*}\otimes\det\check{E}\right)$

The isotropy data of $\check{L}$ is determined by that of $\check{E}$ in the
sense that $b_{i}\in\\{b_{i}^{-},b_{i}^{+}\\}$ for all $i$. Define the vector
$\epsilon=(\epsilon_{1},\cdots,\epsilon_{n})$ by the conditions

$\epsilon_{i}=\begin{cases}0&b_{i}^{-}=b_{i}^{+}\\\ -1&b_{i}=b_{i}^{-}\\\
+1&b_{i}=b_{i}^{+}\end{cases}$

and the integers

$\begin{cases}n_{0}=\\#\\{i\mid\epsilon_{i}=0\\}\\\
n_{\pm}=\\#\\{i\mid\epsilon_{i}=\pm 1\\}\end{cases}$

Then $c_{1}(\check{L})$ satisfies
$c_{1}(\check{L})-\sum_{i=1}^{n}\frac{\epsilon_{i}(b_{i}^{+}-b_{i}^{-})+(b_{i}^{+}+b_{i}^{-})}{2a_{i}}\in\mathbb{Z}$.

The topological isomorphism classes of orbifold line bundles form a group
under tensor product denoted $\text{Pic}_{V}^{t}(\check{\varSigma})$.
According to [42, Corollary 1.7], when $a_{1},\cdots,a_{n}$ are mutually
coprime, there is a unique orbifold line bundle $\check{L}_{0}$ such that
$c_{1}(\check{L}_{0})=\frac{1}{\prod_{i=1}^{n}a_{i}}$ , and any other orbifold
line bundle $\check{L}$ has the form $\check{L}_{0}^{k}$, for some integer
$k$.

Moreover, as explained in [42, p. 47], for $SO(3)$ orbifold bundles there is a
commutative diagram

$\begin{array}[]{ccc}\\{U(2)-\text{orbifold
bundles}\\}&\rightarrow^{\det}&\text{Pic}_{V}^{t}(\check{\varSigma})\\\
\downarrow_{Ad}&&\downarrow\\\ \\{SO(3)-\text{orbifold
bundles}\\}&\rightarrow^{w}&\text{Pic}_{V}^{t}/(\text{Pic}_{V}^{t})^{2}\end{array}$

where the map $w$ is essentially the second Stiefel-Whitney class.

For the examples we will analyze at the end of this paper, it is important to
know when an orbifold $U(2)$ bundle $\check{E}$ admits an irreducible
projectively flat connection.

The proofs can be found in [42], but we found the summary from [70] especially
useful. In particular, we need the following facts:

######  Fact 12.

(See [42], [70, Lemmas 2.2 and 4.2])

a) Suppose $\check{E}$ is an orbifold $U(2)$ bundle with isotropy data
$(b_{1}^{\pm},b_{2}^{\pm},\cdots,b_{n}^{\pm})$ and let $n_{0}=\\#\\{i\mid
b_{i}^{-}=b_{i}^{+}\\}$. If $g=0$ and $n-n_{0}\leq 2$, then $\check{E}$ admits
no irreducible projectively flat connections.

b) The space of irreducible projectively flat connections is connected and
simply connected regardless of the genus of $\varSigma$.

c) The space of irreducible projectively flat connections is empty if and only
if there exists a vector $(\epsilon_{i})$, with $\epsilon_{i}=\pm 1$, such
that $n_{+}+\deg_{B}(\det\check{E})\equiv 1\mod 2$ and
$n_{+}-\sum_{i=1}^{n}\frac{\epsilon_{i}(b_{i}^{+}-b_{i}^{-})}{a_{i}}<1-g$.
Here $n_{\pm}=\\#\\{i\mid\epsilon_{i}=\pm 1\\}$.

##  3\. the $SO(3)$ Vortex Moduli Space

Let $\check{\varSigma}=(\varSigma,p_{1},\cdots,p_{n})$ denote an orbifold
Riemann surface and choose an orbifold $U(2)$ bundle
$\check{E}\rightarrow\check{\varSigma}$. Our policy will be to not make any
particular assumptions on $\check{E}$ at this point. For example, in the
smooth case it is typical to assume $\deg E>4g-4$ [19, Assumption 2], but we
will explain why this choice of degree is made in this section and the next.

Even if the bundle is an orbifold $SU(2)$ bundle, we will regard it as an
orbifold $U(2)$ bundle, in the sense that we will make a choice of reference
connection in $\det\check{E}$, though in the $SU(2)$ case we can just take it
to be the trivial connection.

For notational ease, we will use $\check{}$ to remind the reader that our
bundles and other geometric data should be interpreted in the orbifold sense,
but for an orbi-section of $\check{E}$, we will write $\varUpsilon$ instead of
$\check{\varUpsilon}$. Similar remarks apply to our notation for orbifold
connections.

It is a good moment to justify our unusual choice of notation at this point.
Since we are eventually interested in using these moduli spaces for 3- and
4-manifolds, we want a notation that does not interfere with the one used in
these contexts. In particular, we are trying to follow the notation of
Kronheimer and Mrowka [55], so typically for us

  * •

$X$ denotes a 4-manifold, $A$ a connection of some bundle defined on $X$, and
$\varPhi$ a section of some spinor-type bundle.

  * •

$Y$ denotes a 3-manifold, $B$ a connection of some bundle defined on $Y$, and
$\varPsi$ a section of some spinor-type bundle.

  * •

$\varSigma$ denotes a 2-manifold, $C$ a connection of some bundle defined on
$\varSigma$, and $\varUpsilon$ a section of some spinor-type bundle.

Further remarks regarding our notation will be explained as it becomes
necessary.

######  Definition 13.

Let $\mathcal{A}^{\det}(\check{E})$ denote the space of $U(2)$ connections on
$\check{E}$ which induce a fixed reference $U(1)$ connection $C^{\det}$ on
$\det\check{E}$. The $SO(3)$ vortex equations are equations for a pair
$(C,\varUpsilon)\in\mathcal{A}^{\det}(\check{E})\times\varGamma(\check{E})$
which satisfy

(13) $SO(3)\text{ vortex equations
}\begin{cases}*F_{C}^{0}-i\left[\varUpsilon\varUpsilon^{*}-\frac{1}{2}|\varUpsilon|^{2}I_{E}\right]=0\\\
\bar{\partial}_{C}\varUpsilon=0\end{cases}$

where $F_{C}^{0}$ denotes the traceless part of the curvature of $C$.

######  Remark 14.

Observe that the first equation makes sense since
$\varUpsilon\varUpsilon^{*}-\frac{1}{2}|\varUpsilon|^{2}I_{E}$ is a traceless
hermitian endomorphism of $E$, so
$i\left[\varUpsilon\varUpsilon^{*}-\frac{1}{2}|\varUpsilon|^{2}I_{E}\right]$
is skew-hermitian, and thus can be regarded as an element of
$\mathfrak{su}(2)$.

If we define
$\mathcal{C}(\check{\varSigma},\check{E})=\mathcal{A}^{\det}(\check{E})\times\varGamma(\check{E})$
as the configuration space and $\mathcal{G}^{\det}(\check{E})$ the gauge group
(those automorphisms of $\check{E}$ with determinant one), we want to study
the solutions to the $SO(3)$ vortex equations modulo gauge, i.e, the moduli
space
$\mathcal{M}(\check{\varSigma},\check{E})\subset\mathcal{C}(\check{\varSigma},\check{E})/\mathcal{G}^{\det}(\check{E})$.
Recall that the gauge group $u\in\mathcal{G}^{\det}(E)$ acts on
$(C,\varUpsilon)$ as

$u\cdot(C,\varUpsilon)=(C-(d_{C}u)u^{-1},u\varUpsilon)$

The $SO(3)$ vortex map is the map

(14) $\displaystyle\mathfrak{F}_{SO(3)}:$
$\displaystyle\mathcal{A}^{\det}(\check{E})\times\varGamma(\check{E})\rightarrow\varGamma(\check{\varSigma};\mathfrak{su}(E))\oplus\varGamma(K_{\check{\varSigma}}^{-1}\otimes
E)$ $\displaystyle(C,\varUpsilon)$
$\displaystyle\rightarrow\left(*F_{C}^{0}-i\left[\varUpsilon\varUpsilon^{*}-\frac{1}{2}|\varUpsilon|^{2}I_{E}\right],\bar{\partial}_{C}\varUpsilon\right)$

An easy computation shows that:

######  Lemma 15.

The linearization of the $SO(3)$ vortex map is

$\mathcal{D}\mathfrak{F}_{SO(3),(C,\varUpsilon)}(\dot{c},\dot{\varUpsilon})=\left(*d_{C}\dot{c}-i\left[\varUpsilon\dot{\varUpsilon}^{*}+\dot{\varUpsilon}\varUpsilon^{*}-\text{Re}\left\langle\varUpsilon,\dot{\varUpsilon}\right\rangle
I_{E}\right],\bar{\partial}_{C}\dot{\varUpsilon}+\dot{c}^{\prime\prime}\otimes\varUpsilon\right)$

where we have decomposed
$\dot{c}\in\varOmega^{1}(\check{\varSigma},\mathfrak{su}(E))\subset\varOmega^{1}(\check{\varSigma},\mathfrak{su}(E)\otimes\mathbb{C})$
as $\dot{c}=\dot{c}^{\prime}+\dot{c}^{\prime\prime}$.

######  Remark 16.

Recall that on a Riemann surface the Hodge star operator acting on one-forms
satisfies $*^{2}=-\text{Id}$, so it can be used to define an almost complex
structure on $\varOmega^{1}(\mathfrak{g}_{E})$. In particular, whenever we
write $c=c^{\prime}+c^{\prime\prime}$, then
$*c=-ic^{\prime}+ic^{\prime\prime}$.

###### Proof.

It follows easily after observing that for $t\in\mathbb{R}$,

$\begin{cases}|\varUpsilon+t\dot{\varUpsilon}|^{2}=|\varUpsilon|^{2}+t^{2}|\dot{\varUpsilon}|^{2}+2\text{Re}\left\langle\varUpsilon,\dot{\varUpsilon}\right\rangle\\\
(\varUpsilon+t\dot{\varUpsilon})(\varUpsilon+t\dot{\varUpsilon})^{*}=\varUpsilon\varUpsilon^{*}+t^{2}\dot{\varUpsilon}\dot{\varUpsilon}^{*}+t\varUpsilon\dot{\varUpsilon}^{*}+t\dot{\varUpsilon}\varUpsilon^{*}\end{cases}$

∎

######  Lemma 17.

For $\xi\in\varOmega^{0}(\check{\varSigma};\mathfrak{su}(\check{E}))$, define

$d_{(C,\varUpsilon)}^{0}(\xi)=(-d_{C}\xi,\xi\varUpsilon)$

Then at each solution $(C,\varUpsilon)$ to the $SO(3)$ vortex equations (14),
there is a complex

$\varOmega^{0}(\check{\varSigma};\mathfrak{su}(\check{E}))\rightarrow^{d_{(C,\varUpsilon)}^{0}}\varOmega^{1}(\check{\varSigma};\mathfrak{su}(\check{E}))\oplus\varGamma(\check{E})\rightarrow^{D\mathfrak{F}_{SO(3),(C,\varUpsilon)}}\varOmega^{0}(\check{\varSigma};\mathfrak{su}(\check{E}))\oplus\varGamma(K_{\check{\varSigma}}^{-1}\otimes\check{E})$

###### Proof.

Write
$\mathfrak{F}_{SO(3)}=\left(\mathfrak{F}_{SO(3)}^{1},\mathfrak{F}_{SO(3)}^{2}\right)$.
Then just as in [35, p.312], we have that

$\mathcal{D}\mathfrak{F}_{SO(3),(C,\varUpsilon)}\circ
d_{(C,\varUpsilon)}^{0}(\xi)=\left([\xi,\mathfrak{F}_{SO(3)}^{1}(C,\varUpsilon)],\xi\mathfrak{F}_{SO(3)}^{2}(C,\varUpsilon)\right)$

so the result follows since at a solution to the vortex equations we have
$\mathfrak{F}_{SO(3)}(C,\varUpsilon)=(0,0)$. ∎

######  Definition 18.

Define the $L^{2}$ _real_ inner product _[55, Lemma 9.3.3]_

(15)
$\left\langle(\dot{c}_{1},\dot{\varUpsilon}_{1}),(\dot{c}_{2},\dot{\varUpsilon}_{2})\right\rangle_{L^{2}}\equiv\int_{\check{\varSigma}}\left\langle\dot{c}_{1},\dot{c}_{2}\right\rangle+\text{Re}\left\langle\dot{\varUpsilon}_{1},\dot{\varUpsilon}_{2}\right\rangle$

_where the point-wise inner product on $\mathfrak{su}(\check{E})$ is given by_

$\left\langle\dot{c}_{1},\dot{c}_{2}\right\rangle=\frac{1}{2}\text{Tr}(\dot{c}_{1}^{\dagger}\dot{c}_{2})$

_Moreover, define_

(16)
$d_{(C,\varUpsilon)}^{0,*}(\dot{c},\dot{\varUpsilon})=-d_{C}^{*}\dot{c}+[\dot{\varUpsilon}\varUpsilon^{*}-\varUpsilon\dot{\varUpsilon}^{*}-i\text{Re}<i\varUpsilon,\dot{\varUpsilon}>I_{E}]$

######  Lemma 19.

If $d_{(C,\varUpsilon)}^{0,*}(c,\dot{\varUpsilon})=0$, then
$(\dot{c},\dot{\varUpsilon})$ is $L^{2}$ orthogonal to
$\text{Im}d_{(C,\varUpsilon)}^{0}$ .

###### Proof.

Observe that

$\displaystyle\left\langle(-d_{C}\xi,\xi\varUpsilon),(\dot{c},\dot{\varUpsilon})\right\rangle_{L^{2}}$
$\displaystyle=$ $\displaystyle\left\langle-
d_{C}\xi,\dot{c}\right\rangle_{L^{2}}+\left\langle\xi\varUpsilon,\dot{\varUpsilon}\right\rangle_{L^{2}}$
$\displaystyle=$
$\displaystyle\left\langle\xi,-d_{C}^{*}\dot{c}\right\rangle_{L^{2}}+\left\langle\xi\varUpsilon,\dot{\varUpsilon}\right\rangle_{L^{2}}$
$\displaystyle=$
$\displaystyle\int_{\varSigma}\left\langle\xi,-d_{C}^{*}\dot{c}\right\rangle+\text{Re}\left\langle\xi\varUpsilon,\dot{\varUpsilon}\right\rangle$

So now we need to verify a pointwise identity. First of all, notice that for
two matrices $M_{1},M_{2}$ of $\mathfrak{su}(E)$, their hermitian inner
product is real in the sense that

$\left\langle
M_{1},M_{2}\right\rangle=\frac{1}{2}\text{tr}(M_{1}^{\dagger}M_{2})=-\frac{1}{2}\text{tr}(M_{1}M_{2})=-\frac{1}{2}\text{tr}(M_{2}M_{1})=\frac{1}{2}\text{tr}(M_{2}^{\dagger}M_{1})=\left\langle
M_{2},M_{1}\right\rangle$

which justifies why there is no need to put Re in front of
$\left\langle\xi,-d_{C}^{*}\dot{c}\right\rangle$. On the other hand, we can
re-write $\text{Re}\left\langle\xi\varUpsilon,\dot{\varUpsilon}\right\rangle$
as

$\frac{1}{2}\left\langle\xi\varUpsilon,\dot{\varUpsilon}\right\rangle+\frac{1}{2}\left\langle\dot{\varUpsilon},\xi\varUpsilon\right\rangle$

Therefore, the integrand we are considering becomes [recall
$d_{(C,\varUpsilon)}^{0,*}(c,\dot{\varUpsilon})=0$ so we can substitute
$d_{C}^{*}\dot{c}$]

$\displaystyle\left\langle\xi,-[\dot{\varUpsilon}\varUpsilon^{*}-\varUpsilon\dot{\varUpsilon}^{*}-i\text{Re}<i\varUpsilon,\dot{\varUpsilon}>I_{E}]\right\rangle+\frac{1}{2}\left\langle\xi\varUpsilon,\dot{\varUpsilon}\right\rangle+\frac{1}{2}\left\langle\dot{\varUpsilon},\xi\varUpsilon\right\rangle$
$\displaystyle=$
$\displaystyle\left\langle\xi,\varUpsilon\dot{\varUpsilon}^{*}\right\rangle-\left\langle\xi,\dot{\varUpsilon}\varUpsilon^{*}\right\rangle+\frac{1}{2}\left\langle\xi\varUpsilon,\dot{\varUpsilon}\right\rangle+\frac{1}{2}\left\langle\dot{\varUpsilon},\xi\varUpsilon\right\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left\langle\xi\varUpsilon,\dot{\varUpsilon}\right\rangle-\left\langle\xi,\dot{\varUpsilon}\varUpsilon^{*}\right\rangle+\left\langle\xi,\varUpsilon\dot{\varUpsilon}^{*}\right\rangle-\frac{1}{2}\left\langle\xi\dot{\varUpsilon},\varUpsilon\right\rangle$

So really we need to compare the previous quantities. For example, we want to
know whether

(17)
$\frac{1}{2}\left\langle\xi\varUpsilon,\dot{\varUpsilon}\right\rangle=^{?}\left\langle\xi,\dot{\varUpsilon}\varUpsilon^{*}\right\rangle$

This would guarantee the vanishing of the first two terms, and the other two
are essentially the same. We compute this locally. Observe that we can write

$\displaystyle\xi=$ $\displaystyle\left(\begin{array}[]{cc}it&-\bar{z}\\\
z&-it\end{array}\right)$ $\displaystyle\varUpsilon=$
$\displaystyle\left(\begin{array}[]{c}\varPhi_{1}\\\
\varPhi_{2}\end{array}\right)$ $\displaystyle\dot{\varUpsilon}=$
$\displaystyle\left(\begin{array}[]{c}\phi_{1}\\\ \phi_{2}\end{array}\right)$

Start with the right hand side

$\displaystyle\left\langle\xi,\dot{\varUpsilon}\varUpsilon^{*}\right\rangle$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\text{tr}\left(\xi\dot{\varUpsilon}\varUpsilon^{*}\right)$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\text{tr}\left(\left(\begin{array}[]{cc}it&-\bar{z}\\\
z&-it\end{array}\right)\left(\begin{array}[]{c}\phi_{1}\\\
\phi_{2}\end{array}\right)\left(\begin{array}[]{cc}\bar{\varPhi}_{1}&\bar{\varPhi}_{2}\end{array}\right)\right)$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\text{tr}\left(\left(\begin{array}[]{c}it\phi_{1}-\bar{z}\phi_{2}\\\
z\phi_{1}-it\phi_{2}\end{array}\right)\left(\begin{array}[]{cc}\bar{\varPhi}_{1}&\bar{\varPhi}_{2}\end{array}\right)\right)$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\text{tr}\left(\begin{array}[]{cc}it\phi_{1}\bar{\varPhi}_{1}&\text{*}\\\
&-it\phi_{2}\bar{\varPhi}_{2}\end{array}\right)$

where $*$ are things we don’t care about since we are just computing a trace!
On the other hand, the left hand side is

$\displaystyle\frac{1}{2}\left\langle\xi\varUpsilon,\dot{\varUpsilon}\right\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{2}\varUpsilon^{*}\xi^{*}\dot{\varUpsilon}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\left(\begin{array}[]{cc}\bar{\varPhi}_{1}&\bar{\varPhi}_{2}\end{array}\right)\left(\begin{array}[]{cc}it&-\bar{z}\\\
z&-it\end{array}\right)\left(\begin{array}[]{c}\phi_{1}\\\
\phi_{2}\end{array}\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(\begin{array}[]{cc}\bar{\varPhi}_{1}&\bar{\varPhi}_{2}\end{array}\right)\left(\begin{array}[]{c}it\phi_{1}-\bar{z}\phi_{2}\\\
z\phi_{1}-it\phi_{2}\end{array}\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\text{tr}\left(\begin{array}[]{cc}\bar{\varPhi}_{1}it\phi_{1}&**\\\
*&-it\bar{\varPhi}_{2}\phi_{2}\end{array}\right)$

So the traces of both matrices will be the same. ∎

######  Definition 20.

Suppose that $(C,\varUpsilon)$ is a solution to the $SO(3)$ vortex equations.
Define the cohomology groups

$\begin{cases}\mathbf{H}_{(C,\varUpsilon)}^{0}\equiv\ker
d_{(C,\varUpsilon)}^{0}\\\
\mathbf{H}_{(C,\varUpsilon)}^{1}\equiv\ker(d_{(C,\varUpsilon)}^{0,*}\oplus\mathcal{D}\mathfrak{F}_{SO(3),(C,\varUpsilon)})\\\
\mathbf{H}_{(C,\varUpsilon)}^{2}\equiv\text{coker}\mathcal{D}\mathfrak{F}_{SO(3),(C,\varUpsilon)}\end{cases}$

######  Lemma 21.

Suppose that $(C,\varUpsilon)$ is a solution to the $SO(3)$ vortex equations
satisfying:

i) $\mathbf{H}_{(C,\varUpsilon)}^{0}\equiv 0$.

ii)
$\bar{\partial}_{C}:\varGamma(\check{E})\rightarrow\varGamma(K_{\check{\varSigma}}^{-1}\otimes\check{E})$
is surjective or alternatively $(C,\varUpsilon)$ is irreducible, in the sense
that $\varUpsilon$ is not identically zero and $C$ is an irreducible
connection.

Then $\mathbf{H}_{(C,\varUpsilon)}^{2}\equiv 0$ and near
$[C,\varUpsilon]\in\mathcal{M}(\check{\varSigma},\check{E})$ the moduli space
has the structure of a smooth manifold of real dimension

(18)
$\dim_{\mathbb{R}}\mathcal{M}(\check{\varSigma},\check{E})=\dim\mathbf{H}_{[C,\varUpsilon]}^{1}=2(g-1)+2c_{1}(\det\check{E})+2(n-n_{0})-2\sum_{i=1}^{n}\frac{b_{i}^{-}+b_{i}^{+}}{a_{i}}$

###### Proof.

In order to show that

$\mathcal{D}\mathfrak{F}_{SO(3),(C,\varUpsilon)}:\varOmega^{1}(\check{\varSigma};\mathfrak{su}(\check{E}))\oplus\varGamma(\check{E})\rightarrow\varOmega^{0}(\check{\varSigma};\mathfrak{su}(\check{E}))\oplus\varGamma(K_{\check{\varSigma}}^{-1}\otimes\check{E})$

is surjective we follow a similar argument to the one given by Hitchin in [48,
Section 5]. First of all, we write

$\mathcal{D}\mathfrak{F}_{SO(3),(C,\varUpsilon)}=\left(\mathcal{D}\mathfrak{F}_{SO(3),(C,\varUpsilon)}^{1},\mathcal{D}\mathfrak{F}_{SO(3),(C,\varUpsilon)}^{2}\right)$

A simple calculation shows that if we define
$\mathbf{J}_{(C,\varUpsilon)}(\dot{c},\dot{\varUpsilon})=(*\dot{c},i\dot{\varUpsilon})$,
then

$\mathcal{D}\mathfrak{F}_{SO(3),(C,\varUpsilon)}^{1}(\dot{c},\dot{\varUpsilon})=-d_{(C,\varUpsilon)}^{0,*}\mathbf{J}(\dot{c},\dot{\varUpsilon})$

This computation is implicitly done in Lemma (29), where we show that
$\mathbf{J}$ defines an almost complex structure on the Zariski tangent space.
Therefore, the formal adjoint of
$\mathcal{D}\mathfrak{F}_{SO(3),(C,\varUpsilon)}$ is (recall
$\mathbf{J}^{*}=-\mathbf{J}$)

$\mathcal{D}\mathfrak{F}_{SO(3),(C,\varUpsilon)}^{*}=\left(\mathbf{J}\circ
d_{(C,\varUpsilon)}^{0}\right)\oplus,\mathcal{D}\mathfrak{F}_{SO(3),(C,\varUpsilon)}^{2,*}:\varOmega^{0}(\check{\varSigma};\mathfrak{su}(\check{E}))\oplus\varGamma(K_{\check{\varSigma}}^{-1}\otimes\check{E})\rightarrow\varOmega^{1}(\check{\varSigma};\mathfrak{su}(\check{E}))\oplus\varGamma(\check{E})$

Now suppose that $(\tilde{\xi},\tilde{\varUpsilon})$ is such that

$\begin{cases}\mathbf{J}\circ d_{(C,\varUpsilon)}^{0}\tilde{\xi}=0\\\
\mathcal{D}\mathfrak{F}_{SO(3),(C,\varUpsilon)}^{2,*}\tilde{\varUpsilon}=0\end{cases}$

Since $\mathbf{J}^{2}=-\text{Id}$, the first equation says that
$\tilde{\xi}\in\ker d_{(C,\varUpsilon)}^{0}$, which vanishes by assumption i).
Therefore $\tilde{\xi}=0$.

Likewise, the second equation says that for all
$(\dot{c},\dot{\varUpsilon})\in\varOmega^{1}(\check{\varSigma};\mathfrak{su}(\check{E}))\oplus\varGamma(\check{E})$
,

$<\bar{\partial}_{C}\dot{\varUpsilon}+\dot{c}^{\prime\prime}\otimes\varUpsilon,\tilde{\varUpsilon}>_{L^{2}}=0$

Taking $\dot{\varUpsilon}=\varUpsilon$ this means that
$<\dot{c}^{\prime\prime}\otimes\varUpsilon,\tilde{\varUpsilon}>_{L^{2}}=0$
since $\bar{\partial}_{C}\varUpsilon=0$. Hence, we obtain the condition

$<\bar{\partial}_{C}\dot{\varUpsilon},\tilde{\varUpsilon}>_{L^{2}}=0\iff<\dot{\varUpsilon},\bar{\partial}_{C}^{*}\tilde{\varUpsilon}>_{L^{2}}=0\iff\tilde{\varUpsilon}\in\ker\bar{\partial}_{C}^{*}$

Therefore $\tilde{\varUpsilon}$ is zero if we assume that $\bar{\partial}_{C}$
is surjective. If we no longer assume that $\bar{\partial}_{C}$ is surjective,
notice that from equation (17)
$<\dot{c}^{\prime\prime}\otimes\varUpsilon,\tilde{\varUpsilon}>_{L^{2}}=0$ is
equivalent to
$<\dot{c}^{\prime\prime},(\tilde{\varUpsilon}\varUpsilon^{*})_{0}>_{L^{2}}=0$,
thus we find that

$\tilde{\varUpsilon}\varUpsilon^{*}=\frac{1}{2}\text{tr}(\tilde{\varUpsilon}\varUpsilon^{*})$

Locally, we can write $\varUpsilon=\left(\begin{array}[]{c}\varPhi_{1}\\\
\varPhi_{2}\end{array}\right)$ and
$\tilde{\varUpsilon}=\left(\begin{array}[]{c}\phi_{1}\\\
\phi_{2}\end{array}\right)$ , the previous equation is equivalent to

$\left(\begin{array}[]{cc}\phi_{1}\bar{\varPhi}_{1}&\phi_{1}\bar{\varPhi}_{2}\\\
\phi_{2}\bar{\Phi}_{1}&\phi_{2}\bar{\varPhi}_{2}\end{array}\right)=\left(\begin{array}[]{cc}\frac{1}{2}\left(\phi_{1}\bar{\varPhi}_{1}+\phi_{2}\bar{\varPhi}_{2}\right)&0\\\
0&\frac{1}{2}\left(\phi_{1}\bar{\varPhi}_{1}+\phi_{2}\bar{\varPhi}_{2}\right)\end{array}\right)$

Therefore, we conclude that

$\begin{cases}\phi_{1}\bar{\varPhi}_{2}=0\\\
\phi_{2}\bar{\Phi}_{1}=0\end{cases}$

Now, if $\varUpsilon$ is not a section of some splitting holomorphic sub-
bundle of $\check{E}$, then we must conclude that $\phi_{1}$ and $\phi_{2}$
must vanish on the open sets where $\varPhi_{1},\varPhi_{2}$ are non-
vanishing. But since $*\tilde{\varUpsilon}$ is a holomorphic section, this can
only happen if $\tilde{\varUpsilon}$ vanishes.

To find the expected dimension of the moduli space at the solution
$[C,\varUpsilon]$, we need to compute

$\text{ind}(d_{(C,\varUpsilon)}^{0,*}\oplus\mathcal{D}\mathfrak{F}_{SO(3),(C,\varUpsilon)})$

This operator is homotopic to the operator

$(d_{C}^{0,*}\oplus*d_{C})\oplus\bar{\partial}_{C}$

so the index is equal to

$\text{ind}(d_{C}^{0,*}\oplus*d_{C})+\text{ind}\bar{\partial}_{C}$

From the Riemann-Roch formula (11), we know that

$\text{ind}_{\mathbb{C}}\bar{\partial}_{C}=2(1-g)+c_{1}(\det\check{E})-\sum_{i=1}^{n}\frac{b_{i}^{-}+b_{i}^{+}}{a_{i}}$

To compute $\text{ind}(d_{C}^{0,*}\oplus*d_{C})$, we use the Dolbeault model,
and compute this index as

$\chi(\check{\varSigma};\mathfrak{su}(\check{E})\otimes\mathbb{C})$

Just as we found Riemann-Roch for $U(2)$ orbifold bundles, suppose that

$\check{E}=\check{L}^{-}\oplus\check{L}^{+}$

Then

$\mathfrak{su}(\check{E})\otimes\mathbb{C}\simeq\mathbb{C}\oplus(\check{L}^{-}\otimes\check{L}^{+*})\oplus(\check{L}^{-*}\otimes\check{L}^{+})$

Therefore,

$\chi(\check{\varSigma};\mathfrak{su}(\check{E})\otimes\mathbb{C})=\chi(\check{\varSigma};\mathbb{C})+\chi(\check{\varSigma};\check{L}^{-}\otimes\check{L}^{+*})+\chi(\check{\varSigma};\check{L}^{-*}\otimes\check{L}^{+})$

Recall that the orbifold cohomology groups with complex coefficients are
isomorphic to the ordinary de Rham cohomology groups, in particular, this
means that

$\chi(\check{\varSigma};\mathbb{C})=1-g$

To compute
$\chi(\check{\varSigma};\check{L}^{-}\otimes\check{L}^{+*})+\chi(\check{\varSigma};\check{L}^{-*}\otimes\check{L}^{+})$
, we need to understand the isotropies of the line bundles
$L=\check{L}^{-}\otimes\check{L}^{+*}$ and
$L^{*}=\check{L}^{-*}\otimes\check{L}^{+}$.

Suppose that at a particular marked point, $b_{i}^{+}\neq b_{i}^{-}$. Then $L$
and $L^{*}$ have non-trivial isotropies. If $L$ has isotropy $b_{i,L}$ at the
marked point $p_{i}$, then we can take the isotropy $b_{i,L^{*}}$ of $L^{*}$
to be $b_{i,L^{*}}=a_{i}-b_{i,L}$. Therefore, in the formula (10) for Riemann-
Roch for orbifold line bundles, the terms $\frac{b_{i,L}}{a_{i}}$ and
$\frac{b_{i,L^{*}}}{a_{i}}$ will add up to $1$.

On the other hand, if one looks at a marked point where $b_{i}^{+}=b_{i}^{-}$,
then, the isotropy of $L$ and $L^{*}$ become zero at this point.

In this way we obtain

$\chi(\check{\varSigma};L)+\chi(\check{\varSigma};L^{*})=2(1-g)-(n-n_{0})$

adding the Euler characteristics we find

$\chi(\check{\varSigma};\mathfrak{su}(\check{E})\otimes\mathbb{C})=3(1-g)-(n-n_{0})$

So the _real_ expected dimension of the moduli space is

$\displaystyle\dim_{\mathbb{R}}\mathcal{M}(\check{\varSigma},\check{E})$
$\displaystyle=$ $\displaystyle
6(g-1)+2(n-n_{0})+4(1-g)+2c_{1}(\det\check{E})-2\sum_{i=1}^{n}\frac{b_{i}^{-}+b_{i}^{+}}{a_{i}}$
$\displaystyle=$ $\displaystyle
2(g-1)+2c_{1}(\det\check{E})+2(n-n_{0})-2\sum_{i=1}^{n}\frac{b_{i}^{-}+b_{i}^{+}}{a_{i}}$

∎

######  Remark 22.

a) In [19], Bradlow and Daskalopoulos restrict themselves to the stable pairs
$(\mathcal{E},\varUpsilon)$ where the bundle $\mathcal{E}$ is semi-stable
(here $\mathcal{E}$ denotes $E$ with a particular holomorphic structure on
it). Then they assume $\deg E$ is sufficiently large so that
$\bar{\partial}_{C}$ becomes surjective [16, Proposition 1.8]. In fact the
proof of our previous lemma should be compared with their proof of Proposion
2.1, which is quite similar. Thaddeus also verifies a similar condition in
[90, (1.10)].

b) As a sanity check, notice that the real dimension for the moduli space of
stable pairs in Thaddeus paper is [90, Equation (2.2)] $2g-4+2\deg E$, while
in our case the expected dimension formula seems to be $2g-2+2\deg E$. In
other words, our dimension formula is larger than the one Thaddeus gives, so
why is that? Basically, we still haven’t taken the quotient by a residual
$S^{1}$ action on the moduli space of stable pairs, which will bring our
dimension down to $2g-3+2\deg E$. Furthermore, Thaddeus is working with a
_fixed_ value for the stability parameter $\tau$, which as we explain in the
next section roughly corresponds to fixing a level set for the moment map
function $\mu$ which is defined at the end of this section.

c) In the next section, we will explain why further assumptions on
$c_{1}(\check{E})$ are needed in order to make the moduli space smooth at the
reducibles (fixed points of the circle action).

As mentioned in the introduction, we can define a circle action on the moduli
space given by

(19) $e^{i\theta}\cdot(C,\varUpsilon)=(C,e^{i\theta}\varUpsilon)$

Just as in the case of [48, Proposition 7.1], or [37, Proposition 3.1] the
fixed points _modulo_ gauge correspond to projectively flat connections or
solutions to the abelian vortex equations.

######  Lemma 23.

A fixed point $[C,\varUpsilon]$ to the circle action takes the form:

i) A projectively flat connection $[C,0]$.

ii) A reducible solution $[C_{L}\oplus(C_{L}^{*}\otimes C^{\det}),\alpha\oplus
0]$ compatible with a splitting
$\check{E}=\check{L}\oplus(\check{L}^{*}\otimes\det\check{E})$, where
$(C_{L},\alpha)$ satisfies the abelian vortex equation

$\text{abelian vortex
equation}\begin{cases}*F_{C_{L}}-\frac{i}{2}|\alpha|^{2}=*\frac{1}{2}F_{C^{\det}}\\\
\bar{\partial}_{C_{L}}\alpha=0\end{cases}$

Moreover, $\alpha$ cannot vanish identically if either of the following
conditions is satisfied:

I) In the smooth case, $c_{1}(\det E)$ is of odd degree.

II) In the orbifold case $\check{\varSigma}=(\varSigma,p_{1},\cdots,p_{n})$
with the multiplicities $a_{i}$ of $p_{i}$ mutually coprime, we have
$\det\check{E}$ is an odd integer power of the fundamental orbifold line
bundle $\check{L}_{0}$ satisfying $c_{1}(\check{L}_{0})=\frac{1}{a_{1}\cdots
a_{n}}$ .

######  Remark 24.

When we have a splitting
$\check{E}=\check{L}\oplus(\check{L}^{*}\otimes\det\check{E})$, it is assumed
implicitly that $\check{L}$ has the correct isotropy as a subbundle of
$\check{E}$ as discussed before.

###### Proof.

Case i) is clear.

ii) If $C$ is reducible, then there is a splitting
$\check{E}=\check{L}\oplus(\check{L}^{*}\otimes\det\check{E})$ of the $U(2)$
bundle compatible with the reduction $C=C_{L}\oplus(C_{L}^{*}\otimes
C^{\det})$, where $C_{L}$ denotes a $U(1)$ connection on $\check{L}$.
Likewise, we can decompose $\varUpsilon$ into components
$\varUpsilon=\alpha\oplus\beta$. The curvature decomposes as

$F_{C}=\left(\begin{array}[]{cc}F_{C_{L}}&0\\\
0&F_{C^{\det}}-F_{C_{L}}\end{array}\right)$

so the traceless part is

$F_{C}^{0}=F_{C}-\frac{1}{2}\text{tr}(F_{C})1_{E}=\left(\begin{array}[]{cc}F_{C_{L}}-\frac{1}{2}F_{C^{\det}}&0\\\
0&\frac{1}{2}F_{C^{\det}}-\frac{1}{2}F_{C_{L}}\end{array}\right)$

The other term of the curvature equation is

$\left(\begin{array}[]{c}\alpha\\\
\beta\end{array}\right)\left(\begin{array}[]{cc}\bar{\alpha}&\bar{\beta}\end{array}\right)=\left(\begin{array}[]{cc}|\alpha|^{2}&\alpha\bar{\beta}\\\
\beta\bar{\alpha}&|\beta|^{2}\end{array}\right)$

So

$\varUpsilon\varUpsilon^{*}-\frac{1}{2}|\varUpsilon|^{2}I_{E}=\left(\begin{array}[]{cc}\frac{1}{2}|\alpha|^{2}-\frac{1}{2}|\beta|^{2}&\alpha\bar{\beta}\\\
\beta\bar{\alpha}&\frac{1}{2}|\beta|^{2}-\frac{1}{2}|\alpha|^{2}\end{array}\right)$

Therefore, the $SO(3)$ vortex equations (14) at reducible solutions can be
expressed as

$\begin{cases}*F_{C_{L}}-*\frac{1}{2}F_{C^{\det}}-\frac{i}{2}|\alpha|^{2}+\frac{i}{2}|\beta|^{2}=0\\\
\alpha\bar{\beta}=0\\\ \bar{\partial}_{C_{L}}\alpha=0\\\
\bar{\partial}_{C_{L}^{*}\otimes C^{\det}}\beta=0\end{cases}$

The last two equations say that $\alpha,\beta$ are holomorphic sections, so by
the unique continuation principle, if they vanish on an open set, they must
vanish identically. Using the second equation we can conclude that _at least_
one of the two sections must vanish. Without loss of generality we can assume
that $\beta$ vanishes (see next remark). Therefore, $\alpha$ must solve the
vortex equation

(20) $U(1)\text{ vortex equations
}\begin{cases}*F_{C_{L}}-\frac{i}{2}|\alpha|^{2}=*\frac{1}{2}F_{C^{\det}}\\\
\bar{\partial}_{C_{L}}\alpha=0\end{cases}$

Up to this point, it is not impossible for $\alpha$ to vanish identically. If
this happens, then we obtain $2F_{C_{\check{L}}}=F_{C^{\det}}$, and so Chern-
Weil theory says that

(21) $c_{1}(\det\check{E})=2c_{1}(\check{L})$

Recall that in this case we are referring to the orbifold Chern classes, which
are _rational_ cohomology classes in general.

a) In the smooth case, the condition (21) is not satisfied if we assume that
$c_{1}(\det E)$ is of _odd degree._

b) In the orbifold case, the classification of topological line bundles is
more complicated as explained before (see also [42, Proposition 1.4]).
However, if we assume that $a_{1},\cdots,a_{n}$ are coprime, then we know that
any line bundle has the form $\check{L}_{0}^{k}$ for some integer $k$.
Therefore, $\det\check{E}=\check{L}_{0}^{k}$ for some $k$ and
$\check{L}=\check{L}_{0}^{l}$ for some $l$. Since $c_{1}$ continues to be a
homomorphism, $c_{1}(\det\check{E})=kc_{1}(\check{L}_{0})$ and
$c_{1}(\check{L})=lc_{1}(\check{L}_{0})$, so (21) is equivalent to

$k=2l$

which again does not have any solution if we assume that $k$ is an odd
integer. ∎

######  Remark 25.

Notice that in the proof of the previous lemma we said that we could assume
$\beta$ to vanish. In reality this depends on the fact that we are ultimately
interested in studying the solutions _up to gauge._

As mentioned before, a gauge transformation which swaps the factors
$\check{L},(\check{L}^{*}\otimes\det\check{E})$ is locally of the form
$\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right)$, which is _not_ of
determinant one. However, the section $\left(\begin{array}[]{c}\alpha\\\
0\end{array}\right)$ is gauge equivalent to the section
$\left(\begin{array}[]{c}0\\\ -\alpha\end{array}\right)$ if we use the matrix
$\left(\begin{array}[]{cc}0&1\\\ -1&0\end{array}\right)$, which is of
determinant one. Since eventually we are interested in the moduli space after
the quotient by the residual $S^{1}$ action is taken, then
$\left(\begin{array}[]{c}0\\\ -\alpha\end{array}\right)$ will be equivalent to
$\left(\begin{array}[]{c}0\\\ \alpha\end{array}\right)$, so it is possible to
swap the factors after the full symmetry is taken into account. Therefore, for
our purposes no generality is lost by assuming that $\beta$ is the section
which vanishes.

######  Lemma 26.

If $(C,\varUpsilon)$ is a solution to the $SO(3)$ vortex equations (14) then
$\mathbf{H}_{(C,\varUpsilon)}^{0}$ vanishes identically except when $C$ is a
reducible connection and $\varUpsilon$ vanishes identically.

###### Proof.

Recall that $d_{(C,\varUpsilon)}^{0}(\xi)=(-d_{C}\xi,\xi\varUpsilon)$ and that
$\mathbf{H}_{(C,\varUpsilon)}^{0}=\ker d_{(C,\varUpsilon)}^{0}$. It is clear
that when $C$ is a reducible connection and $\varUpsilon$ vanishes
identically, then $\dim\mathbf{H}_{(C,\varUpsilon)}^{0}\geq 1$ [it has at
least an $S^{1}$ stabilizer].

If $C$ is an irreducible connection, then $\ker d_{C}=\\{0\\}$ so
$\mathbf{H}_{(C,\varUpsilon)}^{0}$ vanishes.

If $C$ reduces to $C=C_{L}\oplus(C_{L}^{*}\otimes C^{\det})$, then $C_{L}$ has
at least an $S^{1}$ stabilizer which acts on $\varUpsilon=\alpha\oplus 0$ by
multiplication. Therefore, if $\alpha$ does not vanish identically, the action
of $S^{1}$ on $(C_{L},\alpha)$ will be trivial, and the result follows. ∎

In the next sections the following function (moment map) will be very
important.

######  Definition 27.

Define the Higgs strength function

$\displaystyle\mu:$
$\displaystyle\mathcal{A}^{\det}(\check{E})\times\varGamma(\check{E})\rightarrow\mathbb{R}$
$\displaystyle(C,\varUpsilon)\rightarrow\frac{1}{2}\|\varUpsilon\|_{L^{2}(\check{\varSigma})}^{2}$

######  Lemma 28.

Suppose that $(C_{L},\alpha)$ is an abelian vortex, i.e, it solves equation
(20). Then

$\mu(C_{L},\alpha)=\pi(c_{1}(\check{E})-2c_{1}(\check{L}))$

In particular, an abelian vortex can only appear in a reduction of the bundle
$\check{E}$ if $c_{1}(\check{L})\leq\frac{1}{2}c_{1}(\check{E})$.

###### Proof.

The curvature equation
$*F_{C_{L}}-\frac{i}{2}|\alpha|^{2}=*\frac{1}{2}F_{C^{\det}}$ is equivalent to

$\frac{*iF_{C_{L}}}{2\pi}+\frac{1}{4\pi}|\alpha|^{2}=\frac{*iF_{C^{\det}}}{4\pi}$

and after integrating of $\check{\varSigma}$ we conclude that

$\frac{1}{4\pi}\int|\alpha|^{2}=\frac{1}{2}c_{1}(\check{E})-c_{1}(\check{L})$

from which the result follows. ∎

Now we discuss the interpretation of $\mu$ as a moment map. For this we need
to introduce other structures on the moduli space. Recall that in equation
(15) we defined the real inner product

$\mathbf{g}_{(C,\varUpsilon)}((\dot{c}_{1},\dot{\varUpsilon}_{1}),(\dot{c}_{2},\dot{\varUpsilon}_{2}))=\int_{\check{\varSigma}}\left\langle\dot{c}_{1},\dot{c}_{2}\right\rangle+\text{Re}\left\langle\dot{\varUpsilon}_{1},\dot{\varUpsilon}_{2}\right\rangle$

With respect to this inner product we have:

######  Lemma 29.

The endomorphism of
$\mathbf{H}_{(C,\varUpsilon)}^{1}=\ker(d_{(C,\varUpsilon)}^{0,*}\oplus\mathcal{D}\mathfrak{F}_{SO(3),(C,\varUpsilon)})$
given by

$\mathbf{J}_{(C,\varUpsilon)}(\dot{c},\dot{\varUpsilon})=(*\dot{c},i\dot{\varUpsilon})$

defines an almost complex structure on $\mathbf{H}_{(C,\varUpsilon)}^{1}$
which is $\mathbf{g}$ -orthogonal, i.e,
$\mathbf{g}(\mathbf{J}\bullet,\mathbf{J}\bullet)=\mathbf{g}(\bullet,\bullet)$.

###### Proof.

Checking that $\mathbf{J}^{2}=-\mathbf{Id}$ is straightforward as well as the
compatibility with $\mathbf{g}$. The only interesting thing to verify is that
$\mathbf{H}_{(C,\varUpsilon)}^{1}$ is preserved by this almost-complex
structure. Recall that $\mathbf{H}_{(C,\varUpsilon)}^{1}$ consisted of the
equations

$\begin{cases}-d_{C}^{*}\dot{c}+[\dot{\varUpsilon}\varUpsilon^{*}-\varUpsilon\dot{\varUpsilon}^{*}-i\text{Re}<i\varUpsilon,\dot{\varUpsilon}>I_{E}]=0\\\
*d_{C}\dot{c}-i\left[\varUpsilon\dot{\varUpsilon}^{*}+\dot{\varUpsilon}\varUpsilon^{*}-\text{Re}\left\langle\varUpsilon,\dot{\varUpsilon}\right\rangle
I_{E}\right]=0\\\
\bar{\partial}_{C}\dot{\varUpsilon}+\dot{c}^{\prime\prime}\otimes\varUpsilon=0\end{cases}$

Since $*\dot{c}^{\prime\prime}=i\dot{c}^{\prime\prime}$ , it is
straightforward to check that the last equation is preserved under
$\mathbf{J}$. For the first equation,

$\displaystyle-
d_{C}^{*}(*\dot{c})+[(i\dot{\varUpsilon})\varUpsilon^{*}-\varUpsilon(i\dot{\varUpsilon})^{*}-i\text{Re}<i\varUpsilon,i\dot{\varUpsilon}>I_{E}]$
$\displaystyle=$
$\displaystyle*d_{C}(**)\dot{c}+i[\dot{\varUpsilon}\varUpsilon^{*}+\varUpsilon\dot{\varUpsilon}{}^{*}-\text{Re}<i\varUpsilon,i\dot{\varUpsilon}>I_{E}]$
$\displaystyle=$
$\displaystyle-\left(*d_{C}\dot{c}-i\left[\varUpsilon\dot{\varUpsilon}^{*}+\dot{\varUpsilon}\varUpsilon^{*}-\text{Re}\left\langle\varUpsilon,\dot{\varUpsilon}\right\rangle
I_{E}\right]\right)$ $\displaystyle=$ $\displaystyle 0$

Conversely,

$\displaystyle*d_{C}(*\dot{c})-i\left[\varUpsilon(i\dot{\varUpsilon})^{*}+(i\dot{\varUpsilon})\varUpsilon^{*}-\text{Re}\left\langle\varUpsilon,i\dot{\varUpsilon}\right\rangle
I_{E}\right]$ $\displaystyle=$ $\displaystyle-
d_{C}^{*}\dot{c}-i\left[-i\varUpsilon\dot{\varUpsilon}+i\dot{\varUpsilon}\varUpsilon^{*}+\text{Re}\left\langle
i\varUpsilon,\dot{\varUpsilon}\right\rangle I_{E}\right]$ $\displaystyle=$
$\displaystyle-
d_{C}^{*}\dot{c}+\left[-\varUpsilon\dot{\varUpsilon}+\dot{\varUpsilon}\varUpsilon^{*}-i\text{Re}\left\langle
i\varUpsilon,\dot{\varUpsilon}\right\rangle I_{E}\right]$ $\displaystyle=$
$\displaystyle 0$

So the claim is verified. ∎

Therefore, we can define the 2-form

$\boldsymbol{\varOmega}(\bullet,\bullet)=\mathbf{g}(\bullet,\mathbf{J}\bullet)$

Since the formulas for $\mathbf{g},\mathbf{J},\boldsymbol{\varOmega}$ are
base-point independent, it is clear that $\boldsymbol{\varOmega}$ will be a
closed two-form, and $\mathbf{J}$ being integrable implies that we have a
Kahler form on the moduli space.

Now consider the vector field

$\boldsymbol{\partial}_{\theta,(C,\varUpsilon)}=(0,i\varUpsilon)$

This is clearly the vector field associated to the $S^{1}$ action defined
before in equation (19).

Moreover, notice that

$\displaystyle(\imath_{\boldsymbol{\partial}_{\theta}}\boldsymbol{\varOmega})(\dot{c},\dot{\varUpsilon})$
$\displaystyle=$
$\displaystyle\boldsymbol{\varOmega}((0,i\varUpsilon),(\dot{c},\dot{\varUpsilon}))$
$\displaystyle=$ $\displaystyle\int\text{Re}\left\langle
i\varUpsilon,i\dot{\varUpsilon}\right\rangle$ $\displaystyle=$
$\displaystyle\int\text{Re}\left\langle\varUpsilon,\dot{\varUpsilon}\right\rangle$
$\displaystyle=$ $\displaystyle
d\mu_{(C,\varUpsilon)}(\dot{c},\dot{\varUpsilon})$

Where $\mu$ was the Higgs strength function. This is precisely the condition
that guarantees that $\mu$ is the moment map associated to the $S^{1}$ action
[46, Chapter 2].

However, notice that $(0,i\varUpsilon)$ does not belong to
$\mathbf{H}_{(C,\varUpsilon)}^{1}$! Indeed, if one looks at whether
$(0,i\varUpsilon)$ satisfies the first equation for
$\mathbf{H}_{(C,\varUpsilon)}^{1}$, we see that

$\displaystyle(i\varUpsilon)\varUpsilon^{*}-\varUpsilon(i\varUpsilon)^{*}-i\text{Re}<i\varUpsilon,i\varUpsilon>I_{E}$
$\displaystyle=$ $\displaystyle
2i\varUpsilon\varUpsilon^{*}-i|\varUpsilon|^{2}I_{E}$

and this clearly need not vanish. However, since $(0,i\varUpsilon)$ does
satisfy the other two equations defining $\mathbf{H}_{(C,\varUpsilon)}^{1}$,
this just means that the representative for $(0,i\varUpsilon)$ in
$\mathbf{H}_{(C,\varUpsilon)}^{1}$ is of the form
$(0,i\varUpsilon)+d_{(C,\varUpsilon)}^{0}\xi$, for some
$\xi\in\varOmega^{0}(\check{\varSigma},\mathfrak{su}(E))$.

Therefore, in order to check that the moment map condition holds with respect
to the vector $(0,i\varUpsilon)+d_{(C,\varUpsilon)}^{0}\xi$, it suffices to
know that for any
$(\dot{c},\dot{\varUpsilon})\in\mathbf{H}_{(C,\varUpsilon)}^{1}$, we have

$\boldsymbol{\varOmega}(d_{(C,\varUpsilon)}^{0}\xi,(\dot{c},\dot{\varUpsilon}))=0$

See also the discussion in [27, Section iv), page 294]. In our case, this
condition is satisfied, since

$\displaystyle\boldsymbol{\varOmega}(d_{(C,\varUpsilon)}^{0}\xi,(\dot{c},\dot{\varUpsilon}))$
$\displaystyle=$
$\displaystyle\boldsymbol{g}(d_{(C,\varUpsilon)}^{0}\xi,\mathbf{J}(\dot{c},\dot{\varUpsilon}))$
$\displaystyle=$
$\displaystyle\left\langle\xi,d_{(C,\varUpsilon)}^{0,*}\mathbf{J}(\dot{c},\dot{\varUpsilon})\right\rangle$
$\displaystyle=$ $\displaystyle 0$

The last equality is true since
$\mathbf{J}(\dot{c},\dot{\varUpsilon})\in\mathbf{H}_{(C,\varUpsilon)}^{1}$,
and therefore it satisfies the Coulomb condition. Summarizing our findings we
have found:

######  Theorem 30.

The two-form $\boldsymbol{\varOmega}$ is a Kahler form on the smooth points of
the $SO(3)$ vortex moduli space, and $\mu$ is a moment map for the $S^{1}$
action defined in (19).

In the next few sections we will study some properties of the map $\mu$ using
this moment map interpretation.

##  4\. $SO(3)$ Vortices and Stable Pairs

In order to study the moment map $\mu$ more easily, we will now compare in
this section and the next the $SO(3)$ vortex equations to the stable pairs
equations [90, 18, 19, 20, 11].

A stable pair $(\bar{\partial}_{C},\varUpsilon)$ consists for us of a
holomorphic structure $\bar{\partial}_{C}$ on the orbifold bundle $\check{E}$,
together with a holomorphic section $\varUpsilon$ of $\check{E}$, that is,
$\bar{\partial}_{C}\varUpsilon=0$. When convenient, we will sometimes write
the stable pair as $(\check{\mathcal{E}},\varUpsilon)$, where
$\check{\mathcal{E}}$ refers to the bundle $\check{E}$ endowed with the
holomorphic structure given by $\bar{\partial}_{C}$.

The complex gauge group $\mathcal{G}_{\mathbb{C}}^{\det}(\check{E})$ acts on
the space of stable pairs via

$u\cdot(\bar{\partial}_{C},\varUpsilon)=(u\circ\bar{\partial}_{C}\circ
u^{-1},u\varUpsilon)=(\bar{\partial}_{C}-(\bar{\partial}_{C}u)u^{-1},u\varUpsilon)$

Following [30, Section 6.4], we can identify the tangent space to the space of
$\bar{\partial}$ operators with $\varOmega^{0,1}(\text{End}_{0}\check{E})$,
where as usual $\text{End}_{0}\check{E}$ represents the bundle of trace free
endomorphisms of $\check{E}$ (this is because we are working with bundles
having a fixed determinant). The linearization of the derivative of the
complex gauge group action is $\bar{\partial}_{C}$, and so we have:

######  Lemma 31.

The deformation theorem for the stable pair equation is determined by the
cohomolology groups of the complex

$0\rightarrow\varOmega^{0}(\check{\varSigma},\text{End}_{0}\check{E})\rightarrow^{d_{1}}\varOmega^{0,1}(\check{\varSigma},\text{End}_{0}\check{E})\oplus\varOmega^{0}(\check{\varSigma},\check{E})\rightarrow^{d_{2}}\varOmega^{0,1}(\check{E})\rightarrow
0$

where

$\begin{cases}d_{1}\xi=(-\bar{\partial}_{C}\xi,\xi\varUpsilon)\\\
d_{2}(\dot{c}^{\prime\prime},\dot{\varUpsilon})=\bar{\partial}_{C}\dot{\varUpsilon}+\dot{c}^{\prime\prime}\varUpsilon\end{cases}$

###### Proof.

This is a simple computation, and essentially it is the orbifold version of
[19, Proposition 2.1] ∎

Now we review the relevant version of the Hitchin-Kobayashi correspondence
between solutions to the $SO(3)$ vortex equations and stable pairs. This
correspondence has found different instantiations, and the orbifold version is
a trivial extension of the argument given in [63, 31, Theorem 1.4.2], since
integration by parts and the Kahler identities continue to work on orbifolds.

Observe first of all that if $(C,\varUpsilon)$ is an $SO(3)$ vortex, then $C$
naturally defines a holomorphic structure $\bar{\partial}_{C}$, and
$(\bar{\partial}_{C},\varUpsilon)$ is a stable pair. Since
$\mathcal{G}^{\det}(\check{E})$ is a subgroup of
$\mathcal{G}_{\mathbb{C}}^{\det}(\check{E})$, this means that there is a well-
defined map

(22)
$\displaystyle\mathcal{M}^{vor}(\check{\varSigma},\check{E})\rightarrow\mathcal{M}^{st}(\check{\varSigma},\check{E})$
$\displaystyle{}[C,\varUpsilon]_{\mathbb{R}}\rightarrow[\bar{\partial}_{C},\varUpsilon]_{\mathbb{C}}$

where $[\cdot]_{\mathbb{R}}$ denotes the gauge equivalence class with respect
to the “real” gauge group $\mathcal{G}^{\det}(\check{E})$ , while
$[\cdot]_{\mathbb{C}}$ denotes the gauge equivalence class with respect to the
complexified gauge group $\mathcal{G}_{\mathbb{C}}^{\det}(\check{E})$. We need
to understand the surjectivity of this map.

We will focus in the case that $C$ is irreducible and $\varUpsilon$ is nowhere
vanishing, since the case of $(C,0)$ can be found in [83], while the case of a
$U(1)$ vortex on an orbifold can be found in [67, Theorem 5].

We recall first the definition of stability for a pair
$(\bar{\partial}_{C},\varUpsilon)$ [31, Section 1.3], [63, Chapter 1, p.3].
Let $\check{\mathcal{E}}$ denote the bundle $\check{E}$ with the holomorphic
structure induced by $\bar{\partial}_{C}$. Denote by
$\mathcal{R}(\check{\mathcal{E}})$ the set of subsheaves of
$\check{\mathcal{E}}$ with torsion free quotients. For any sheaf $\check{F}$
define the slope $\mu_{\check{g}}(\check{F})$ as

$\mu_{\check{g}}(\check{F})=\frac{\int_{\check{\varSigma}}c_{1}(\check{F})}{\text{rk}(\check{F})}\in\mathbb{Q}$

######  Definition 32.

The pair $(\check{\mathcal{E}},\varUpsilon)$ is stable if
$\mu_{\check{g}}(\check{\mathcal{F}})<\mu_{\check{g}}(\check{\mathcal{E}})$
whenever $\check{\mathcal{F}}\in\mathcal{R}(\check{\mathcal{E}})$ and
$\varUpsilon\in H^{0}(\check{\mathcal{F}})$. In other words,

$c_{1}(\check{L})<\frac{1}{2}c_{1}(\check{E})$

whenever $\varUpsilon\in H^{0}(\check{L})$.

The same arguments to those in [31, Section 2.3] show that a pair
$(\bar{\partial}_{C},\varUpsilon)$ which solves the projective vortex equation

$*F_{C}^{0}-i\left[\varUpsilon\varUpsilon^{*}-\frac{1}{2}|\varUpsilon|^{2}I_{E}\right]=0$

must be stable, provided that $C$ is irreducible and $\varUpsilon$ non-
vanishing. Moreover, if the pair $(\bar{\partial}_{C},\varUpsilon)$ is stable
then it will solve the projective vortex equations (up to gauge of course).
Therefore, we have

######  Theorem 33.

The irreducible solutions to the $SO(3)$ vortex equations $(C,\varUpsilon)$
[so $C$ is irreducible and $\varUpsilon$ non-vanishing], are in bijection with
the stable pairs. In other words, the Kobayashi-Hitchin map (22) is a
bijection when restricted to the irreducible $SO(3)$ vortices.

The reader may be more familiarized with a version of the stable pair
equations where a stability parameter explicitly appears [18, 19, 90, 20].
This occurs if one uses the $U(2)$ vortex equations instead of the $SO(3)$
vortex equations, so we now proceed to explain their relationship. We can
think of the $U(2)$ vortex equations as equations for a pair $(C,\varUpsilon)$
which now satisfy

$\begin{cases}*F_{C}-i\varUpsilon\varUpsilon^{*}+i\tau\text{Id}_{E}=0\\\
\bar{\partial}_{C}\varUpsilon=0\end{cases}$

The main difference with equations (13) is that we are no longer requiring
that $C$ induce a fixed connection $C^{\det}$ on $\det\check{E}$, which is why
we now have an equation for the entire curvature of $F_{C}$ and not just its
trace-free part. Here $\tau$ is a constant which is related to a stability
parameter. Clearly we can break the curvature equation into its traceless and
trace parts, so that the $U(2)$ equations read

(23) $\begin{cases}2*F_{C^{\det}}-i|\varUpsilon|^{2}+2i\tau=0\\\
*F_{C}^{0}-i\left[\varUpsilon\varUpsilon^{*}-\frac{1}{2}|\varUpsilon|^{2}I_{E}\right]=0\\\
\bar{\partial}_{C}\varUpsilon=0\end{cases}$

Notice that if we integrate the first equation, we obtain the relation

$\text{vol}(\check{\varSigma})\tau=\mu(\varUpsilon)+2\pi c_{1}(\det\check{E})$

Therefore, fixing a value of $\tau$ is equivalent to fixing a level set for
the moment map $\mu$. Clearly, any solution $(C,\varUpsilon)$ to the $U(2)$
vortex equations will yield a solution to the $SO(3)$ vortex equations,
provided $C$ happened to induce our predetermined connection $C^{\det}$ on
$\det\check{E}$. On the other hand, if we start with a solution
$(C,\varUpsilon)$ to our $SO(3)$ vortex equations, there is a priori no reason
why the first equation in (23) has to be satisfied, so the moduli space of
$SO(3)$ vortices is not just the moduli space of $U(2)$ vortices which induce
a particular connection on $\det\check{E}$.

##  5\. A Morse-Bott Function on the Moduli Space

As explained in Theorem (30), the Higgs strength function can be interpreted
as a moment map for the $S^{1}$ action on the $SO(3)$ vortex moduli space.

This means that whenever we choose the $U(2)$ bundle $\check{E}$ with the
property that it guarantees the moduli space to be smooth (at the reducibles),
Frankel’s theorem holds [40], which implies in our case the $\mu$ is a Morse-
Bott function with the critical set being equal to the fixed points of the
$S^{1}$ action. In other words, the critical sets of $\mu$ consist of the
abelian vortices or the moduli space of flat connections.

Moreover, the index of a particular critical set is the dimension of the
subspace on which the circle action acts with _negative weight._ As mentioned
before, a version of Frankel’s theorem for almost hermitian manifolds was used
recently by Feehan and Leness in order to compute the Morse indices of the
analogue of $\mu$ in the case of the $SO(3)$ monopole moduli spaces (see
Theorem 1 in [34] and the discussion that follows). In the gauge theory
context, probably the first use of this idea appeared in Hitchin’s paper [48,
Section 7]. For the case of $U(2)$ vortices some calculations in this spirit
are done in [20, Sections 3 and 4], and Thaddeus also computes the dimensions
of the subspaces with positive and negative weights in [91, Section 8].

First we understand the deformation theory at an abelian vortex. Recall that
the linearization of the $SO(3)$ vortex map was given in Lemma 15

$\mathcal{D}\mathfrak{F}_{SO(3),(C,\varUpsilon)}(\dot{c},\dot{\varUpsilon})=\left(*d_{C}\dot{c}-i\left[\varUpsilon\dot{\varUpsilon}^{*}+\dot{\varUpsilon}\varUpsilon^{*}-\text{Re}\left\langle\varUpsilon,\dot{\varUpsilon}\right\rangle
I_{E}\right],\bar{\partial}_{C}\dot{\varUpsilon}+\dot{c}^{\prime\prime}\otimes\varUpsilon\right)$

and likewise from equation (16)

$d_{(C,\varUpsilon)}^{0,*}(\dot{c},\dot{\varUpsilon})=-d_{C}^{*}\dot{c}+[\dot{\varUpsilon}\varUpsilon^{*}-\varUpsilon\dot{\varUpsilon}^{*}-i\text{Re}<i\varUpsilon,\dot{\varUpsilon}>I_{E}]$

At an abelian vortex we can write
$C=C_{\check{L}}\oplus(C_{\check{L}}^{*}\otimes C^{\det})$ and
$\varUpsilon=\alpha\oplus 0$. Since
$\check{E}=\check{L}\oplus(\check{L}^{*}\otimes\det\check{E})$, we have
$\mathfrak{g}_{\check{E}}=i\mathbb{R}\oplus(\check{L}^{2}\otimes(\det\check{E})^{-1})$.
If $\dot{\varUpsilon}=(\dot{\varUpsilon}_{t}\oplus\dot{\varUpsilon}_{n})$ and
$\dot{c}=\dot{c}_{t}\oplus\dot{c}_{n}$ (here $t$ and $n$ stand for tangential
and normal respectively) then

$\begin{cases}\dot{\varUpsilon}\varUpsilon^{*}=\left(\begin{array}[]{c}\dot{\varUpsilon}_{t}\\\
\dot{\varUpsilon}_{n}\end{array}\right)\left(\begin{array}[]{cc}\alpha^{*}&0\end{array}\right)=\left(\begin{array}[]{cc}\dot{\varUpsilon}_{t}\alpha^{*}&0\\\
\dot{\varUpsilon}_{n}\alpha^{*}&0\end{array}\right)\\\
\varUpsilon\dot{\varUpsilon}^{*}=\left(\begin{array}[]{c}\alpha\\\
0\end{array}\right)\left(\begin{array}[]{cc}\dot{\varUpsilon}_{t}^{*}&\dot{\varUpsilon}_{n}^{*}\end{array}\right)=\left(\begin{array}[]{cc}\alpha\dot{\varUpsilon}_{t}^{*}&\alpha\dot{\varUpsilon}_{n}^{*}\\\
0&0\end{array}\right)\\\
\varUpsilon\dot{\varUpsilon}^{*}+\dot{\varUpsilon}\varUpsilon^{*}-\text{Re}\left\langle\varUpsilon,\dot{\varUpsilon}\right\rangle
I_{E}=\left(\begin{array}[]{cc}\text{Re}\left\langle\alpha,\dot{\varUpsilon}_{t}\right\rangle&\alpha\dot{\varUpsilon}_{n}^{*}\\\
\dot{\varUpsilon}_{n}\alpha^{*}&-\text{Re}\left\langle\alpha,\dot{\varUpsilon}_{t}\right\rangle\end{array}\right)\end{cases}$

We can interpret $\dot{c}$ as a matrix by writing
$\dot{c}=\left(\begin{array}[]{cc}\dot{c}_{t}&\dot{c}_{n}\\\
-\dot{c}_{n}^{*}&-\dot{c}_{t}\end{array}\right)$. Then if we write
$\check{N}=\check{L}^{2}\otimes(\det\check{E})^{-1}$ so that
$\dot{c}_{n}\in\varOmega^{1}(\check{\varSigma};\check{N})$, and
$C_{\check{N}}$ represents the induced connection $C_{\check{L}}^{*}\otimes
C^{\det}$,

$*d_{C}\dot{c}-i\left[\varUpsilon\dot{\varUpsilon}^{*}+\dot{\varUpsilon}\varUpsilon^{*}-\text{Re}\left\langle\varUpsilon,\dot{\varUpsilon}\right\rangle
I_{E}\right]=\left(\begin{array}[]{c}*d\dot{c}_{t}-i\text{Re}\left\langle\alpha,\dot{\varUpsilon}_{t}\right\rangle\\\
d_{C_{\check{N}}}\dot{c}_{n}-i\alpha\dot{\varUpsilon}_{n}^{*}\end{array}\right)$

Likewise,

$\bar{\partial}_{C}\dot{\varUpsilon}+\dot{c}^{\prime\prime}\otimes\varUpsilon=\left(\begin{array}[]{c}\bar{\partial}_{C_{\check{L}}}\dot{\varUpsilon}_{t}+\dot{c}_{t}^{\prime\prime}\otimes\alpha\\\
\bar{\partial}_{C_{\check{L}}^{*}\otimes
C^{\det}}\dot{\varUpsilon}_{n}-\dot{c}_{n}^{*\prime\prime}\alpha\end{array}\right)$

It is easy to recognize the first row in each of these vectors as
corresponding to the linearization to the $U(1)$ vortex equations. The
remaining rows together with the Coulomb conditions leads us to define the
normal operator

$\boxed{\mathscr{D}_{(C_{\check{L}},\alpha)}^{n}(\dot{c}_{n},\dot{\varUpsilon}_{n})=\left(\begin{array}[]{c}-d_{C_{\check{N}}}^{*}\dot{c}_{n}-\alpha\dot{\varUpsilon}_{n}^{*}\\\
d_{C_{\check{N}}}\dot{c}_{n}-i\alpha\dot{\varUpsilon}_{n}^{*}\\\
\bar{\partial}_{C_{\check{L}}^{*}\otimes
C^{\det}}\dot{\varUpsilon}_{n}-\dot{c}_{n}^{*\prime\prime}\alpha\end{array}\right)}$

Observe that the normal bundle to the abelian vortex moduli space inside the
$SO(3)$ vortex moduli space then corresponds to
$\ker\mathscr{D}_{(C_{\check{L}},\alpha)}^{n}$. For the case of the
projectively flat connections, the story is a bit different. Here the normal
operator is easily seen to be

$\boxed{\mathscr{D}_{(C,0)}^{n}(\dot{c},\dot{\varUpsilon})=\left(\begin{array}[]{c}-d_{C}^{*}\dot{c}\\\
\bar{\partial}_{C}\dot{\varUpsilon}\end{array}\right)}$

A simple adaptation of (21) shows:

######  Lemma 34.

Suppose that $c_{1}(\check{E})>2c_{1}(K_{\check{\varSigma}})$ and that for the
smooth case $\check{E}$ is of odd odd degree or when all the
$a_{1},\cdots,a_{n}$ are co-prime we are in the case where $\det\check{E}$ is
an odd power of $\check{L}_{0}$. Then the moduli space
$\mathcal{M}(\check{\varSigma},\check{E})$ is smooth at the abelian vortices
and the projectively flat connections.

###### Proof.

It is easier to explain why $\mathscr{D}_{(C,0)}^{n}$ is surjective. Since
$\ker d_{C}=\\{0\\}$ by assumption, we just need to worry about the
surjectivity of $\bar{\partial}_{C}$. Thus we need to guarantee that
$H^{1}(\check{E})$ vanishes.

If $H^{1}(\check{E})\text{$\neq 0$}$, then
$H^{0}(K_{\check{\varSigma}}E^{*})\neq 0$ by Serre duality, so following the
argument in [90, 1.10], we conclude that there is an injection

$0\rightarrow K_{\check{\varSigma}}^{-1}\otimes\det\check{E}\otimes
L_{\check{D}}\rightarrow\check{E}$

for some effective divisor $\check{D}$ (recall $L_{\check{D}}$ was the
orbifold line bundle corresponding to $\check{D}$).

By the famous theorem of Narasimhan and Seshadri [64, 69] in the orbifold (or
more generally parabolic) version, the bundle $\check{E}$ equipped with the
holomorphic structure provided by $\bar{\partial}_{C}$ is stable. Thus

$c_{1}\left(K_{\check{\varSigma}}^{-1}\otimes\det\check{E}\otimes
L_{\check{D}}\right)<\frac{1}{2}c_{1}(\check{E})$

At the same time, the left hand side is greater than or equal to
$c_{1}(\check{E})-c_{1}(K_{\check{\varSigma}})$, so we conclude that

$\frac{1}{2}c_{1}(\check{E})<c_{1}(K_{\check{\varSigma}})$

and thus

$c_{1}(\check{E})<2c_{1}(K_{\check{\varSigma}})$

The converse $c_{1}(\check{E})>2c_{1}(K_{\check{\varSigma}})$ implies that
$\bar{\partial}_{C}$ is surjective follows from the same argument given by
Thaddeus in [90, 1.10]).

The argument given for the surjectivity of $\mathscr{D}_{(C_{L},\alpha)}^{n}$
is indistinguishable from the one we gave in Lemma (21) . Alternatively, one
can check that

$H^{1}(\check{L}^{*}\otimes\det\check{E})\simeq
H^{0}(K_{\check{\varSigma}}\otimes\check{L}\otimes(\det\check{E})^{*})$

and since $c_{1}(\check{L})<\frac{1}{2}c_{1}(\check{E})$ and we are assuming
$\frac{1}{2}c_{1}(\check{E})>c_{1}(K_{\check{\varSigma}})$ we have

$c_{1}(K_{\check{\varSigma}}\otimes\check{L}\otimes(\det\check{E})^{*})=c_{1}(K_{\check{\varSigma}})+c_{1}(\check{L})-c_{1}(\check{E})<c_{1}(K_{\check{\varSigma}})-\frac{1}{2}c_{1}(\check{E})<0$

and thus $H^{1}(\check{L}^{*}\otimes\det\check{E})$ vanishes, guaranteeing the
surjectivity of the $\bar{\partial}_{C_{\check{L}}^{*}\otimes C^{\det}}$
operator. ∎

From the stable pair perspective, the Zariski tangent space can be identified
with the hypercohomology

$\mathbf{H}^{1}(\text{End}_{0}\check{E}\rightarrow^{\varUpsilon}\check{E})$

where $\text{End}_{0}\check{E}=\mathfrak{su}(2)\otimes\mathbb{C}$. The reason
for this is that the deformation theory at $(\bar{\partial}_{C},\varUpsilon)$
is obtained from the complex

$0\rightarrow\varOmega^{0}(\check{\varSigma},\text{End}_{0}\check{E})\rightarrow^{d_{1}}\varOmega^{0,1}(\check{\varSigma},\text{End}_{0}\check{E})\oplus\varOmega^{0}(\check{\varSigma},\check{E})\rightarrow^{d_{2}}\varOmega^{0,1}(\check{\varSigma},\check{E})$

where

$\begin{cases}d_{1}\xi=(-\bar{\partial}_{C}\xi,\xi\varUpsilon)\\\
d_{2}(\dot{c}^{\prime\prime},\dot{\varUpsilon})=(\bar{\partial}_{C}\dot{\varUpsilon}+\dot{c}^{\prime\prime}\varUpsilon)\end{cases}$

When $\varUpsilon=\alpha\oplus 0$, we know that the matrices [37, Section 3.1]

$G(\theta)=\left(\begin{array}[]{cc}1&0\\\ 0&e^{2i\theta}\end{array}\right)$

belong to the stabilizer of $(\check{E},\alpha)$. Thus $G(\theta)$ acts with
trivial weight on the $\check{L}$ factor, and with positive weight on the
$\check{M}\equiv\check{L}^{*}\otimes\det\check{E}$ factor. Since

$\mathfrak{su}(2)\simeq\mathbb{R}\oplus(\check{L}\otimes\check{M}^{*})$

After complexifying and writing an element $A\in\mathfrak{sl}(2,\mathbb{C})$
as

$A=\left(\begin{array}[]{cc}a_{11}&a_{12}\\\ a_{21}&-a_{11}\end{array}\right)$

with

$\begin{cases}a_{11}\in\hom(\check{L},\check{L})\simeq\mathbb{C}\\\
a_{12}\in\hom(\check{M},\check{L})\simeq\check{L}\otimes\check{M}^{*}\\\
a_{21}\in\hom(\check{L},\check{M})\simeq\check{M}\otimes\check{L}^{*}\end{cases}$

we get the decomposition

$\text{End}_{0}\check{E}\simeq\underbrace{\mathbb{C}}_{\text{trivial
weight}}\oplus(\underbrace{\check{L}\otimes\check{M}^{*}}_{\text{negative
weight}})\oplus(\underbrace{\check{M}\otimes\check{L}^{*}}_{\text{positive
weight}})$

Decompose the maps $d_{1},d_{2}$ as

$\begin{cases}d_{1}\xi=\left(\left(\begin{array}[]{cc}-\bar{\partial}\xi_{11}&-\bar{\partial}_{\check{L}\otimes\check{M}^{*}}\xi_{12}\\\
-\bar{\partial}_{\check{M}\otimes\check{L}^{*}}\xi_{21}&\bar{\partial}\xi_{11}\end{array}\right),\left(\begin{array}[]{c}\xi_{11}\alpha\\\
\xi_{21}\alpha\end{array}\right)\right)\\\
d_{2}\left(\left(\begin{array}[]{cc}a_{11}&a_{12}\\\
a_{21}&-a_{11}\end{array}\right),\left(\begin{array}[]{c}\dot{\varUpsilon}_{\check{L}}\\\
\dot{\varUpsilon}_{\check{M}}\end{array}\right)\right)=\left(\begin{array}[]{c}\bar{\partial}_{\check{L}}\dot{\varUpsilon}_{\check{L}}+a_{11}\alpha\\\
\bar{\partial}_{\check{M}}\dot{\varUpsilon}_{M}+a_{21}\alpha\end{array}\right)\end{cases}$

Comparing the equations, we see that for a stable pair the tangential complex
is

$\displaystyle\varOmega^{0}(\check{\varSigma},\mathbb{C})$
$\displaystyle\varOmega^{0,1}(\check{\varSigma},\mathbb{C})\oplus\varOmega^{0}(X,\check{L})$
$\displaystyle\varOmega^{0,1}(\check{\varSigma},\check{L})$
$\displaystyle\xi_{11}$ $\displaystyle\rightarrow$
$\displaystyle(-\bar{\partial}\xi_{11},\xi_{11}\alpha)$
$\displaystyle(a_{11},\dot{\varUpsilon}_{\check{L}})$
$\displaystyle\rightarrow$
$\displaystyle\bar{\partial}_{\check{L}}\dot{\varUpsilon}_{\check{L}}+a_{11}\alpha$

while the normal complex is the direct sum of the negative weight complex

(24)
$\begin{array}[]{cccc}\varOmega^{0}(\check{\varSigma},\check{L}\otimes\check{M}^{*})&&\varOmega^{0,1}(\check{\varSigma},\check{L}\otimes\check{M}^{*})\\\
\xi_{12}&\rightarrow&-\bar{\partial}_{\check{L}\otimes\check{M}^{*}}\xi_{12}\\\
&&a_{12}\rightarrow&0\end{array}$

and the positive weight complex

$\displaystyle\varOmega^{0}(\check{\varSigma},\check{M}\otimes\check{L}^{*})$
$\displaystyle\varOmega^{0,1}(\check{\varSigma},\check{M}\otimes\check{L}^{*})\oplus\varOmega^{0}(X,\check{M})$
$\displaystyle\varOmega^{0,1}(\check{\varSigma},\check{M})$
$\displaystyle\xi_{21}$ $\displaystyle\rightarrow$
$\displaystyle(-\bar{\partial}_{\check{M}\otimes\check{L}^{*}}\xi_{21},\xi_{21}\alpha)$
$\displaystyle(a_{21},\dot{\varUpsilon}_{\check{M}})$
$\displaystyle\rightarrow$
$\displaystyle\bar{\partial}_{\check{M}}\dot{\varUpsilon}_{M}+a_{21}\alpha$

Since a slice for the gauge group action
$\mathcal{G}_{\mathbb{C}}^{\det}(\check{E})$ can be taken to be
$\ker\bar{\partial}_{C}^{*}$ ([41, Section 4.1], [89]), the normal bundle at
$(\bar{\partial}_{C_{\check{L}}},\alpha)$ can be identified with
$\ker\bar{\partial}_{C}^{*}$ together with the kernel of the map
$\dot{\varUpsilon}_{\check{M}}\rightarrow\bar{\partial}_{\check{M}}\dot{\varUpsilon}_{M}+a_{21}\alpha$.

So to compare the normal bundles we complexify
$\ker\mathscr{D}_{(C_{\check{L}},\alpha)}^{n}$. This allows us to write
$\dot{c}_{n}$ in terms of its $(1,0)$ and $(0,1)$ types. For example, the
second equation of $\mathscr{D}_{(C_{\check{L}},\alpha)}^{n}$ becomes

$\displaystyle*d_{C_{\check{N}}}\dot{c}_{n}-i\alpha\dot{\varUpsilon}_{n}^{*}$
$\displaystyle=$
$\displaystyle-*d_{C_{\check{N}}}**\dot{c}_{n}-i\alpha\dot{\varUpsilon}_{n}^{*}$
$\displaystyle=$ $\displaystyle
d_{C_{\check{N}}}^{*}*\dot{c}_{n}-i\alpha\dot{\varUpsilon}_{n}^{*}$
$\displaystyle=$ $\displaystyle
d_{C_{\check{N}}}^{*}(-i\dot{c}_{n}^{\prime}+i\dot{c}_{n}^{\prime\prime})-i\alpha\dot{\varUpsilon}_{n}^{*}$

while the first equation can be rewritten as

$-d_{C_{\check{N}}}^{*}(\dot{c}_{n}^{\prime}+\dot{c}_{n}^{\prime\prime})-\alpha\dot{\varUpsilon}_{n}^{*}$

So the first two equations defining
$\ker\mathscr{D}_{(C_{\check{L}},\alpha)}^{n}\otimes\mathbb{C}$ become
equivalent to

$\begin{cases}d_{C_{\check{N}}}^{*}(\dot{c}_{n}^{\prime}-\dot{c}_{n}^{\prime\prime})+\alpha\dot{\varUpsilon}_{n}^{*}=0\\\
d_{C_{\check{N}}}^{*}(\dot{c}_{n}^{\prime}+\dot{c}_{n}^{\prime\prime})+\alpha\dot{\varUpsilon}_{n}^{*}=0\end{cases}$

which is the same as

$\begin{cases}d_{C_{\check{N}}}^{*}\dot{c}_{n}^{\prime}+\alpha\dot{\varUpsilon}_{n}^{*}=0\\\
d_{C_{\check{N}}}^{*}\dot{c}_{n}^{\prime\prime}=0\iff\dot{c}_{n}^{\prime\prime}\in\ker\bar{\partial}_{C_{\check{N}}}^{*}\end{cases}$

Since $\ker\bar{\partial}_{C_{\check{N}}}^{*}$ was one of the equations
defining the normal bundle for the stable pair equation, this means that the
Zariski tangent space for the moduli space of $SO(3)$ vortices can be regarded
as a subspace of the Zariski tangent space to the moduli space of stable
pairs. Since both moduli spaces have the same dimensions, there is an
isomorphism between the Zariski tangent spaces, so the moduli spaces are
isomorphic as well (not just homeomorphic).

Once we know this, we can compute the dimension of the negative weight complex
(24). A simple application of the index formula shows that this is the same as

$\dim_{\mathbb{R}}\mathbf{H}^{1}(\check{L}\otimes\check{M}^{*}\rightarrow
0)=\dim_{\mathbb{R}}\mathbf{H}^{1}(\check{\varSigma},\check{L}\otimes\check{M}^{*})=-2\chi_{\mathbb{C}}(\check{\varSigma},\check{L}^{2}\otimes(\det\check{E})^{-1})$

where in the last step we used that
$\mathbf{H}^{0}(\check{\varSigma},\check{L}\otimes\check{M}^{*})$ vanishes.
This happens since

$\displaystyle\deg_{B}(\check{L}\otimes\check{M}^{*})$ $\displaystyle\leq$
$\displaystyle c_{1}(\check{L}\otimes\check{M}^{*})$ $\displaystyle=$
$\displaystyle 2\deg\check{L}-\deg\check{E}$ $\displaystyle<$ $\displaystyle
0$

where we used Lemma (28) (strictly speaking we need to assume that $\check{E}$
was chosen so that there are no abelian vortices of degree
$\frac{1}{2}\deg\check{E}$).

To apply Riemann-Roch (10) we need to know what the isotropy of
$\check{L}^{2}\otimes(\det\check{E})^{-1}$ is. Recall that if the $U(2)$
bundle $\check{E}$ has isotropy
$\left(\begin{array}[]{cc}\sigma_{i}^{b_{i}^{-}}\\\
&\sigma_{i}^{b_{i}^{+}}\end{array}\right)$ , then the isotropy of the
determinant line bundle is $\sigma_{i}^{b_{i}^{-}+b_{i}^{+}}$, while the
isotropy of $\check{L}^{\otimes 2}$ will be $\sigma_{i}^{2b_{i}}$ where
$b_{i}=\frac{\epsilon_{i}(b_{i}^{+}-b_{i}^{-})+(b_{i}^{+}+b_{i}^{-})}{2}$ ,
$\epsilon_{i}=1$ if $b_{i}=b_{i}^{+}$ and $\epsilon_{i}=-1$ if
$b_{i}=b_{i}^{-}$. Therefore the isotropy of
$\check{L}^{2}\otimes(\det\check{E})^{-1}$ is

$\epsilon_{i}(b_{i}^{+}-b_{i}^{-})+(b_{i}^{+}+b_{i}^{-})-(b_{i}^{-}+b_{i}^{+})=\epsilon_{i}(b_{i}^{+}-b_{i}^{-})$

if $\epsilon_{i}=1$ then this gives $b_{i}^{+}-b_{i}^{-}$ and this is already
between $0$ and $a_{i}$ so there is nothing to worry about.

If $\epsilon_{i}=-1$ then this gives $b_{i}^{-}-b_{i}^{+}$ which is non-
positive (smaller or equal to $0$ and definitely bigger than $-a_{i}$), so we
should in fact have to consider a shift by $a_{i}$, i.e, the isotropy is

$\begin{cases}b_{i}^{+}-b_{i}^{-}&\epsilon_{i}=1\\\
a_{i}+b_{i}^{-}-b_{i}^{+}&\epsilon_{i}=-1\end{cases}$

Therefore, (minus) the Euler characteristic according to Rieman-Roch is

$g-1+c_{1}(\det\check{E})-2c_{1}(\check{L})+\sum_{i\mid\epsilon_{i}=1}\frac{b_{i}^{+}-b_{i}^{-}}{a_{i}}+n_{-}+\sum_{i\mid\epsilon_{i}=-1}\frac{b_{i}^{-}-b_{i}^{+}}{a_{i}}$

and thus the index is twice this amount, i.e, we found

######  Theorem 35.

Suppose that a bundle $\check{E}$ is chosen on $\check{\varSigma}$ so
$c_{1}(\check{E})>2c_{1}(K_{\check{\varSigma}})$ and that for the smooth case
we are in the case $\check{E}$ with odd degree or when all the
$a_{1},\cdots,a_{n}$ are co-prime we are in the case where $\det\check{E}$ is
an odd power of $\check{L}_{0}$.

Consider the moduli space of abelian vortices associated to the reduction
$\check{E}=\check{L}\oplus\check{L}^{*}\otimes\det\check{E}$. Then the Morse-
Bott index of $\mu$ at this submanifold of abelian vortices is given by the
formula

(25)
$\text{index}(\check{E},\check{L})=2\left[g-1+c_{1}(\det\check{E})-2c_{1}(\check{L})+\sum_{i\mid\epsilon_{i}=1}\frac{b_{i}^{+}-b_{i}^{-}}{a_{i}}+n_{-}+\sum_{i\mid\epsilon_{i}=-1}\frac{b_{i}^{-}-b_{i}^{+}}{a_{i}}\right]$

where $\check{L}$ has isotropy $b_{i}$, and $\epsilon_{i}=1$ if
$b_{i}=b_{i}^{+}$, $\epsilon_{i}=-1$ if $b_{i}=b_{i}^{-}$. Here
$n_{-}=\\#\\{i\mid b_{i}=b_{i}^{-}<b_{i}^{+}\\}$.

######  Remark 36.

From the previous formula we can observe the following:

In the smooth case the formula for the index reduces to

$2(g-1+\underbrace{\deg E-2\deg L}_{\geq 0})$

Therefore the index of the abelian vortices will always be non-negative
whenever $g\geq 1$. If we choose $\deg E$ odd, then the index is strictly
positive whenever $g\geq 1$ so this means that the minima of $\mu$ is not
achieved at the abelian vortices.

Moreover, observe that when $\deg L=0$, then the index gives $2g-2+2\deg E$,
which is the dimension formula for the moduli space of stable pairs we found
in (18). This vortex moduli space thus corresponds to the critical set which
is the absolute maximum for $\mu$, and since the vortex moduli space can be
identified with $\text{Sym}^{2\deg L}\varSigma$, this means that the maximum
occurs at a single abelian vortex (it cannot be more than one since the level
sets of $\mu$ are connected [7, Theorem IV.3.1]).

In the orbifold case when the $a_{i}$ are coprime, if
$\det\check{E}=\check{L}_{0}^{k}$ and $\check{L}=\check{L}_{0}^{l}$ then the
previous quantity is the same as

$2\left[g-1+\underbrace{\frac{k-2l}{a_{1}\cdots
a_{n}}}_{>0}+\underbrace{\sum_{i\mid\epsilon_{i}=1}\frac{b_{i}^{+}-b_{i}^{-}}{a_{i}}}_{>0}+\underbrace{n_{-}+\sum_{i\mid\epsilon_{i}=-1}\frac{b_{i}^{-}-b_{i}^{+}}{a_{i}}}_{>0}\right]$

So the only chance this index has of being non-positive is if we take $g=0$.
This is consistent with the fact that on $S^{2}$ with marked points the moduli
space of parabolic stable bundles can be empty depending on the choice of
weights [69, 64].

######  Example 37.

To illustrate the index calculation, consider on $T^{2}$ the $U(2)$ bundle $E$
with $\deg E=1$. The expected dimension of the $SO(3)$ monopole moduli space
is given in equation (18) and happens to be $2$ in this case. Since $\deg
E>4g-4$ the moduli space of stable pairs is smooth [90].

The abelian moduli spaces can only appear when $\deg L\leq\frac{1}{2}\deg
E=\frac{1}{2}$, so $\deg L=0$ and hence there is a unique solution up to gauge
(because the abelian vortex can be interpreted as a holomorphic function and
must thus be constant). The index of this abelian monopole is $2$, so it
corresponds to the absolute maximum. Since the level sets of $\mu$ are
connected [7, Theorem IV.3.1], there is only a minimum, which must be a flat
connection. After we take the quotient by the $S^{1}$ action, this gives an
interpretation of the moduli space of stable pairs as a cobordism between the
abelian vortex and the flat connections.

##  6\. $SO(3)$ Monopoles on 3-Manifolds

We now review the $SO(3)$ monopole equations on 3-manifolds, which can be
regarded as the dimensional reduction of the $SO(3)$ monopole equations on
4-manifolds introduced by Pidstrigach and Tyurin and studied extensively by
Feehan and Leness [35, 33, 37, 36, 38, 39]. In particular, we will follow
mostly their conventions (see [37, Section 2.1.2]).

In the same way that the Seiberg-Witten equations require a choice of spin-c
structure $\mathfrak{s}$ in order to be defined on a 3 or 4-manifold, the
$SO(3)$ monopole equations require the choice of a _spin-u_ structure
$\mathfrak{t}$. The most expedient way to define them on 3-manifolds $Y$ is as
follows.

Consider a spin-c structure $\mathfrak{s}=(S,\rho)$ which is represented by a
spinor bundle $S$ with corresponding Clifford multiplication map $\rho$. Let
$E$ denote a $U(2)$ bundle on $Y$. Then $(S,\rho)$ and $E$ determine the
spin-u structure

$\mathfrak{t}=(V,\rho_{V})=(S\otimes E,\rho\otimes 1_{E})$

By abuse of notation we will write $\rho$ instead of $\rho_{V}$. The above
construction shows that spin-u structures always exist on a 3-manifold, so the
next question is the topological classification of these. Notice first of all
that if $L$ is an arbitrary complex line bundle then

$S\otimes E\simeq(S\otimes L)\otimes(E\otimes L^{-1})$

and since $\mathfrak{s}_{L}=(S\otimes L,\rho\otimes 1_{L})$ represents another
spin-c structure, the decomposition of $V$ as $S\otimes E$ is not intrinsic.
Since

$\begin{cases}c_{1}(\mathfrak{s})+c_{1}(E)=c_{1}(\mathfrak{s}_{L})+c_{1}(E\otimes
L^{-1})\\\ w_{2}(\mathfrak{su}(E))=w_{2}(\mathfrak{su}(E\otimes
L^{-1}))\end{cases}$

the three-dimensional version of the discussion in [37, Section 2.1.3] shows
that the spin-u structure $\mathfrak{t}$ determines the characteristic classes

$\begin{cases}c_{1}(\mathfrak{t})\equiv&c_{1}(\mathfrak{s})+c_{1}(E)\\\
w_{2}(\mathfrak{t})\equiv&w_{2}(\mathfrak{su}(E))\end{cases}$

On 3-manifolds $w_{2}(\mathfrak{su}(E))$ can always be lifted to an integral
cohomology class because of the Bockstein long exact sequence and the fact
that $H^{3}(Y;\mathbb{Z})\simeq\mathbb{Z}$ has no two-torsion, so any pair
$(c,w)\in H^{2}(Y;\mathbb{Z})\times H^{2}(Y;\mathbb{Z}_{2})$ determines a
spin-u structure.

If we think of the $SO(3)$ monopole equations as a system of equations used to
relate the monopole [55] and instanton Floer homologies on a specific
3-manifold [29], we should regard $w_{2}(\mathfrak{t})$ as being the
characteristic class provided the $SO(3)$ bundle used to define the instanton
Floer homology of that 3-manifold.

Since every $SO(3)$ bundle can be lifted to a $U(2)$ bundle on a 3-manifold,
there is no harm in choosing a reference spin-c structure to be a torsion
spin-c structure, i.e, one for which $c_{1}(\mathfrak{s})=0$, so that we can
think of the spin-u structure $\mathfrak{t}$ as being completely specified by
a choice of a $U(2)$ bundle and nothing else.

Choosing the reference spin-c structure $\mathfrak{s}$ to be torsion also has
the added advantage that there is a _canonical_ lift of the Levi-Civita
connection on $TY$ to a spin-c connection on $S$ [61, Section 5.1]. With this
reference spin-c connection, we can define the configuration space where the
$SO(3)$ monopole equations are defined as

$\mathcal{C}(Y,\mathfrak{t})=\mathcal{A}_{\det}(E)\times\varGamma(S\otimes E)$

where the notation $\mathcal{A}_{\det}(E)$ indicates that we are considering
the space of unitary connections $B$ on $E$ which induce a fixed connection
$B^{\det}$ on $\det E$. The _unperturbed_ $SO(3)$_monopole equations_ are then
equations for a pair $(B,\varPsi)$ which satisfy the equations:

(26)
$\mathit{SO(3)\;monopole\;equations\;}\begin{cases}*F_{B}^{0}+2\rho^{-1}(\varPsi\varPsi^{*})_{00}=0\\\
D_{B}\varPsi=0\end{cases}$

Here $(\varPhi\varPhi^{*})_{00}$ represents the component of
$\varPhi\varPhi^{*}\in\varGamma(S\otimes E\otimes S^{*}\otimes
E^{*})\simeq\varGamma(\text{End}S\otimes\text{End}E)$ which is traceless on
each of the factors of $\text{End}(S)\otimes\text{End}(E)$, that is,
$(\varPhi\varPhi^{*})_{00}$ is a section of
$\text{End}_{0}(S)\otimes\text{End}_{0}(E)$. We are using
$\rho:TY\rightarrow\hom(S,S)$ to identify $TY$ isometrically with the sub-
bundle $\mathfrak{su}(S)$ of traceless, skew-adjoint endomorphisms equipped
with the inner product $\frac{1}{2}\text{tr}(a^{*}b)$. As usual, $*$ denotes
the Hodge star operator on $Y$. Finally, $D_{B}$ represents the twisted Dirac
operator acting on sections of $S\otimes E$, which strictly speaking should be
written as $D_{B_{S}\otimes B}$, where $B_{S}$ denotes the canonical spin-c
connection we are using on $S$, coming from the fact that we are using a
torsion spin-c structure for describing $\mathfrak{t}$.

One can first consider the solutions to the $SO(3)$ monopole equations on $Y$
modulo the gauge group $\mathcal{G}_{\det}(E)$. We will denote such
equivalence classes as pairs $[B,\varPsi]$. In the space

$\mathcal{B}(Y,\mathfrak{t})=\mathcal{C}(Y,\mathfrak{t})/\mathcal{G}_{\det}(E)$

there is still a residual $S^{1}$ action given by

$e^{i\theta}\cdot[B,\varPsi]\rightarrow[B,e^{i\theta}\varPsi]$

######  Lemma 38.

The fixed points of this circle action which solve (26) are of two types:

a) The pairs $[B,0]$ with vanishing spinor, which correspond to the
projectively flat connections.

b) The pairs of the form $[B_{L}\oplus(B^{\det}\otimes
B_{L}^{*}),\varPsi=\psi\oplus 0]$ corresponding to a reduction of the bundle
$E$ into two line bundles $E=L\oplus(\det E\otimes L^{*})$, which solve the
perturbed __ Seiberg-Witten equations

(27)
$\mathit{perturbed\;Seiberg\;Witten\;equations}\begin{cases}*F_{B_{L}}+\rho^{-1}(\psi\psi^{*})_{0}-\frac{1}{2}*F_{B^{\det}}=0\\\
D_{B}\psi=0\end{cases}$

associated to the spin-c structure $S\otimes L$.

###### Proof.

This is basically a restatement of [37, Lemma 3.12], with some simplifications
in the case of a 3-manifold, given the choice of our reference spin-c
structure. The only thing we really need to check is our choice of conventions
for the constants. We have chosen them so that they agree with [55, Eq 4.4].
At a reducible connection $B=B_{L}\oplus(B^{\det}\otimes B_{L}^{*})$, the
traceless part of the curvature $F_{B}^{0}$ can be written as

$\left(\begin{array}[]{cc}F_{B_{L}}-\frac{1}{2}F_{B^{\det}}&0\\\
0&\frac{1}{2}F_{B^{\det}}-F_{B_{L}}\end{array}\right)$

so the curvature equation becomes

$*F_{B_{L}}-\frac{1}{2}*F_{B^{\det}}+\rho^{-1}(\psi\psi^{*})_{0}=0$

We used the fact that for $\varPsi=\psi\otimes 0$, we have [37, Eq. 3.13]