# A Fueter operator for 3/2-spinors

Ahmad Reza Haj Saeedi Sadegh  Northeastern University
<EMAIL_ADDRESS>and  Minh Lam Nguyen  Washington University
in St. Louis<EMAIL_ADDRESS>
###### Abstract.

We show the non-compactness of moduli space of solutions of the monopole
equations for $3/2$-spinors on a closed 3-manifold is equivalent to the
existence of ‘3/2-Fueter sections’ that are solutions of an overdetermined
non-linear elliptic differential equation. These are sections of a fiber
bundle whose fiber is a special 4-dimensional submanifold of the hyperkähler
manifold of center-framed charged one $SU(2)$-instantons on $\mathbf{R}^{4}$.
This fiber bundle does not inherit a hyperkähler structure.

Keywords: Rarita-Schwinger operator, Seiberg-Witten equations, gauge theory,
Fueter sections

###### 2020 Mathematics Subject Classification:

Primary 53Cxx, 57Rxx, 58Jxx, 57Kxx

###### Contents

  1. 1 Preliminaries
    1. 1.1 Connections on fiber bundles
    2. 1.2 3/2-spinors
    3. 1.3 Hyperkäler manifolds
    4. 1.4 Hyperkähler reduction
    5. 1.5 Hyperkähler bundles
    6. 1.6 The Fueter operator
  2. 2 An aquaternionic moduli space
    1. 2.1 $\text{SO}(3)\times\text{Spin}^{c}(3)$ action
    2. 2.2 An aquaternionic quotient
    3. 2.3 The tangent bundle $TW^{\mu}$
    4. 2.4 A normal line bundle on $W^{\mu}$
    5. 2.5 Two examples of hyperkähler bundles
    6. 2.6 Special subbundles of hyperkähler bundles
    7. 2.7 The 3/2-Fueter operator
  3. 3 A Haydys’ correspondence for $3/2$-spinors
  4. 4 The overdetermination of $\mathfrak{Q}_{A}$

## Introduction

### Main result

Let $Y$ be a closed, oriented smooth Riemannian $3$-manifold. Suppose
$P_{spin^{c}}\to Y$ is a $spin^{c}$ structure on $Y$ and
$\not{S}:=P_{spin^{c}}\times_{Spin^{c}(3)}\bf{H}$ is denoted by the associated
spinor bundle over $Y$. $\not{S}$ is a Clifford module; and thus there is a
Dirac operator $D_{A}$ on it. $D_{A}$ is determined by

  * •

a Clifford multiplication induced by the Riemannian metric on $Y$,

  * •

the covariant derivative $\nabla_{A}$ induced by the Levi-Civita connection on
$Y$ and a $U(1)$-connection $A\in\mathcal{A}(det\,P_{spin^{c}})$ on
$det\,P_{spin^{c}}=P_{spin^{c}}/Spin(3)$.

The Dirac operator $D_{A}$ acts on sections of $\not{S}$, which are referred
to as $1/2$-spinors in Physics. There is also a cousin to the Dirac operator
called the Rarita-Schwinger (RS) operator [11, 8, 2]. Associated to a
$spin^{c}$ structure on $Y$, a RS operator is also defined based on a choice
of a $U(1)$-connection on the determinant line bundle of the $spin^{c}$
structure. Very briefly, an RS operator $Q_{A}$ is defined by

$Q_{A}=\pi_{3/2}\circ D^{TY}_{A}|_{\Gamma(X,ker\,c)},$

where

  * •

$ker\,c$ denotes the sub-bundle of $T^{*}Y\otimes\not{S}$ that is the kernel
of the Clifford multiplication $c:T^{*}Y\otimes\not{S}\to\not{S}$,

  * •

$D^{TY}_{A}$ is a Dirac operator on $T^{*}Y\otimes\not{S}$,

  * •

$\pi_{3/2}$ is the orthogonal projection of $T^{*}Y\otimes\not{S}\to ker\,c$.

The operator $Q_{A}$ acts on sections of $ker\,c$, which are referred to as
$3/2$-spinors. $Q_{A}$ is the first example of many higher-spin Dirac
operators associated with a $spin^{c}$ structure on $Y$. Having set up the
terminologies, the Rarita-Schwinger-Seiberg-Witten (RSSW) equations is a
system of geometric PDEs on $Y$ that look for unknowns
$(A,\psi)\in\mathcal{A}(det\,P_{spin^{c}})\times\Gamma(Y,ker\,c)$ satisfying

(0.1) $\begin{cases}Q_{A}\psi=0,\\\ \star_{3}F_{A}-\mu(\psi)=0.\end{cases}$

where $\mu:ker\,c\subset\Gamma(Y,T^{*}Y\otimes\not{S})\to i\mathfrak{su}(2)$
is a quadratic map that maps each $3/2$-spinor $\psi$ to the traceless part of
the endomorphism $\psi\psi^{*}$ on $\not{S}$.

By blowing-up along the locus of the reducible solutions $\psi\equiv 0$ of
(0.1), we obtain the equations with an extra unknown $\epsilon\in(0,\infty)$

(0.2) $\begin{cases}\left\lVert\psi\right\rVert_{L^{4}}=1,\\\ Q_{A}\psi=0,\\\
\epsilon^{2}\star_{3}F_{A}-\mu(\psi)=0.\end{cases}$

The equation (0.1) has an abelian gauge symmetry given by
$\mathcal{G}=Maps(Y,S^{1})$. The moduli space is defined to be the solution
space of (0.1) mod $\mathcal{G}$. Note that the solutions to (0.1) can also be
thought of as critical points of a certain modified Chern-Simons-Dirac
functional $\mathcal{L}^{RS}$, where (0.1) is interpreted as the minimizing
condition for $\mathcal{L}^{RS}$ (see [12] for more details).

In dimension four, the RSSW equations were first introduced by the second-
named author in [10]. The moduli space of the $4$-D RSSW equations is known to
be not compact in general–a feature that is different from the classical
Seiberg-Witten theory. There is a topological obstruction in terms of an
inequality relating the second betti number $b_{2}$ and the signature of the
four manifolds that guarantees the non-compactness of the moduli space. Thus,
on those four manifolds that satisfy such topological conditions, we are
guaranteed to have non-trivial solutions of the RSSW equations.

The story is a bit different in dimension three. In [12], we studied the
behavior of convergence of solutions of (0.1) after the blow-up procedure. In
particular, we proved that if $\\{(A_{n},\psi_{n},\epsilon_{n})\\}$ is a
sequence of solutions of (0.2) such that $F_{A_{n}}$ are uniformly bounded in
$L^{6}$ and $\epsilon_{n}$ gets arbitrarily small, then away from a closed
nowhere-dense subset $Z\subset Y$, after passing through a subsequence and up
to gauge transformations, $A_{n}$ converges weakly to $A$ in $L^{2}_{1,loc}$,
$\psi_{n}$ converges weakly to $\psi$ in $L^{2}_{2,loc}$, and $(A,\psi)$ must
satisfy the following degenerate situation

(0.3) $Q_{A}\psi=0,\quad\mu(\psi)=0.$

The limiting solutions of the moduli space are called $3/2$-Fueter sections.
Formally, they solve (0.2) by setting $\epsilon=0$. Then one of the
consequences of the above statement is that the existence of $3/2$-Fueter
sections is obstruction to the compactness of moduli space of (0.1).
Indirectly, if one can show that (0.2) with $\epsilon=0$ has no solution in
certain situations, then potentially the moduli space of (0.1) would be
compact!

In this paper, we study the degenerate situation of (0.1). Viewing the
Clifford module $T^{*}Y\otimes\not{S}$ as a (linear) hyperkähler fiber bundle
over $Y$ and $\mu$ as an associated hyperkähler moment, the hyperkähler
reduction $\mu^{-1}(0)/S^{1}$ in turn is another hyperkähler fiber bundle over
$Y$ [7]. We consider a subbundle of $\mu^{-1}(0)/S^{1}$ by intersecting the
entire total space with $ker\,c$. Denote such a subbundle by $\mathbb{W}_{0}$.
Note that $\mathbb{W}_{0}$ no longer necessarily inherits a hyperkähler
structure. However, there is a one-to-one correspondence between solutions of
(0.3) and solutions of a certain non-linear differential operator
$\mathfrak{Q}_{B}$ defined on $\mathbb{W}_{0}$. In particular, the main result
of this paper shows that

######  Theorem.

(ref. Theorem 3.1 and Theorem 4.1) Let $\psi\in\Gamma(Y,ker\,c)$ be a nowhere
vanishing 3/2-spinor section that satisfies the equations

$\begin{cases}Q_{A}\psi=0,\\\ \mu(\psi)=0.\end{cases}$

Then there exists a section $\Phi_{0}\in\Gamma(Y,\mathbb{W}_{0})$ and a
connection $B$ satisfying $\mathfrak{Q}_{B}\Phi_{0}=0$. The converse also
holds: For every $\Phi_{0}\in\Gamma(Y,\mathbb{W}_{0})$ satisfying
$\mathfrak{Q}_{B}\Phi_{0}=0$, there is a solution $(A,\psi)$ of (0.3).
Furthermore, $\mathfrak{Q}_{B}$ is a non-linear over-determined elliptic
operator.

We call $\mathfrak{Q}_{B}$ a 3/2-Fueter operator. To the best of our
knowledge, this is the first time such an operator is defined in the
literature. Its construction mirrors that of the Fueter operator, which can be
thought of as a generalized (non-linear) Dirac operator defined on a
hyperkähler fiber bundle over $Y$ associated with a $spin^{c}$ structure (see
[5, 6, 13]). Whenever there is a Dirac operator, there is also an RS operator.
Following this philosophy, it is not unexpected to have a counterpart of the
Fueter operator for the $3/2$-spinors. Our 3/2-Fueter operator
$\mathfrak{Q}_{B}$ fulfills exactly this role. In other words,
$\mathfrak{Q}_{B}$ is the gauged-sigma model for the RS operator.

######  Remark 0.1.

One should think of the main theorem above as a so-called Haydys’
correspondence for 3/2-spinors (see [5, 6]). However, there is a notable
difference in our situation: $\mathfrak{Q}_{B}$ is over-determined, which
indicates there might be certain situations of three-manifolds where there
should be no solutions of (0.3). Combined with our result in [12], possibly on
certain three-manifolds, the moduli space of RSSW equations is compact. This
begs further investigation and will be addressed in our future works.

######  Remark 0.2.

We have a rather explicit local description of the fibers of $\mathbb{W}_{0}$
(ref. Proposition 2.1). The fibers are $4$-dimensional manifolds and can be
viewed as a variety, which means that the total space $\mathbb{W}_{0}$ is
$7$-dimensional. It would be interesting to know more about the topology and
geometry of $\mathbb{W}_{0}$.

The RSSW equations can be of interest to high-energy Physics. From a physical
perspective, the equations describe a minimally coupled $U(1)$ gauge field
with spin $3/2$-charged matter. In Physics, spin $3/2$-fields are usually
considered to be a part of the supergravity multiplet, but those would not be
charged under $U(1)$ gauge symmetry. Thus, solutions of (0.1) should be
referred to as matters. However, it is possible to swap out $U(1)$ with some
other compact Lie group $G$ to account for twisted supergravity phenomenon and
derive a similar behavior of convergence result as in [12] and a version of
Haydys’ correspondence analogous to the main result of this paper. This circle
of ideas will be addressed elsewhere.

As a concluding remark for this subsection, we state the main motivation for
us to study (0.1). Our larger goal is to derive a new $3$-manifold invariant
using RS operator. The equations we study fall largely within a framework of
Langragian QFTs. Thus, one could either define a numerical invariant via
Rozansky-Witten’s approach by taking a formal path integral of
$\exp(\text{const}\cdot\mathcal{L}^{RS})$ against the moduli space of (0.1),
or define a homological invariant via a certain Floer theory approach that
should categorify the previously mentioned numerical invariant. The results in
our previous paper [12] and in this current one serve as a first step toward
this program.

### Organization of the paper

In Section 1 of the paper, we discuss background materials. We begin by
recalling the notions of connection and covariant derivative on fiber bundles,
3/2-spinors, and hyperkähler structure and its reduction. We then discuss the
notion of hyperkähler bundles and their associated generalized Dirac
operators.

In Section 2, we introduce an 8-dimensional hyperkähler manifold with an
action of $\text{SO}(3)\times\text{Spin}^{c}(3)$ and special invariant
4-dimensional submanifold $W_{0}$. We study this 4-manifold in more detail in
the subsequent subsections. We show the existence of a canonical line bundle
over this manifold. We then construct a rank four fiber with fiber $W_{0}$ and
introduce a nonlinear differential operator on this bundle that is called the
3/2-Fueter operator.

In Sections 3 and 4, we prove the paper’s main result by dividing it into two
parts. In the first part, we prove the existence of solutions to the
degeneracy equations (0.3) is equivalent to the existence of solutions to the
3/2-Fueter operator. In the second part, we prove that the 3/2-Fueter equation
is an overdetermined elliptic equation.

######  Acknowledgement.

We would like to thank Vafa Cumrum and Sergei Gukov for a discussion about the
broad physical framework that our work could fit into. The second-named author
would like to thank Gregory Parker for some of the discussions and suggestions
regarding the construction of the 3/2-Fueter operator $\mathfrak{Q}_{B}$.

##  1\. Preliminaries

###  1.1. Connections on fiber bundles

We first recall some basic facts about Ehressman connections on a fiber
bundle. Consider a fiber bundle bundle $\pi:\mathbb{M}\to Y$. The _vertical
bundle_ $\mathcal{V}\mathbb{M}\to\mathbb{M}$ is defined as the kernel of the
differential map $d\pi:T\mathbb{M}\to TY$. A choice of (Ehressman) connection
on $\mathbb{M}\to Y$ determines a bundle decomposition
$T\mathbb{M}=\mathcal{V}\mathbb{M}\oplus\mathcal{H}\mathbb{M}$ where we call
$\mathcal{H}\mathbb{M}$ is called the horizontal bundle whose fibers naturally
identify with tangent spaces of $Y$, i.e., for all $p\in\mathbb{M}$:

$\mathcal{H}\mathbb{M}|_{p}\buildrel\pi\over{\simeq}T_{\pi(p)}Y.$

The covariant derivative (associated with the connection) of a section of the
fiber bundle is defined as follows. Let $\phi\in\Gamma(\mathbb{M})$ be a
section, i.e., a map $\phi:Y\to\mathbb{M}$ such that
$\pi\circ\phi=\textup{id}_{Y}.$ Taking a differential gives a bundle map
$d\phi:TY\to T\mathbb{M}.$ Composing with the projection map
$T\mathbb{M}\to\mathcal{V}\mathbb{M}$ gives the covariant derivative of the
section:

$\nabla\phi:TY\to\mathcal{V}\mathbb{M}$

and equivalently, a $\phi^{*}\mathcal{V}\mathbb{M}$-valued one-form

(1.1) $\nabla\phi\in\Omega^{1}(Y,\phi^{*}\mathcal{V}\mathbb{M}).$

######  Remark 1.1.

When $\pi:\mathbb{M}\to Y$ is a vector bundle, one has a natural isomorphism
$\mathbb{M}\simeq\phi^{*}\mathcal{V}\mathbb{M}$ and hence the covariant
derivative above is an $\mathbb{M}$-valued one-form.

When $P\to Y$ is principal $G$-bundle, by a connection, we mean an equivariant
decomposition $TP=\mathcal{V}P\oplus\mathcal{H}P$. Fixing a connection on the
principal bundle $P\to Y$ gives a covariant derivative on any associated
bundle $\mathbb{M}=P\times_{G}M$ where $M$ is a $G$-manifold; any section
$\phi\in\Gamma(\mathbb{M})$ can be considered as an equivariant function
$\phi:P\to M$. The restriction of the differential $d\phi:TP\to TM$ to the
horizontal subbundle $\mathcal{H}P$ gives an bundle map

$d\phi|_{\mathcal{H}P}:\mathcal{H}P\to TM.$

Since $\mathcal{H}P\to TY$ is a principal $G$-bundle and using the canonical
isomorphism$V\mathbb{M}\simeq\mathcal{H}P\times_{G}TM$ ($\simeq
P\times_{G}TM$) we obtain a bundle map

$\nabla\phi:TY\to\mathcal{V}\mathbb{M}\simeq P\times_{G}TM,$

and equivalently, a vector bundle valued one-form

$\nabla\phi\in\Omega^{1}(Y,\phi^{*}\mathcal{V}\mathbb{M}).$

###  1.2. 3/2-spinors

In this article, $\bf{H}$ denotes the division algebra of quaternions with its
natural inner product, and $\textup{Im}{\bf H}$ denotes its subspace of pure
imaginary elements. Let $\bf{H}^{\pm}$ be the positive and negative half-spin
representation of the Clifford algebra $\text{C}\ell(4)$ with their natural
identification with quaternions $\bf{H}$. Both $\bf{H}^{\pm}$ give the two
spin representation of the Clifford algebra $\text{C}\ell(3)$; however, they
are equivalent as the representation of the spin group
$\text{Spin}(3)\subset\text{C}\ell(3)$.

The Clifford action on the spinors $\bf{H}\simeq\bf{H}^{+}$ gives the linear
map

$c:\textup{Hom}(\mathbf{C}\otimes\textup{Im}{\bf H},\bf{H})\to\bf{H}$

whose kernel we call the space of _3/2-spinors_ :

$W:=\ker(c).$

The complement of $W$ in $\textup{Hom}(\mathbf{C}\otimes\textup{Im}{\bf
H},\bf{H})$ is a copy of spinor space $\bf{H}$. Indeed, consider the map

(1.2) $\iota:\bf{H}\to\textup{Hom}(\mathbf{C}\otimes\textup{Im}{\bf
H},\bf{H})$ $\iota(s)=I^{*}\otimes Is+J^{*}\otimes Js+K^{*}\otimes Ks.$

One has the orthogonal decomposition

$\textup{Hom}(\mathbf{C}\otimes\textup{Im}{\bf
H},\mathbf{H})=\iota(\boldsymbol{H})\oplus W.$

By “orthogonal”, we refer to the hermitian structure on
$\textup{Hom}(\mathbf{C}\otimes\textup{Im}{\bf H},\bf{H})$ given by
$\langle\phi,\psi\rangle:=\frac{1}{2}\textup{tr}(\phi\psi^{*})$. This
Hermitian structure coincides with the tensor hermitian structure on
$(\mathbf{C}\otimes\textup{Im}{\bf
H})^{*}\otimes\bf{H}\simeq\textup{Hom}(\mathbf{C}\otimes\textup{Im}{\bf
H},\bf{H})$.

We also put a Riemannian structure on
$S:=\textup{Hom}(\mathbf{C}\otimes\textup{Im}{\bf H},\bf{H})$, by putting the
inner product on the tangent spaces $T_{\phi}S\simeq S$ given by

$(A,B):=\frac{1}{2}\textup{Re}(\textup{tr}(AB^{*})).$

###  1.3. Hyperkäler manifolds

Let $(M,g,I,J,K)$ be a hyperkähler manifold that is a Riemannian manifold
$(M,g)$ with three integrable almost complex structures on the tangent bundle
that are $g$-kähler and satisfy the quaternionic relations:

$I^{2}=J^{2}=K^{2}=IJK=-1.$

One has three Kähler forms

$\omega_{I}(u,v):=g(Iu,v),\ \ \omega_{J}(u,v):=g(Ju,v),\ \
\omega_{K}(u,v):=g(Ku,v).$

######  Proposition 1.2.

[7] With respect to the complex structure $I$, one has the holomorphic
symplectic form

$\Omega=\omega_{J}+i\omega_{K}.$

Conversely, any Kähler manifold with a holomorphic symplectic form is
hyperkähler.

The 2-sphere of complex structure

(1.3) $\mathfrak{b}(M,g)=\\{aI+bJ+cK:a^{2}+b^{2}+c^{2}=1\\}$

consists of the integrable almost complex structures on $M$ that are Kähler
with respect to the metric $g$. When the Riemannian metric is fixed we drop
the reference to the metric for brevity of the notation. Hence for any
orthonormal basis $u_{1},u_{2},u_{3}$ of $\mathbf{R}^{3}$, the triple
$(I^{\prime},J^{\prime},K^{\prime})$ with

$I^{\prime}:=u_{1}\cdot(I,J,K),\ J^{\prime}:=u_{2}\cdot(I,J,K),\
K^{\prime}=u_{3}\cdot(I,J,K),$

give a hyperkähler structure on $(M,g)$ equivalent to the original one
(obtained by a rotation). If we drop the oriented condition, we need to
consider the triple $(I^{\prime},J^{\prime},-K^{\prime})$.

The most basic examples of a hyperkähler manifolds are quaternionic vector
spaces, including $\not{S}$ or $\textup{End}(E,\not{S})$ where $E$ is a
hermitian vector space. One obtains more such examples via hyperkähler
quotients.

###  1.4. Hyperkähler reduction

Consider a hyperkähler manifold $(M,g,I,J,K)$ carrying action of a compact Lie
group $G$ under preserving the symplectic forms
$\omega_{I},\omega_{J},\omega_{K}$. Such an action of $G$ on $M$ is called a
hyperkähler action.

For any complex structure $S\in\\{I,J,K\\}$ and any $\xi\in\mathfrak{g}$, by
the Cartan’s magic formula, we have

$0=L_{X^{M}_{\xi}}\omega_{S}=d(\iota_{X^{M}_{\xi}}\omega_{S})+\iota_{X^{M}_{\xi}}(d\omega_{S}),$

where $\iota_{v}$ denotes the usual contraction operator of a differential
form by a vector field on $M$, and $X^{M}_{\xi}$ denotes the killing vector
field on $M$ associated with $\xi$. Since $\omega_{S}$ is a symplectic form,
the above identity tells us that $\iota_{X^{M}_{\xi}}\omega_{S}$ is a closed
$1$-form. Furthermore, if we assume that $H^{1}_{dR}(M;\mathbf{R})$ is
trivial, then every closed $1$-form is exact. Thus, there exists a function

$\mu^{S}_{X^{M}_{\xi}}:M\to\mathbf{R},\quad
d\mu^{S}_{X^{M}_{\xi}}=\iota_{X^{M}_{\xi}}\omega_{S}.$

For this, we may define (assume that there is an $Ad$-invariant metric on
$\mathfrak{g}$) a map $\mu^{S}:M\to\mathfrak{g}^{*}$ where

$\langle\mu^{S}(m),\xi\rangle=\mu^{S}_{X^{M}_{\xi}}(m).$

We can combine each of the moment maps defined above as

$\mu=\mu^{I}i+\mu^{J}j+\mu^{K}k:M\to\mathfrak{g}^{*}\otimes\text{Im}\mathbf{H}\cong\mathfrak{g}^{*}\otimes\mathfrak{su}(2).$

Note that by construction, such a map $\mu$ is $G$-equivariant. This prompts
the following definition.

######  Definition 1.3.

A hyperkähler moment map of a hyperkähler action $G\curvearrowright M$ is a
map $\mu:M\to\mathfrak{g}^{*}\otimes\mathfrak{su}(2)$ such that

  1. (1)

$\mu$ is $G$-equivariant,

  2. (2)

$d\mu=\iota_{X^{M}_{\xi}}\omega$, for all $\xi\in\mathfrak{g}$ and $\omega$ is
the hyperkähler form associated with the hyperkähler structure on $M$.

If a hyperkähler $G$-action possesses a hyperkähler moment map $\mu$, we say
that $G\curvearrowright M$ is a tri-Hamiltonian action.

To get a new hyperkähler manifold from an old one, we suppose that there is a
tri-Hamiltonian action of $G$ and consider the associated hyperkähler moment
map

$\mu:M\to\mathfrak{g}^{*}\otimes\mathfrak{su}(2).$

######  Theorem 1.4.

[7] If $c\in\mathfrak{g}^{*}\otimes\mathfrak{su}(2)$ is an invariant regular
value, then the quotient $\mu^{-1}(c)/G$ is a hyperkähler manifold of
dimension $\dim(M)-4\dim(G).$

When the choice of the invariant regular value $c$ is clear, we simply denote
by $M\mathbin{/\mkern-5.0mu/\mkern-5.0mu/}G$ the hyperkähler reduction
$\mu^{-1}(c)/G$. Denote by $\mathcal{L}\subset TM$ the subbundle spanned by
the Killing vector field of the action of $G$. So, the fibers of $\mathcal{L}$
are isomorphic to the Lie algebra $\mathfrak{g}$. We will need the following
lemma in the subsequent sections:

######  Lemma 1.5.

[7] One has the following $G$-equivariant orthogonal decomposition:

$TM|_{\mu^{-1}(c)}=\mathcal{H}\oplus\mathcal{L}\oplus I\mathcal{L}\oplus
J\mathcal{L}\oplus K\mathcal{L}$

where $\mathcal{H}$ is a vector bundle whose fiber isometrically isomorphic to
tangent spaces of $M\mathbin{/\mkern-5.0mu/\mkern-5.0mu/}G$ under the quotient
map $\tau:M\to M\mathbin{/\mkern-5.0mu/\mkern-5.0mu/}G$ and one also has

$T\mu^{-1}(c)=\mathcal{H}\oplus\mathcal{L}.$

######  Example 1.6.

The hyperkähler manifold $M:=\textup{Hom}(E,\mathbf{H})\cong\mathbf{H}\otimes
E$, where $E$ is an $n$-dimensional hermitian vector space, comes with a
$U(1)$-moment map associated with a $U(1)\curvearrowright M$ induced by the
left multiplication of $U(1)$ on $\mathbf{H}$

$\mu:\textup{Hom}(E,\mathbf{H})\to\mathfrak{u}(1)\otimes\mathfrak{su}(2)$
(1.4) $\psi\mapsto\psi\psi^{*}-\frac{1}{2}|\psi|^{2}\textup{id}_{\mathbf{H}}.$

Although $0$ is a critical point of $\mu$, the set $\mu^{-1}\backslash\\{0\\}$
consist of only regular points. Hence one has the hyperkähler reduction

(1.5)
$\Big{(}\textup{Hom}(E,\mathbf{H})\backslash\\{0\\}\Big{)}\mathbin{/\mkern-5.0mu/\mkern-5.0mu/}U(1):=\Big{(}\mu^{-1}(0)\backslash\\{0\\}\Big{)}/U(1).$

It follows from the ADHM construction [1, 4] that the hyperkähler quotient
(1.5) can be identified with the moduli space $\mathcal{M}_{\textup{fr}}(1,n)$
of framed charge one $\text{SU}(n)$-instantons on $\mathbf{R}^{4}$ which is a
$4(n-1)$-dimensional non-compact hyperkählaer manifold. The action of
$\text{Spin}(3)$ on $\mathbf{H}$ and the action of $\text{SU}(n)$ on $E$
induce an action of the Lie group $\text{SU}(n)\times\text{Spin}(3)$ on
$\mathcal{M}_{\textup{fr}}(1,n)$.

###  1.5. Hyperkähler bundles

The hyperkähler structure gives a parallel action of $\textup{Im}{\bf H}$ on
the tangent bundle $M$:

$\gamma:\textup{Im}{\bf H}\to\textup{End}(TM)$ $ai+bj+ck\mapsto aI+bJ+cK$

which satisfies the Clifford relations:

$\gamma(v)^{2}+|v|^{2}=0.$

(Note that $\gamma$ can be equivalently given by an isometry between the unit
sphere $\textup{S}(\textup{Im}{\bf H})$ and $\mathfrak{b}(M)$.) Hence, one
obtains the Clifford action

(1.6) $\gamma:\text{C}\ell(3)\to\textup{End}(TM)$

that is parallel with respect to the Riemannian metric. Conversely, any
parallel Clifford action (1.6) is given by an oriented isometry

$\gamma:\textup{S}(\textup{Im}{\bf H})\to\mathfrak{b}(M).$

This observation extends to hyperkähler bundles over 3-manifolds due to Taubes
[13]. Fix an oriented Riemannian 3-manifold $(Y,g_{Y})$ which is always
parallelizable. We then can write

$TY\simeq TY^{*}\simeq P_{\text{SO}}\times_{\text{SO}(3)}\textup{Im}{\bf H}.$

Here $P_{\text{SO}}$ is the bundle of oriented orthonormal frames of $Y$.

######  Definition 1.7.

By a _hyperkähler bundle_ over $Y$ we mean a fiber bundle $\pi:\mathbb{M}\to
Y$ with fiber a hyperkähler manifold $(M,g,I,J,K)$ with an isometry of bundles
$\gamma:\textup{S}(TY^{*})\buildrel\simeq\over{\to}\mathfrak{b}(\mathbb{M})$.
Here $\textup{S}(TY^{*})$ is the unit tangent bundle and
$\mathfrak{b}(\mathbb{M})$ is defined fiberwise as in (1.3). By the discussion
above we then obtain an action

$\gamma:\pi^{*}TY^{*}\to\textup{End}(\mathcal{V}\mathbb{M})$

that extends to a bundle map
$\gamma:\pi^{*}\text{C}\ell(TY^{*})\to\textup{End}(\mathcal{V}\mathbb{M})$.
Equivalently, one obtains a section of the tensor bundle

(1.7) $\gamma\in\Gamma(\pi^{*}TY\otimes\textup{Hom}(\mathcal{V}\mathbb{M})).$

In this paper, the main examples of hyperkähler bundles are associated
bundles. For some compact Lie group $\mathscr{K}$, a principal
$\mathscr{K}$-bundle $P$, and a hyperkähler manifold $M$ with a
$\mathscr{K}$-action, we form the hyperkähler bundle
$\mathbb{M}:=P\times_{\mathscr{K}}M$. A special case treated here is for

$M=\textup{Hom}(\mathbf{C}^{n},\mathbf{H})\backslash\\{0\\}\ \textup{or}\
\Big{(}\textup{Hom}(\mathbf{C}^{n},\mathbf{H})\backslash\\{0\\}\Big{)}\mathbin{/\mkern-5.0mu/\mkern-5.0mu/}U(1),$
$\mathscr{K}=\text{SU}(n)\times\text{Spin}(3),$

and $P$ is a principal $\text{SU}(n)\times\text{Spin}(3)$-bundle.

###  1.6. The Fueter operator

Let $\mathbb{M}\to Y$ be a hyperkähler bundle and fix a connection. For a
section $\phi\in\Gamma(\mathbb{M})$, we can pull back the Clifford section
(1.7) to obtain:

(1.8)
$\phi^{*}\gamma:\Gamma(Y,TY\otimes\phi^{*}\textup{End}(\mathcal{V}\mathbb{M})).$

Then we may compose this pullback section with the covariant derivative
$\nabla\phi\in\Omega^{1}(Y,\phi^{*}\mathcal{V}\mathbb{M})$ to obtain the
_Fueter operator_ $\mathfrak{F}$:

(1.9)
$\mathfrak{F}\phi:=\phi^{*}\gamma\circ\nabla\phi\in\Gamma(Y,\phi^{*}\mathcal{V}\mathbb{M}).$

The Fueter operator is a nonlinear generalization of Dirac operators first
introduced in [13]. When $N$ is a representation of the Clifford algebra
$\text{C}\ell(3)$, one obtains a Clifford module
$\mathbb{N}:=P_{\text{Spin}^{c}}\times_{\text{Spin}^{c}(3)}N$. Since
$\mathbb{N}$ is a vector bundle, for any section $\phi\in\Gamma(\mathbb{N})$,
one has the natural identification
$\phi^{*}\mathcal{V}\mathbb{N}\simeq\mathbb{N}$, and hence the Fueter operator
is the usual Dirac operator

$\mathfrak{F}\phi\in\Gamma(\mathbb{N}).$

We can say a bit about the linearization of the Fueter operator. Let
$\phi\in\Gamma(Y,\mathbb{M})$ be section of the hyperkähler bundle. The
pullback of the vertical bundle gives a _Clifford module_

(1.10) $\phi^{*}\mathcal{V}\mathbb{M}\to Y.$

To see why this bundle is a Clifford module, first note that the action of
quaternions on $\mathcal{V}\mathbb{M}$ can be interpreted as an action of the
tangent bundle $TY\simeq Y\times\textup{Im}{\bf H}$. Hence, one obtains a
linear Dirac operator $D^{\phi}$ on the Clifford module (1.10).

The following proposition is well-known. However, we could not pinpoint an
exact reference to a proof of it in the literature. Below, we give a detailed
proof of the statement. In the appendix, we contextualize the linearization of
connections in the case of associated hyperkähler fiber bundles.

######  Proposition 1.8.

The linearization $L_{\phi}\mathfrak{F}$ of the Fueter operator (1.9) at
$\phi$ is given by $D^{\phi}$. In particular, the linearization is elliptic.

###### Proof.

We first describe the induced connection on the Clifford module
$\phi^{*}\mathcal{V}\mathbb{M}$. We may identify the tangent space
$T_{\phi}\Gamma(Y,\mathbb{M})$ with $\Gamma(Y,\phi^{*}\mathcal{V}\mathbb{M})$.
Hence the linearization of the connection $\nabla$ at $\phi$ gives a map

$\nabla^{\phi}:=L_{\phi}\nabla:\Gamma(Y,\phi^{*}\mathcal{V}\mathbb{M})\to\Gamma(Y,TY^{*}\otimes\phi^{*}\mathcal{V}\mathbb{M})$
$\psi\mapsto\frac{d}{dt}|_{t=0}P_{\phi_{t}}\nabla\phi_{t}$

where $\phi_{t}\in\Gamma(T,\mathbb{M})$ is a curve of sections such that
$\frac{d}{dt}|_{t=0}\phi_{t}=\psi$ and
$P_{\phi_{t}}:\phi_{t}^{*}\mathcal{V}\mathbb{M}\to\phi_{0}^{*}\mathcal{V}\mathbb{M}$
is the bundle isomorphism given, for every $y\in Y$, by the parallel transport
along the curve $s\mapsto\phi_{s}(y)$ in $\mathbb{M}_{y}$ via the Levi-Civita
connection on the fiber $\mathbb{M}_{y}$.

The linearization $\nabla^{\phi}$ gives a connection on the vector bundle
$\phi^{*}\mathcal{V}\mathbb{M}\to Y$. To see why this is a connection, take
$\psi\in\Gamma(Y,\phi^{*}\mathcal{V}\mathbb{M})$ which can be represented by a
curve of sections $\phi_{t}\in\Gamma(Y,\mathbb{M})$, i.e.,
$\frac{\partial}{\partial t}\phi_{t}=\psi$. We need to show
$\nabla^{\phi}f\cdot\psi=f\nabla^{\phi}\psi+df\cdot\psi$ for $f\in
C^{\infty}(Y)$. First, note the “tangent vector” $f\cdot\psi$ corresponds to
the curve $\phi_{f\cdot t}$. So, we may write

$\nabla^{\phi}f\cdot\psi=\frac{d}{dt}|_{t=0}P_{\phi_{f.t}}\nabla\phi_{f\cdot
t}=\frac{d}{dt}|_{t=0}(P_{\phi_{f.t}}{\pi^{v}d\phi_{f\cdot t}})$
$=\frac{d}{dt}|_{t=0}(P_{\phi_{f.t}}\pi^{v}(d\phi)|_{f\cdot t}+t\cdot df\cdot
P_{\phi_{f.t}}\pi^{v}\frac{\partial}{\partial s}|_{s=f\cdot
t}\phi_{s})=f\nabla^{\phi}\psi+df\cdot\psi.$

Here $\pi^{v}:T\mathbb{M}\to\mathcal{V}\mathbb{M}$ is the projection along the
horizontal bundle associated with the connection onto the vertical bundle, and
$\nabla=\pi^{v}\circ d.$

The Dirac operator $D^{\phi}$ on $\phi^{*}\mathcal{V}\mathbb{M}$ is then
defined by

$D^{\phi}=\phi^{*}\gamma\circ\nabla^{\phi}.$

Since $\phi^{*}\gamma$ is linear, it commutes with the linearization map; we
then have

$L_{\phi}(\mathfrak{F})=L_{\phi}(\phi^{*}\gamma\circ\nabla)=\phi^{*}\gamma\circ
L_{\phi}\nabla=D^{\phi},$

as claimed. ∎

######  Remark 1.9.

The connection $\nabla^{\phi}$ on $\phi^{*}\mathcal{V}\mathbb{M}\to Y$ does
not need to be a Clifford connection [3], so the Dirac operator $D^{\phi}$
does not share many of the geometric properties of the classical Dirac
operators (such as the Lichnerowicz-Weitzenböck formula, etc.). However, its
symbol is still given by the Clifford multiplication by the corresponding
covectors.

##  2\. An aquaternionic moduli space

In this section, we fix the following notations:

${M}:=\textup{Hom}(\mathbf{C}\otimes\textup{Im}{\bf
H},\bf{H})\backslash\\{0\\},$

We also denote by $M^{\mu}$ the inverse image $\mu^{-1}(0)\backslash\\{0\\}$
and the hyperkähler quotient by

${M}_{0}:=\mathcal{M}_{\textup{fr}}(1,3).$

###  2.1. $\text{SO}(3)\times\text{Spin}^{c}(3)$ action

The observation above was crucial in the description of the noncompactness
theorem for the multi-spinor Seiberg-Witten equation [6] in terms of nonlinear
harmonic spinors. We are interested in the case of
$E=\mathbf{C}\otimes\textup{Im}{\bf H}$. The special orthogonal group
$\text{SO}(3)$ acts on $\textup{Im}{\bf H}$ and by complexifying, we then
obtain an action of $\text{SO}(3)$ on $E=\mathbf{C}\otimes\textup{Im}{\bf H}$.
Hence $\textup{Hom}(\mathbf{C}\otimes\textup{Im}{\bf H},\mathbf{H})$ carries
actions of the Lie groups $\text{SO}(3)\times\text{Spin}(3)$ and
$\text{SO}(3)\times\text{Spin}^{c}(3)$ where the $\text{SO}(3)$-factor acts on
$\mathbf{C}\otimes\textup{Im}{\bf H}$, and $\text{Spin}(3)$ and
$\text{Spin}^{c}(3)$ act on $\mathbf{H}$. This action descends to the moduli
space $\mathcal{M}_{\textup{fr}}(1,3)$.

###  2.2. An aquaternionic quotient

Identifying $E=E^{*}$, we have the orthogonal splitting

$\textup{Hom}(\mathbf{C}\otimes\textup{Im}{\bf
H},\mathbf{H})\simeq(\mathbf{C}\otimes\textup{Im}{\bf
H})\otimes\mathbf{H}=\iota(\mathbf{H})\oplus W$

preserved by the diagonal action of
$\text{Spin}(3)$($\buildrel\textup{diag.}\over{\hookrightarrow}\text{Spin}(3)\times\text{Spin}((3)$).
Denote the image of $W^{\mu}:=W\cap\Big{(}\mu^{-1}(0)\backslash\\{0\\}\Big{)}$
under the projection $M\to M_{0}$ by $W_{0}$ which carries an action of
$\text{Spin}(3)$. We call the $W_{0}$ the moduli space of _aquaternionic
$\text{SU}(3)$-instantons_. We will show this is a 4-dimensional manifold.

######  Proposition 2.1.

The moduli space $W_{0}$ is a 4-dimensional submanifold of $M_{0}$.

######  Lemma 2.2.

We have a decomposition $W=W_{1}\oplus W_{2}$ where the subspace $W_{1},W_{2}$
are in the zero set of the quadratic map $\mu$. One choice for such
decomposition is given by

$W_{1}=\\{I\otimes Is-J\otimes Js:s\in\not{S}\\},$

and

$W_{2}=\\{I\otimes Is-K\otimes Ks:s\in\not{S}\\}.$

###### Proof.

Clearly $W_{1},W_{2}\subset W$ and also $W_{1}\cap W_{2}=\\{0\\}.$ Hence by
dimension counting we deduce $W=W_{1}\oplus W_{2}$. We only show
$\mu(W_{1})=0$, since a similar argument applies to $W_{2}$. Consider
$\psi=I\otimes Is-J\otimes Js\in W_{1},$ and using the identification
$\mathbf{H}\simeq\mathbf{C}^{2}$ denote the spinor vector $s$ by

$\begin{bmatrix}a+bi\\\ c+di\end{bmatrix}.$

Under the identification $\textup{Hom}(\mathbf{C}\otimes\textup{Im}{\bf
H},\mathbf{H})\simeq\textup{Hom}(\mathbf{C}^{3},\mathbf{C}^{2})$, the
3/2-spinor vector $\psi$ is given by the matrix

$\psi=\begin{bmatrix}-b+ai&c-di&0\\\ -d+ic&-a+bi&0\end{bmatrix}.$

Now a direct calculation shows
$\mu(\psi)=\psi\psi^{*}-\frac{1}{2}\textup{tr}(\psi\psi^{*})\textup{Id}$
vanishes. ∎

###### Proof of Proposition 2.1.

Since the action of $S^{1}$ on $W\backslash\\{0\\}$ and
$\mu^{-1}(0)\backslash\\{0\\}$ is free, we only need to show that the
intersection ${W}^{\mu}=W\cap\Big{(}\mu^{-1}(0)\backslash\\{0\\}\Big{)}$ is a
5-dimensional manifold.

Using the decomposition in Lemma 2.2, we may write $\psi\in W=W_{1}\oplus
W_{2}$ as $\psi_{1}+\psi_{2}$ where

$\psi_{1}=I\otimes Is_{1}-J\otimes Js_{1}$

and

$\psi_{2}=I\otimes Is_{2}-K\otimes Ks_{2}.$

Under the identification $\not{S}\simeq\mathbf{C}^{2}$, we denote
$s_{1}=\begin{bmatrix}a_{1}+b_{1}i\\\ c_{1}+d_{1}i\end{bmatrix}$ and
$s_{2}=\begin{bmatrix}a_{2}+b_{2}i\\\ c_{2}+d_{2}i\end{bmatrix}$. So, under
the identification $\textup{Hom}(\mathbf{C}\otimes\textup{Im}{\bf
H},\mathbf{H})\simeq\textup{Hom}(\mathbf{C}^{3},\mathbf{C}^{2})$, the
3/2-spinor vector $\psi$ is given by the matrix

(2.1)
$\psi=\begin{bmatrix}-(b_{1}+b_{2})+(a_{1}+a_{2})i&c_{1}-d_{1}i&d_{2}+c_{2}i\\\
-(d_{1}+d_{2})+(c_{1}+c_{2})i&-a_{1}+b_{1}i&-b_{2}-a_{2}i\end{bmatrix}.$

From the equation $\mu(\psi)=0$ one obtains the system of equations

$\begin{cases}a_{1}a_{2}+b_{1}b_{2}-c_{1}c_{2}-d_{1}d_{2}=0\\\
c_{1}a_{2}+d_{1}b_{2}+a_{1}c_{2}+b_{1}d_{2}=0\\\
d_{1}a_{2}-c_{1}b_{2}-b_{1}c_{2}+a_{1}d_{2}=0\end{cases}$

or equivalently,

(2.2) $\begin{bmatrix}a_{1}&b_{1}&-c_{1}&-d_{1}\\\ c_{1}&d_{1}&a_{1}&b_{1}\\\
d_{1}&-c_{1}&-b_{1}&a_{1}\end{bmatrix}\begin{bmatrix}a_{2}\\\ b_{2}\\\
c_{2}\\\ d_{2}\end{bmatrix}=0.$

The $3\times 4$ matrix in (2.2) is of rank $3$ unless $\psi_{1}=0$ (i.e., if
$a_{1},b_{1},c_{1},d_{1}$ vanish simultaneously); indeed, the rows of the
matrix are orthogonal, and if one vanishes, all rows vanish. Consequently,
there is a one dimensional solution set for $\psi_{2}$ (identified with
$\begin{bmatrix}a_{2}\\\ b_{2}\\\ c_{2}\\\ d_{2}\end{bmatrix}$). One has an
explicit formula for the solutions:

$\begin{bmatrix}a_{2}\\\ b_{2}\\\ c_{2}\\\
d_{2}\end{bmatrix}=\lambda\begin{bmatrix}b_{1}\\\ -a_{1}\\\ d_{1}\\\
-c_{1}\end{bmatrix},\ \ \lambda\in\mathbf{R}.$

So if $\psi_{1}\neq 0$, $W\cap\mu^{-1}(0)$ is a line bundle over a
neighborhood of $\psi_{1}$ in $W_{1}$, as a subbundle of $W_{1}\times W_{2}\to
W_{1}$. Hence $W\cap\mu^{-1}(0)$ is a 5-dimensional manifold in a neighborhood
of $\psi$; one indeed has the local parameterization on $W^{\mu}$ near $\psi$
given by

(2.3) $(a_{1},b_{1},c_{1},d_{1},\lambda)\mapsto\begin{bmatrix}-(b_{1}-\lambda
a_{1})+(a_{1}+\lambda b_{1})i&c_{1}-d_{1}i&-\lambda c_{1}+\lambda d_{1}i\\\
-(d_{1}-\lambda c_{1})+(c_{1}+\lambda d_{1})i&-a_{1}+b_{1}i&\lambda
a_{1}-\lambda b_{1}i\end{bmatrix}.$

By the symmetry between $\psi_{1}$ and $\psi_{2}$, we obtain a 5-dimensional
manifold structure for $W^{\mu}$ in a neighborhood of $\psi$, when
$\psi_{2}\neq 0$. From this, the proposition follows. ∎

######  Remark 2.3.

It is worth mentioning that $0\in i\mathfrak{su}(2)$ is a regular value for
the restriction of the quadratic map $\mu$ to the space of 3/2-spinors $W$. To
see this, first note that for any
$\psi\in\textup{Hom}(\mathbf{C}^{3},\mathbf{C}^{2})$ one has

$d_{\psi}\mu(h)=\psi h^{*}+h\psi^{*}-\frac{1}{2}\textup{tr}(\psi
h^{*}+h\psi^{*}).$

When $\psi\in W^{\mu}$, by the proof of the Proposition 2.1, we may write
$\psi$ as the matrix

$\psi=\begin{bmatrix}-(b_{1}-\lambda a_{1})+(a_{1}+\lambda
b_{1})i&c_{1}-d_{1}i&-\lambda c_{1}+\lambda d_{1}i\\\ -(d_{1}-\lambda
c_{1})+(c_{1}+\lambda d_{1})i&-a_{1}+b_{1}i&\lambda a_{1}-\lambda
b_{1}i\end{bmatrix}.$

If one choose $h_{1},h_{2},h_{3}\in T_{\psi}W\simeq W$ given by

$h_{1}=\begin{bmatrix}-b_{1}+a_{1}i&0&-d_{1}+c_{1}i\\\
d_{1}-c_{1}i&0&-b_{1}-a_{1}i\end{bmatrix},$
$h_{2}=\begin{bmatrix}-d_{1}+c_{1}i&0&b_{1}+a_{1}i\\\
-b_{1}+a_{1}i&0&-d_{1}-c_{1}i\end{bmatrix},$

and

$h_{3}=\begin{bmatrix}c_{1}+d_{1}i&0&a_{1}-b_{1}i\\\
-a_{1}-b_{1}i&0&c_{1}-d_{1}i\end{bmatrix},$

then the three vectors $d_{\psi}\mu(h_{1})$, $d_{\psi}\mu(h_{2})$, and
$d_{\psi}\mu(h_{3})$ are linearly independent.

###  2.3. The tangent bundle $TW^{\mu}$

In the proof of Proposition 2.1, we described the 5-dimensional manifold
$W^{\mu}$ rather explicitly. Using the local coordinate (2.3), one has the
local trivialization of the tangent tangent space
$T_{(a,b,c,d,\lambda_{0})}W^{\mu}$ by the vector linearly independent vector
fields

$\frac{\partial}{\partial a}:=\begin{bmatrix}\lambda+i&0&0\\\
0&-1&\lambda\end{bmatrix},\ \ \frac{\partial}{\partial
b}:=\begin{bmatrix}-1+\lambda i&0&0\\\ 0&i&-\lambda i\end{bmatrix},\ \
\frac{\partial}{\partial c}:=\begin{bmatrix}0&1&-\lambda\\\
\lambda+i&0&0\end{bmatrix},$ $\frac{\partial}{\partial
d}:=\begin{bmatrix}0&-i&\lambda i\\\ -1+\lambda i&0&0\end{bmatrix},\ \
\frac{\partial}{\partial\lambda}:=\begin{bmatrix}a+bi&0&-c+di\\\
c+di&0&a-bi\end{bmatrix}.$

The first four vectors $\frac{\partial}{\partial a},\frac{\partial}{\partial
b},\frac{\partial}{\partial c},\frac{\partial}{\partial d}$ are mutually
orthogonal, but they are not orthogonal to $\frac{\partial}{\partial\lambda}$.

The tangent bundle has a line sub-bundle $\mathcal{L}\subset TW^{\mu}$ spanned
by the Killing vector field associated with the $U(1)$\- action. This line
bundle in the trivialization above is spanned by the rotation vector field

(2.4) $a\frac{\partial}{\partial b}-b\frac{\partial}{\partial
a}+c\frac{\partial}{\partial d}-d\frac{\partial}{\partial
c}=\begin{bmatrix}-b\lambda-a+(-b+a\lambda)i&-d-ci&\lambda d+\lambda ci\\\
-\lambda d-c+(-d+\lambda c)i&b+ai&-\lambda b-\lambda ai\end{bmatrix}.$

This line bundle is the restriction of the line bundle on
$M:=\textup{Hom}(\mathbf{C}\otimes\textup{Im}{\bf
H},\mathbf{H})\backslash\\{0\\}$ generated by the $U(1)$-action which we
denote still by $\mathcal{L}$. Under the hyperkähler quotient map
$\tau:M^{\mu}\to M_{0}$ one has the orthogonal decomposition (Lemma 1.5)

$TM|_{M^{\mu}}\simeq\tau^{*}TM_{0}\oplus(\mathcal{L}\oplus\textup{Im}{\bf
H}\cdot\mathcal{L})$

where the components $\tau^{*}TM_{0}$ and $(\mathcal{L}\oplus\textup{Im}{\bf
H}\cdot\mathcal{L})$ are invariant under the Clifford action ([7]). Under the
decomposition above, one has

(2.5) $TM^{\mu}\simeq\tau^{*}TM_{0}\oplus\mathcal{L}$

and similarly,

(2.6) $TW^{\mu}\simeq\tau^{*}TW_{0}\oplus\mathcal{L}.$

###  2.4. A normal line bundle on $W^{\mu}$

. Remember the orthogonal decomposition

$\textup{Hom}(\mathbf{C}\otimes\textup{Im}{\bf
H},\mathbf{H})=\iota(\mathbf{H})\oplus W$

where $\iota:\mathbf{H}\to\textup{Hom}(\mathbf{C}\otimes\textup{Im}{\bf
H},\mathbf{H})$ is the embedding (1.2). Clearly, $\iota(\mathbf{H})$ is
orthogonal to the fibers of $TW^{\mu}$, so in particular to the fibers of
$\mathcal{L}|_{W^{\mu}}.$ The matrix form of the elements of
$\iota(\mathbf{H})$ is given by

(2.7) $\begin{bmatrix}-b+ai&-c+di&-d-ci\\\ -d+ci&a-bi&b+ai\end{bmatrix}.$

One may consider $\iota(\mathbf{H})$ as a trivial subbundle of the tangent
bundle of $\textup{Hom}(\mathbf{C}\otimes\textup{Im}{\bf H},\mathbf{H})$. We
then obtain a canonical line bundle:

######  Proposition 2.4.

The intersection of the bundles
$\mathcal{N}:=TM^{\mu}|_{W^{\mu}}\cap\iota(\mathbf{H})$ is an
$S^{1}$-invariant line bundle over $W^{\mu}$ orthogonal to the line bundle
$\mathcal{L}|_{W^{\mu}}$. Also $\mathcal{N}$ carries an equivariant
$\text{Spin}^{c}(3)$-action.

###### Proof.

Since $M^{\mu}$ is given as an inverse image of a regular point of $\mu$, the
tangent bundle $TM^{\mu}$ is given by the equation $d\mu=0$. So in order to
find the intersection $TM^{\mu}|_{W^{\mu}}\cap\iota(\mathbf{H})$, we need to
solve the equation $d_{w}\mu|_{\iota({\mathbf{H}})}=0$ for every $w\in
W^{\mu}$. Without loss of generality, we may assume, as in (2.3), $w$ is of
the form

$\begin{bmatrix}-(b_{1}-\lambda a_{1})+(a_{1}+\lambda
b_{1})i&c_{1}-d_{1}i&-\lambda c_{1}+\lambda d_{1}i\\\ -(d_{1}-\lambda
c_{1})+(c_{1}+\lambda d_{1})i&-a_{1}+b_{1}i&\lambda a_{1}-\lambda
b_{1}i\end{bmatrix}$

with $(a_{1},b_{1},c_{1},d_{1})\neq(0,0,0,0)$ and take $s$ of the form (2.7).
From the equation

$d_{w}\mu(s)=w^{*}s+s^{*}w-\frac{1}{2}\textup{tr}(w^{*}s+s^{*}w)=0,$

we obtain the linear system

(2.8)
$\begin{bmatrix}a_{1}+\lambda_{1}b_{1}&b_{1}-\lambda_{1}a_{1}&-c_{1}-\lambda_{1}d_{1}&-d_{1}+\lambda_{1}c_{1}\\\
c_{1}+\lambda_{1}d_{1}&d_{1}-\lambda_{1}c_{1}&a_{1}+\lambda_{1}b_{1}&b_{1}-\lambda_{1}a_{1}\\\
d_{1}-\lambda_{1}c_{1}&-c_{1}-\lambda_{1}d_{1}&-b_{1}+\lambda_{1}a_{1}&-a_{1}+\lambda_{1}b_{1}\end{bmatrix}\begin{bmatrix}a\\\
b\\\ c\\\ d\end{bmatrix}=0.$

The $3\times 4$ matrix (2.8) is of rank $3$; indeed, the rows are nonzero and
mutually orthogonal. So the equation (2.8) has a one-dimensional space of
solutions spanned by the vector

$\begin{bmatrix}a\\\ b\\\ c\\\
d\end{bmatrix}=\begin{bmatrix}-b_{1}+\lambda_{1}a_{1}\\\
a_{1}+\lambda_{1}b_{1}\\\ -d_{1}+\lambda_{1}c_{1}\\\
c_{1}+\lambda_{1}d_{1}\end{bmatrix}.$

From this, the proposition follows. ∎

######  Remark 2.5.

Since $\mathcal{N}$ is $S^{1}$-invariant is orthogonal to
$\mathcal{L}|_{W^{\mu}}$, we obtain a _pushforward subbundle_
$\tau_{*}\mathcal{N}$ of $TM_{0}|_{W_{0}}$. The $S^{1}$-invariance and
$\text{Spin}^{c}(3)$-equivariance properties follow from similar properties of
$TM^{\mu}$ and $\iota(\mathbf{H})$.

######  Definition 2.6.

Denote by $\mathcal{T}$ the orthogonal complement of
$\mathcal{N}\oplus\mathcal{L}|_{W^{\mu}}$ in $TM^{\mu}|_{W^{\mu}}$. We also
denote by $\mathcal{T}_{0}$ the orthogonal complement of $\tau_{*}\mathcal{N}$
in $TM_{0}|_{W_{0}}$.

###  2.5. Two examples of hyperkähler bundles

Let $S=\textup{Hom}(\mathbf{C}\otimes\textup{Im}{\bf H},\bf{H})$ and
$M_{0}=\big{(}S\backslash\\{0\\}\big{)}\mathbin{/\mkern-5.0mu/\mkern-5.0mu/}U(1)$
be as in Section 2. These are hyperkähler manifolds which carry an
$\text{SO}(3)\times\text{Spin}^{c}(3)$ action. Consider an oriented Riemannian
manifold $Y$ with its canonical spin structure. Denote by $P_{\text{SO}}$ and
$P_{\text{Spin}^{c}}$ its associated principal $\text{SO}(3)$\- and
$\text{Spin}^{c}(3)$-bundles and define the principal
$\text{SO}(3)\times\text{Spin}^{c}(3)$-bundle
$P:=P_{\text{SO}}\times_{Y}P_{\text{Spin}^{c}}$. Now the associated bundles
$\mathbb{S}:=P\times_{\text{SO}(3)\times\text{Spin}(3)}S$ and
$\mathbb{M}_{0}:=P\times_{\text{SO}(3)\times\text{Spin}^{c}(3)}M_{0}$ are the
hyperkähler bundles. Note that $\mathbb{S}$ is just the twisted spinor bundle
$TY\otimes\not{S}$.

These bundles can be given with structure group $\text{Spin}(3)$ as follows.
Consider the $\text{Spin}(3)$-action on $S$ and $M_{0}$ given via the
composition

(2.9)
$\text{Spin}^{c}(3)\buildrel\Delta\over{\hookrightarrow}\text{Spin}^{c}(3)\times\text{Spin}^{c}(3)\buildrel(\xi,\textup{Id})\over{\to}\text{SO}(3)\times\text{Spin}^{c}(3)$

where the map $\Delta$ is the diagonal embedding and
$\xi:\text{Spin}^{c}(3)\to\text{SO}(3)$ is the canonical homomorphism. Since
we have a compatible bundle maps
$P_{\text{Spin}^{c}}\buildrel\Delta\over{\hookrightarrow}P_{\text{Spin}^{c}}\times_{Y}P_{\text{Spin}^{c}}\to
P_{\text{SO}}\times_{Y}P_{\text{Spin}^{c}}$, we have the canonical
identifications

(2.10) $\mathbb{S}\simeq P_{\text{Spin}^{c}}\times_{\text{Spin}^{c}(3)}S$

and

(2.11) $\mathbb{M}_{0}\simeq
P_{\text{Spin}^{c}}\times_{\text{Spin}^{c}(3)}M_{0}.$

###  2.6. Special subbundles of hyperkähler bundles

Since $W\subset S$ and $W_{0}\subset M$ are invariant under this diagonal
action (2.9), we obtain the associated subbundles

(2.12) $\mathbb{W}\simeq
P_{\text{Spin}^{c}}\times_{\text{Spin}^{c}(3)}W\subset\mathbb{S}$

and

(2.13) $\mathbb{W}_{0}\simeq
P_{\text{Spin}^{c}}\times_{\text{Spin}^{c}(3)}W_{0}\subset\mathbb{M}_{0}.$

Note that $\mathbb{W}$ is the bundle of 3/2-spinors. The bundles $\mathbb{W}$
and $\mathbb{W}_{0}$ do not inherit hyperkähler bundle structures.

###  2.7. The 3/2-Fueter operator

In Subsection 2.5 we introduced the fiber bundle
$\mathbb{W}_{0}=P_{\text{Spin}^{c}}\times_{\text{Spin}^{c}(3)}W_{0}$ , which
is subbundle of a hyperkähler bundle
$\mathbb{M}_{0}=P_{\text{Spin}^{c}}\times_{\text{Spin}^{c}(3)}M_{0}$ that are
associated bundles for the diagonal action of $\text{Spin}^{c}(3)$(2.9). Under
this action, the bundle $\mathcal{T}_{0}\to W_{0}$ (Definition 2.6 and Remark
2.5) is invariant; from Proposition 2.4, we see that the normal bundle
$\mathcal{N}$ is invariant under the aforementioned action and since
$TM_{0}|_{W_{0}}=\mathcal{T}_{0}\oplus\tau_{*}\mathcal{N}$ is
$\text{Spin}^{c}(3)$ invariant, the invariance of $\mathcal{T}_{0}$ follows.
We then define the associated bundle

(2.14)
$\mathbb{T}_{0}:=P_{\text{Spin}^{c}}\times_{\text{Spin}^{c}(3)}\mathcal{T}_{0}.$

We define a version of the Fueter operator for $\mathbb{W}_{0}$. Let
$p:TM_{0}|_{W_{0}}\to\mathcal{T}_{0}$ be the orthogonal projection map (see
Definition 2.6). Clearly, this induces a projection

(2.15) $p_{0}:\mathcal{V}\mathbb{M}_{0}|_{\mathbb{W}_{0}}\to\mathbb{T}_{0}.$

######  Definition 2.7.

Fix a connection on $P_{\text{Spin}^{c}}$ induced from the Levi-Civita
connection and a connection $A$ on the associated canonical $U(1)$-bundle. Let
$\phi\in\Gamma(\mathbb{W}_{0})$ be a section that can be considered as a
section of the bundle $\mathbb{M}_{0}$. Applying the Fueter operator on
$\mathbb{M}_{0}$, we obtain a section
$\mathfrak{F}_{A}\phi\in\Gamma(Y,\phi^{*}\mathcal{V}\mathbb{M}_{0})$. Applying
the projection map (2.15), we obtain the _3/2-Fueter operator_
$\mathfrak{Q}_{A}$ by

$\mathfrak{Q}_{A}\phi:=p_{0}\circ\mathfrak{F}_{A}\phi\in\Gamma(Y,\phi^{*}\mathbb{T}_{0}).$

##  3\. A Haydys’ correspondence for $3/2$-spinors

In this section, we show the first part of the main Theorem of the paper.

######  Theorem 3.1.

Let $\Phi\in\Gamma(\mathbb{W})$ be a nowhere vanishing 3/2-spinor section that
satisfies the equations

(3.1) $\begin{cases}Q_{A}\Phi=0,\\\ \mu(\Phi)=0.\end{cases}$

Then there exists a section $\Phi_{0}\in\Gamma(\mathbb{W}_{0})$ and a
connection $B$ satisfying

$\mathfrak{Q}_{B}\Phi_{0}=0.$

The converse also holds: For every $\Phi_{0}\in\Gamma(\mathbb{W}_{0})$
satisfying $\mathfrak{Q}_{B}\Phi_{0}=0$, there is a solution $(A,\Phi)$ of
(3.1).

###### Proof.

The proof of this theorem is very similar to but slightly more involved than
the proof of [5, Proposition 4.2]; so we give the proof here.

Consider a nowhere vanishing 3/2-spinor $\Phi$ satisfying $\mu(\Phi)=0$,
meaning $\Phi$ is a section of the bundle
$\mathbb{W}^{\mu}:=P_{\text{Spin}^{c}}\times_{\text{Spin}^{c}(3)}W^{\mu}$.
Under the quotient map

$\tau:\mathbb{W}^{\mu}\to\mathbb{W}_{0},$

one obtains a section $\Phi_{0}:=\tau(\Phi)\in\Gamma(\mathbb{W}_{0}).$ One may
consider the sections $\Phi,\Phi_{0}$ as equivariant functions
$\Phi:P_{\text{Spin}^{c}}\to W^{\mu}$ and $\Phi_{0}:P_{\text{Spin}^{c}}\to
W_{0}$, whose covariant derivative is the restriction of the differentials
$d\Phi:TP_{\text{Spin}^{c}}\to TW^{\mu}$ and
$d\Phi_{0}:TP_{\text{Spin}^{c}}\to TW_{0}$ to the horizontal bundle for the
appropriate connection on $P_{\text{Spin}^{c}}.$ For $p\in
P_{\text{Spin}^{c}}$, the projection of $d_{p}\Phi$ onto
$T_{\phi_{0}(p)}W_{0}$ in the decomposition (2.6) equals $d_{p}\Phi_{0}$.
Hence, there exists a basic form
$b\in\Omega^{1}(P_{\text{Spin}^{c}},\mathfrak{u}(1))$ such that

$d_{p}\Phi-d_{p}\Phi_{0}=K_{b}$

where $K_{b(X)}$ for $X\in T_{p}P_{\text{Spin}^{c}}$ is the Killing vector.

If $Q_{A}\Phi=0$, then
$c(d\Phi|_{\mathcal{H}P_{\text{Spin}^{c}}})\in\iota(\mathbf{H})$ where
$\mathcal{H}P_{\text{Spin}^{c}}\subset TP_{\text{Spin}^{c}}$ is the horizontal
bundle associated with connection $A$. Denoting by
$\mathcal{H}^{\prime}P_{\text{Spin}^{c}}$ the horizontal bundle associated to
$B:=A+b,$ we obtain
$c(d\Phi_{0}|_{\mathcal{H}^{\prime}P_{\text{Spin}^{c}}})=c(d\Phi_{0}|_{\mathcal{H}P_{\text{Spin}^{c}}}+K_{b})=c(d\Phi|_{\mathcal{H}P_{\text{Spin}^{c}}})\in\iota(\mathbf{H})$,
and since $\mu(\Phi)=0$, we deduce
$c(d\Phi_{0}|_{\mathcal{H}^{\prime}P_{\text{Spin}^{c}}})\in\tau_{*}\mathcal{N}=TM_{0}\cap\iota(\mathbf{H})$
(so it is orthogonal to $\mathcal{T}_{0}$), i.e.,
$\mathfrak{Q}_{B}\Phi_{0}=0$. This argument works backward. ∎

##  4\. The overdetermination of $\mathfrak{Q}_{A}$

In this section, we show the symbol of the linearization of $\mathfrak{Q}_{A}$
is injective. For a section $\phi\in\Gamma(\mathbb{W}_{0})$, we may write

$L_{\phi}\mathfrak{Q}_{A}=p_{0}\circ L_{\phi}\mathfrak{F}_{A}$

since the the projection map (2.15) is linear. By Proposition 1.8, we conclude

$L_{\phi}\mathfrak{{Q}}_{A}=p_{0}\circ D^{\phi},$

where $D^{\phi}$ is the Dirac operator on the Clifford module
$\phi^{*}\mathcal{V}\mathbb{M}_{0}\to Y.$

######  Theorem 4.1.

The symbol of $p_{0}\circ
D^{\phi}:\Gamma(Y,\phi^{*}\mathcal{V}\mathbb{W}_{0})\to\Gamma(Y,\mathbb{T}_{0})$
is injective.

###### Proof.

The idea is to lift the operators via the hyperkähler quotient map
$\tau:\mathbb{M}^{\mu}\to\mathbb{M}_{0}$. Consider
$\psi\in\Gamma(\mathbb{M}^{\mu})$ and $\phi\in\Gamma(\mathbb{M}_{0})$ such
that $\tau\psi=\phi$. Denote the corresponding Dirac operators
$D^{\psi}\curvearrowright\Gamma(\psi^{*}\mathcal{V}\mathbb{M})$ and
$D^{\phi}\curvearrowright\Gamma(\phi^{*}\mathcal{V}\mathbb{M}_{0})$; then we
have the equality

(4.1) $\tau_{*}\circ\sigma(D^{\psi})=\sigma(D^{\phi})\circ\tau_{*}$

where
$\tau_{*}:\psi^{*}\mathcal{V}\mathbb{M}|_{\mathbb{M}^{\mu}}\to\phi^{*}\mathcal{V}\mathbb{M}_{0}$
is the bundle map covering the quotient
$\tau:\mathbb{M}^{\mu}\to\mathbb{M}_{0}$ induced from the decomposition in
Lemma 1.5 and $\sigma(.)$ denotes the symbol of the operator. The equality
(4.1) follows simply from the fact that the symbol of the Dirac operator is
given by Clifford multiplication with a covector.

Consider the projection
$p:\mathcal{V}\mathbb{M}|_{\mathbb{W}^{\mu}}\to\mathbb{T}$ where
$\mathbb{T}:=P_{\text{Spin}^{c}}\times_{\text{Spin}^{c}(3)}\mathcal{T}$. Since
we have the equality

$\tau_{*}\circ p=p_{0}\circ\tau_{*}$

on the pullback bundle $\psi^{*}\mathcal{V}\mathbb{M}$, and since the symbol
map commutes with the projections $p,p_{0}$ we have

$\sigma(p_{0}\circ
D^{\phi})\circ\tau_{*}=p_{0}\circ\sigma(D^{\phi})\circ\tau=p_{0}\circ\tau_{*}\circ\sigma(D^{\psi})=\tau_{*}\circ
p\circ\sigma(D^{\psi}).$

Now assume $\psi\in\mathbb{W}^{\mu}$ and $\phi\in\mathbb{W}_{0}$ with
$\tau\psi=\phi$. Since the kernel of
$\tau_{*}|_{\psi^{*}\mathcal{V}\mathbb{W}^{\mu}}:\psi^{*}\mathcal{V}\mathbb{W}^{\mu}\to\phi^{*}\mathcal{V}\mathbb{W}_{0}$
is the given by $\psi^{*}\mathcal{L}$, and since
$\tau_{*}|_{\mathbb{T}}:\mathbb{T}\to\mathbb{T}_{0}$ is isomorphism on the
fibers, we conclude $p\circ\sigma(D^{\psi})$ vanishes on
$\psi^{*}\mathcal{L}$. So we only need to show the fiberwise rank of
$p\circ\sigma(D^{\psi})|_{\psi^{*}\mathcal{V}\mathbb{W}^{\mu}}:\psi^{*}\mathcal{V}\mathbb{W}^{\mu}\to\mathbb{T}$
is equal to $4$.

Consider the mutually orthogonal line bundles $I\mathcal{N}$, $J\mathcal{N}$
and $K\mathcal{N}$ that are formed by the action of the quaternionic elements
$I,J,K$ on $\mathcal{N}$, and they are orthogonal to $\mathcal{N}$. Fix $w\in
W^{\mu}$, and without loss of generality we may assume $w$ is of the following
form (Proposition 2.1):

$\begin{bmatrix}-(b-\lambda a)+(a+\lambda b)i&c-di&-\lambda c+\lambda di\\\
-(d-\lambda c)+(c+\lambda d)i&-a+bi&\lambda a-\lambda bi\end{bmatrix}$

with $(a,b,c,d)\neq(0,0,0,0)$ and by Proposition 2.4 we have

$\mathcal{N}=\textup{span}\left\\{\begin{bmatrix}-a-\lambda b+i(-b+\lambda
a)&d-\lambda c+i(c_{1}-\lambda d)&-c-\lambda d+i(-d+\lambda c)\\\ -c-\lambda
d+i(-d+\lambda c)&-b+\lambda a+i(-a-\lambda b)&a+\lambda b+i(-b+\lambda
a)\end{bmatrix}\right\\}.$

Similarly, we have

$I\mathcal{N}=\textup{span}\left\\{\begin{bmatrix}b-\lambda a+i(a+\lambda
b)&-c-\lambda d+i(d-\lambda c)&-d+\lambda c+i(c+\lambda d)\\\
d-\lambda_{1}c+i(-c-\lambda d)&a+\lambda b+i(-b+\lambda a)&b-\lambda
a+i(a+\lambda b)\end{bmatrix}\right\\},$
$J\mathcal{N}=\textup{span}\left\\{\begin{bmatrix}c+\lambda d+i(-d+\lambda
c)&b-\lambda a+i(a+\lambda b)&-a-\lambda b+i(-b+\lambda a)\\\ -a-\lambda
b+i(b-\lambda a)&d-\lambda c+i(c+\lambda d)&-c-\lambda d+i(-d+\lambda
c)\end{bmatrix}\right\\},$

and

$K\mathcal{N}=\textup{span}\left\\{\begin{bmatrix}d-\lambda c+i(c+\lambda
d)&a+\lambda b+i(b-\lambda a)&b-\lambda a+i(a+\lambda b)\\\ -b-\lambda
a+i(-a-\lambda b)&c+\lambda d+i(d-\lambda c)&d-\lambda c+i(-c-\lambda
d)\end{bmatrix}\right\\}.$

Now a straightforward calculation shows $I\mathcal{N}$, $J\mathcal{N}$, and
$K\mathcal{N}$ are orthogonal to $\mathcal{L}$ and hence $I\mathcal{N}\oplus
J\mathcal{N}\oplus K\mathcal{N}$ is subbundle of $\mathcal{T}$.

Remember from Subsection 2.3 that $TW_{w}^{\mu}$ is spanned by the vectors
$\frac{\partial}{\partial a}$, $\frac{\partial}{\partial b}$,
$\frac{\partial}{\partial c}$, $\frac{\partial}{\partial d}$, and
$\frac{\partial}{\partial\lambda}$, the first four of which are mutually
orthogonal. By doing a Gram-Schmidt process, we may replace
$\frac{\partial}{\partial\lambda}$ with a new one that, up to a constant, is
of the form

$\widetilde{\frac{\partial}{\partial\lambda}}=\begin{bmatrix}a+\lambda
b+i(b-\lambda a)&-\lambda c+i\lambda d&-c+id\\\ c+\lambda d+i(d-\lambda
c)&\lambda a-i\lambda b&a-ib.\end{bmatrix}.$

A direct calculation shows that $\widetilde{\frac{\partial}{\partial\lambda}}$
is orthogonal to $I\mathcal{N}$, $J\mathcal{N}$, and $K\mathcal{N}$ thus
$\mathcal{T}^{\prime}:=I\mathcal{N}\oplus J\mathcal{N}\oplus
K\mathcal{N}\oplus\textup{span}\\{\widetilde{\frac{\partial}{\partial\lambda}}\\}$
is rank 4 subbundle of $\mathcal{T}$. Using the fact that the symbol of the
Dirac operator is given by the Clifford multiplication, by restricting
$\sigma(D^{\psi})|_{w}$ to the 4-dimensional space
$\textup{span}\\{\frac{\partial}{\partial a},\frac{\partial}{\partial
b},\frac{\partial}{\partial c},\frac{\partial}{\partial d}\\}|_{w}$ and
projecting to the rank 4-dimensional space $\mathcal{T}^{\prime}|_{w}$, we
obtain its matrix form as

$\begin{bmatrix}-2(\lambda^{2}+1)b&2(\lambda^{2}+1)a&-2(\lambda^{2}+1)d&2(\lambda^{2}+1)c\\\
2(2\lambda^{2}+1)c-2\lambda d&-2(2\lambda^{2}+1)d-2\lambda
c&-2(2\lambda^{2}+1)a+2\lambda b&2(2\lambda^{2}+1)b+2\lambda a\\\
-2(\lambda^{2}+2)d+2\lambda c&-2(\lambda^{2}+2)c-2\lambda
d&2(\lambda^{2}+2)b-2\lambda a&2(\lambda^{2}+2)a+2\lambda b\\\
a&b&c&d\end{bmatrix}.$

An explicit calculation shows the determinant of this $4\times 4$ matrix is
the following

$2(a^{2}+b^{2}+c^{2}+d^{2})^{2}(1+\lambda^{2})^{2}$

which is clearly positive and the theorem now follows. ∎

## Appendix

In this section, we give a proof of Proposition 1.8 in the case of associated
hyperkähler bundle. The proof presented here relies on the concept of a
connector [9], which is roughly an equivalent way to define a linear
connection on a vector bundle. One of the nice features of connectors is that
they can be readily adapted in infinite-dimensional situations.

To motivate the definition of a connector, we consider the following
situation. Suppose $\pi:V\to B$ is a finite rank vector bundle over a finite-
dimensional manifold $B$ and $s\in\Gamma(B,V)$. We wish to find a
linearization of $s$. Obviously, viewing $s:B\to V$ as a smooth map, since $s$
could be non-linear, the naïve usual differential $ds:TB\to TV$ is not good
enough to be taken as a linearization of $s$. Furthermore, we need the co-
domain of the linearization to $V$ again. Thus, it is natural for us to
compose $ds$ with another map $\mathcal{K}:TV\to V$. However, we do not just
want any map $TV\to V$. In fact, we want certain bundle morphisms that are
compatible with both vector bundle structures on $TV$ in a certain sense.
Specifically, their construction is as follows.

Note that vertical component $\mathcal{V}V$ of $V$ can be identified with
$V\times_{B}V$ via the map

$L_{V}:V\times_{B}V\to\mathcal{V}V,\quad(v,v^{\prime})\mapsto\dfrac{d}{dt}(v+tv^{\prime})\bigg{|}_{t=0}.$

Recall that a linear connection on $V$ can be viewed as a bundle morphism
$\Phi:TV\to TV$ such that

$\pi_{TV}\circ\Phi=\pi_{TV},\quad d\pi_{V}\circ\Phi=d\pi_{V},$

where $\pi_{TV}:TV\to V$ denotes the tangent bundle projection and
$d\pi_{V}:TV\to TB$ denotes the induced map from the bundle projection
$\pi:V\to B$.

######  Definition.

A connector $\mathcal{K}:TV\to V$ associated with $\Phi$ is defined to be
$\mathcal{K}=\pi_{2}\circ L^{-1}_{V}\circ\Phi$, where $\pi_{2}$ denotes the
projection of $V\times_{B}V$ onto the second component.

By composing $\mathcal{K}$ with $ds$, we obtain $\nabla s:=\mathcal{K}\circ
ds\in\Gamma(B,T^{*}B\otimes V)$, which is the usual notion of a covariant
derivative of $s$. As a result, it can be shown that each of the connectors of
a vector bundle uniquely determines a covariant derivative and vice versa. In
the case that $V:=TB$, then the Levi-Civita connection $\nabla^{LC}$
determined uniquely a cononical connector $\mathcal{K}^{B}:TTB\to TB$. Note
that, for any manifold $B$, there is a canonical map $\text{flip}_{B}:TTB\to
TTB$ that is defined as following: Let $c:\mathbf{R}^{2}\to B$ such that
$c_{t,s}\in B$ and the derivative of $c$ at $t,s=0$ is a vector belonging to a
fiber of $TTB$, then

$\text{flip}_{B}\left(\dfrac{d}{dt}\dfrac{d}{ds}c_{t,s}\bigg{|}_{t,s=0}\right)=\dfrac{d}{ds}\dfrac{d}{dt}c_{t,s}\bigg{|}_{t,s=0}.$

It is known that if a connector is torsion-free, i.e., the corresponding
covariant derivative is torsion-free, then
$\mathcal{K}\circ\text{flip}_{B}=\mathcal{K}$. In particular, the canonical
connector $\mathcal{K}^{B}$ is torsion-free and thus satisfies the
aforementioned property. This discussion can be extended naturally to infinite
dimensional setting.

Suppose $P\to Y$ is a principal $\mathscr{K}$-bundle over $Y$. Let
$(M,g,I,J,K)$ be a hyperkähler manifold that admits a $\mathscr{K}$-symmetry.
Then we can form an associated fiber bundle of $P$, denoted by $\mathbb{M}\to
Y$, where

$\mathbb{M}:=P\times_{\mathscr{K}}M.$

Note that $\mathbb{M}$ is a hyperkähler bundle over $Y$ (cf. Definition 1.7).
A section $\phi\in\Gamma(Y,\mathbb{M})$ can be viewed as an
$\mathscr{K}$-equivariant map $\phi:P\to M$. A connection on $\mathbb{M}$ is
induced from a connection $A\in\Omega^{1}(adP)$ as follows

$\nabla_{A}\phi(v)=d\phi(v)+X^{M}_{A(v)}(\phi).$

Here, $v$ is a vector field on $P$, and $X^{M}_{A(v)}$ denotes the killing
vector field associated with the Lie algebra element $A(v)\in\mathfrak{k}$.
Since $\nabla_{A}\phi$ vanishes on the vertical component of $TP$, we can view
$\nabla_{A}\phi\in Maps(P,T^{*}Y\otimes TM)^{\mathscr{K}}$. Consequently, we
can view $\nabla_{A}$ as a section of the following infinite dimensional
vector bundle

(4.2) $Maps(P,T^{*}Y\otimes TM)^{\mathscr{K}}\to Maps(P,M)^{\mathscr{K}}.$

###### Proof of Proposition 1.8.

In the particular setup of Proposition 1.8 above, note that
$T\Gamma(Y,\mathbb{M})$ can be identified with $Maps(P,TM)^{\mathscr{K}}$. So,
given a section $\phi$ of the hyperkähler bundle $\mathbb{M}\to Y$,
$T_{\phi}\Gamma(Y,\mathbb{M})$ is identified with
$Maps(P,\phi^{*}TM)^{\mathscr{K}}$. Let $\phi_{t}\in\Gamma(Y,\mathbb{M})$ for
small $t\in(-\epsilon,\epsilon)$ such that $\phi_{0}=\phi$ and
$\dot{\phi}_{0}\in Maps(P,\phi^{*}TM)^{\mathscr{K}}$. To find the
linearization of $\mathfrak{F}$ at $\phi$, it suffices for us to compute the
linearization of $\nabla_{A}$ at $\phi$.

Following the finite-dimensional prototype, the linearization of $\nabla_{A}$
is then interpreted as a certain covariant derivative of $\nabla_{A}$ as a
section of the above vector bundle (4.2). We now must choose a covariant
derivative of $Maps(P,T^{*}Y\otimes TM)^{\mathscr{K}}\to
Maps(P,M)^{\mathscr{K}}$. There is a canonical choice. Since we have already
fixed a Riemannian metric on $M$, its associated Levi-Civita connection
induces the canonical connector $\mathcal{K}^{M}$ on $TM$. In turn,
$\mathcal{K}^{M}$ determines for us a canonical connector on
$Maps(P,T^{*}Y\otimes TM)^{\mathscr{K}}$. In other words, we have a linear map

$\mathcal{K}^{M}:Maps(P,T^{*}Y\otimes TTM)^{\mathscr{K}}\to
Maps(P,T^{*}Y\otimes TM)^{\mathscr{K}}.$

Then the covariant derivative of $\nabla_{A}$ is given by

$\mathcal{K}^{M}\circ d\nabla_{A}.$

As a result, we obtain (all calculations restricted to the horizontal subspace
$\mathcal{H}P$ of $TP$)

$\displaystyle\mathcal{K}^{M}\circ
d\nabla\dot{\phi}_{0}=\mathcal{K}^{M}\circ\dfrac{d}{dt}d\phi_{t}\bigg{|}_{t=0}=\mathcal{K}^{M}\circ\text{flip}_{M}\circ
d\dot{\phi}_{0}=\mathcal{K}^{M}\circ d\dot{\phi}_{0}.$

Let $\pi_{\mathcal{H}}$ denote the projection of $TP$ onto the horizontal
subspace $\mathcal{H}P$. Therefore, the linearization of $\nabla_{A}$ at
$\phi$ gives us a map $\nabla^{M}:=\mathcal{K}^{M}\circ
d\circ\pi_{\mathcal{H}}$ which can be viewed as a covariant derivative of the
Clifford module $\phi^{*}TM/\mathscr{K}$. This shows that the linearization of
$\mathfrak{F}$ at $\phi$ is a linear Dirac operator $D^{\phi}$ defined
earlier. ∎

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