\section{Introduction}

\subsection{Background}
Let $M$ be an oriented 7-manifold with a ${\mathbb G}_2$-structure, defined by a positive 3-form $\phi.$
Endow $M$ with the corresponding Riemannian metric $g = g_\phi$ and Hodge star operator $* = *_g$ (see \S \ref{ss:npg2} below).

A connection $A$ on a vector bundle $E \to M$ is said to be a \emph{${\mathbb G}_2$-instanton} if its curvature $F_A$ satisfies
\begin{equation}\label{instantondef1}
F_A + * \left( \phi \wedge F_A \right) = 0.
\end{equation}
This equation appeared in the physics literature in the early 1980s \cite{corrigandevchand}. In the mid-1990s, not long after Joyce's construction \cite{joyceg2} of compact Riemannian manifolds with $\mathrm{Hol}(g) ={\mathbb G}_2,$ Donaldson and Thomas \cite{donaldsonthomas} proposed to define an invariant of (torsion-free) ${\mathbb G}_2$-structures
by ``counting'' ${\mathbb G}_2$-instantons in a manner analogous to the Casson invariant in dimension three. Although much progress has been made in the intervening two decades, instantons on compact ${\mathbb G}_2$-holonomy manifolds
are still quite difficult to construct,
and even more so to count.

This article investigates a species of ${\mathbb G}_2$-instanton that arises naturally from the geometry of the 7-sphere.
Although $S^7$ is not a ${\mathbb G}_2$-holonomy manifold, it carries a ``standard'' ${\mathbb G}_2$-structure, $\phi_{std},$ coming from the octonionic structure of ${\mathbb R}^8.$
This is an example of a \emph{nearly parallel ${\mathbb G}_2$-structure}, satisfying
\begin{equation}\label{nearlypar}
d \phi = \tau_0 \psi
\end{equation}
where $\psi = *_{g_{\phi}} \phi$ is the dual 4-form and $\tau_0$ is a
constant.
Nearly parallel ${\mathbb G}_2$-structures
have much in common with the torsion-free case
\cite{friedrichkathmoroianusemmelmann}.

Our
motivation for studying ${\mathbb G}_2$-instantons on
$S^7$ is threefold. First, we would ideally like
to determine the complete moduli space of instantons in this model case, as achieved by ADHM \cite{adhm} for the 4-sphere. This is a very difficult problem; we are limited here to studying the deformations of a given ${\mathbb G}_2$-instanton, in the spirit of Atiyah, Hitchin, and Singer \cite{ahs}. As we shall see, instantons on $S^7$ tend to have large spaces of infinitesimal deformations---one of many ``pathological phenomena'' that may also potentially occur in the torsion-free setting.

The second motivation comes from another difficulty in higher-dimensional gauge theory: the appearance of
instantons with essential singularities. Given a ${\mathrm{Spin} }(7)$-instanton with an isolated singularity on a ${\mathrm{Spin} }(7)$-manifold,
the cross-section of the tangent cone at the singular point
is a ${\mathbb G}_2$-instanton on the standard $S^7.$
Hence, these are the ``building blocks'' for the simplest non-removable singularities in dimension eight.

The third motivation is the relationship with gauge theory in fewer than seven dimensions.
Since ${\mathbb G}_2$-holonomy metrics typically collapse at the boundary of the moduli space,
it is important to understand instantons on model ${\mathbb G}_2$-manifolds coming from lower-dimensional geometries. Recently, Y. Wang \cite{yuanqicyproduct} has shown that
any ${\mathbb G}_2$-instanton on $S^1 \times X,$ for $X$ a Calabi-Yau 3-fold, is gauge-equivalent to the pullback of a Hermitian-Yang-Mills connection.
This is the first case where the full moduli space of ${\mathbb G}_2$-instantons on a given manifold with holonomy contained in ${\mathbb G}_2$ has been identified.

In the present case, a direct link with 4-dimensional gauge theory
is provided by the \emph{quaternionic Hopf fibration}
\begin{equation}\label{quathopf}
S^3 \to S^7 \to S^4.
\end{equation}
With the proper conventions, the pullback
of any anti-self-dual (ASD) instanton on $S^4$
is a ${\mathbb G}_2$-instanton on $S^7,$ giving us an abundant source of examples.
The resulting families of ${\mathbb G}_2$-instantons can be further enlarged by the action of the automorphism group ${\mathrm{Spin} }(7),$
which interacts with the Hopf fibration in a nontrivial way.
Our main results
concern the dimension and completeness properties of these families.

\subsection{Summary}

In \S \ref{sec:conventions}, we set our conventions and derive the basic results concerning instantons on nearly parallel ${\mathbb G}_2$-manifolds, while reviewing the literature in this area.

In \S \ref{sec:quatinst}, we take a concrete approach to the special case of the standard instanton. In quaternionic notation on ${\mathbb R}^8 \cong \H^2,$
the pullback by the (right) Hopf fibration of the standard ASD instanton
may be given by
\begin{equation}\label{standardg2instanton}
A_0(x,y) = \frac{1}{|x|^2 + |y|^2} \Im \left[ |y|^2  x^{-1} \, dx + |x|^2 y^{-1} \, dy \right].
\end{equation}
\begin{comment}
As written here, this defines a smooth connection over
 ${\mathbb R}^8\setminus \{0\}$
with curvature
valued in ${\mathrm{Lie}}({\mathrm{Sp} }(2)) \otimes \Im \H;$ in particular, it is a ${\mathrm{Spin} }(7)$-instanton.
\end{comment}
The restriction of $A_0$ to $S^7$ is dubbed the \emph{standard ${\mathbb G}_2$-instanton}.
Proposition \ref{prop:15deformations} describes a 15-dimensional space of infinitesimal deformations of $A_0,$

which will turn out to be its full space of deformations as a ${\mathbb G}_2$-instanton.

In \S \ref{sec:generaldefos}, we determine the space of infinitesimal deformations of the pullback by the Hopf fibration
of a general irreducible ASD instanton on $S^4.$
The argument relies
on a vanishing result for the vertical component of an infinitesimal deformation in Coulomb gauge, Theorem \ref{thm:vanishing}, which follows from a delicate analysis of the Weitzenbock formula for the deformation operator. Having established that the vertical component vanishes, we use another trick, Lemma \ref{lemma:horiz}, to calculate the horizontal component. This leads directly to our main result, Theorem \ref{thm:generaldefos}, which equates the full space of infinitesimal deformations of the pullback,
as a ${\mathbb G}_2$-instanton,
with three copies of the ASD deformations.

In \S \ref{sec:global}, we briefly discuss the global structure of these families of ${\mathbb G}_2$-instantons.
In the case of charge $\kappa = 1$ and structure group ${\mathrm{SU} }(2),$ which includes the standard instanton,
we have:

\begin{thm}\label{thm:grassmann} The connected component of $A_0$ in the moduli space of ${\mathbb G}_2$-instantons on $S^7$ is diffeomorphic to the tautological 5-plane bundle over the oriented real Grassmannian ${\mathbb G}^{or}(5,7).$
\end{thm}

\noindent
In this description, the base
corresponds to the orbit of the Hopf fibration (\ref{quathopf}) under conjugation by ${\mathrm{Spin} }(7),$ and the fiber corresponds to the pullback of the unit-charge ASD moduli space.
The total space is 15-dimensional, agreeing with
the dimension formula of Theorem \ref{thm:generaldefos}.
By contrast, in the case of higher charge,
we do not expect all of the infinitesimal deformations identified by Theorem \ref{thm:generaldefos}
to be integrable.

Lastly, we state Conjecture \ref{conj:donaldson}, due to Donaldson, which asserts that every ${\mathbb G}_2$-instanton on $S^7$ having integral Chern-Simons value should arise from the pullback construction.

\subsection{Acknowledgements} The author thanks Simon Donaldson
and Thomas Walpuski for discussions, and thanks Aleksander Doan for comments on the manuscript. This research was supported during 2017-18 by the Simons Collaboration on Special Holonomy.

\section{Conventions and basic results}\label{sec:conventions}

\begin{comment}
Let ${\mathbb R}^8 = {\mathbb R}^4 \oplus {\mathbb R}^{4'},$ and choose bases $\omega_i, \omega'_i,$ $i = 1, 2,3,$ for the self-dual 2-forms on ${\mathbb R}^4$ and ${\mathbb R}^{4'},$ respectively. We may take
\begin{equation}
\begin{split}
\omega_1 = dx^{12} + dx^{34}, \qquad \omega'_1 = dx^{56} + dx^{78} \\
\omega_2 = dx^{13} - dx^{24}, \qquad \omega'_2 = dx^{57} - dx^{68} \\
\omega_3 = dx^{14} + dx^{23}, \qquad \omega'_3 = dx^{58} + dx^{67}. \\
\end{split}
\end{equation}
\end{comment}

\subsection{Quaternions, 2-forms, and Fueter maps}\label{ss:quatsandfueter} Let $\H$ denote the 4-dimensional algebra of quaternions, generated by $\mathbf{1}, {\mathbf i}, {\mathbf j},{\mathbf k},$ and subject to the relations
\begin{equation*}
{\mathbf i}^2 = {\mathbf j}^2 = {\mathbf k}^2 = {\mathbf i} {\mathbf j} {\mathbf k} = -\mathbf{1}.
\end{equation*}
The 3-dimensional space of imaginary quaternions $\Im \H,$ with commutator bracket,
is isomorphic to the Lie algebra $\mathfrak{su}(2).$

\begin{comment}
Then the 2-forms (\ref{2formbases}) act on ${\mathbb R}^4,$ in the basis (\ref{quaternionbasis}), by quaternion multiplication on the right:
\begin{equation}\label{quaternionaction}
\cdot \left( - {\mathbf i} \right) =  \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \mathbf{\omega}_1, \qquad
\cdot \left( - {\mathbf j} \right) = \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \mathbf{\omega}_2, \qquad
\cdot \left( - {\mathbf k} \right)  = \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \mathbf{\omega}_3 .
\end{equation}
We define the three complex structures $I_1, I_2, I_3$ on ${\mathbb R}^4.$
Then, in the basis (\ref{quaternionbasis}), one can check that
\begin{equation}\label{quaternionaction}
I_1 = \cdot \left( - {\mathbf i} \right) , \qquad
I_2 = \cdot \left( - {\mathbf j} \right) , \qquad
I_3 = \cdot \left( - {\mathbf k} \right) .
\end{equation}
\end{comment}

We shall identify $\H$ with ${\mathbb R}^4$
as follows:
\begin{equation}\label{quaternionbasis}
\{ e_0, e_1, e_2, e_3 \} = \{ \mathbf{1}, -{\mathbf i}, -{\mathbf j}, -{\mathbf k}\}.
\end{equation}
Denote by $\H_l$ and $\H_r$ the commuting subalgebras of $\mathrm{End} \, {\mathbb R}^4$ corresponding to left- and right-multiplication by $\H,$ respectively, and let ${\mathrm{Sp} }(1)_l$ and ${\mathrm{Sp} }(1)_r$ be the corresponding subgroups of ${\mathrm{SO} }(4)$ generated by unit quaternions.
With this convention, we have
\begin{equation*}
{\mathrm{Lie}}({\mathrm{Sp} }(1)_l) = \Lambda^{2-}, \qquad {\mathrm{Lie}}({\mathrm{Sp} }(1)_r) = \Lambda^{2+}
\end{equation*}
where $\Lambda^{2 \pm} \subset \Lambda^2 {\mathbb R}^4$ denotes the space of (anti-)self-dual 2-forms with respect to the Euclidean metric.

Choose the following standard basis for the self-dual 2-forms on ${\mathbb R}^4:$
\begin{equation}\label{2formbases}
 \mathbf{\omega}_1 = dx^{01} + dx^{23}, \qquad \mathbf{\omega}_2 = dx^{02} - dx^{13}, \qquad \mathbf{\omega}_3 = dx^{03} + dx^{12}.
\end{equation}
Here we abbreviate $dx^{01} = dx^0 \wedge dx^1,$ {\it etc.} Define the complex structures
$I_1, I_2,$ and $I_3,$ by
\begin{equation}\label{quaternionaction}
I_1(v) = v \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \mathbf{\omega}_1, \qquad
I_2(v) = v \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \mathbf{\omega}_2, \qquad
I_3(v)  = v \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \mathbf{\omega}_3
\end{equation}
for $v \in T {\mathbb R}^4.$
Then $I_1, I_2,$ and $I_3$ correspond, respectively, to right-multiplication by $e_1, e_2,$ and $e_3,$ in the basis (\ref{quaternionbasis}).

We define a \textit{Fueter map} $L : {\mathbb R}^4 \to {\mathbb R}^4$ to be an endomorphism satisfying
\begin{equation}\label{fueter}
L + I_1 L I_1 + I_2 L I_2 - I_3 L I_3 = 0.
\end{equation}
The 12-dimensional subspace of Fueter maps $\mathfrak{F} \subset {\text{End}} \, {\mathbb R}^4$ is the direct sum:
\begin{equation}\label{fueterfactors}
\mathfrak{F} = \H_l \oplus \H_l I_1 \oplus \H_l I_2.
\end{equation}
In particular, $\mathfrak{F}$ contains the space of linear maps for the standard complex structure, $I_1.$

\subsection{${\mathrm{Spin} }(7)$ and its subgroups}\label{ss:spin7andg2}
The standard 4-form on ${\mathbb R}^8 = {\mathbb R}^4_x \oplus {\mathbb R}^4_y$ is defined by
\begin{equation}\label{psidef}
\Psi_0 = dx^{0123} + dy^{0123} + \omega^x_1 \wedge \omega^y_1 + \omega^x_2 \wedge \omega^y_2 - \omega^x_3 \wedge \omega^y_3.
\end{equation}
\begin{comment}
\begin{equation}\label{omegaidef}
\begin{split}
\omega^x_1 & = dx^{01} + dx^{23}, \qquad \omega^y_1 = dy^{01} + dy^{23} \\
\omega^x_2 & = dx^{02} - dx^{13}, \qquad \omega^y_2 = dy^{02} - dy^{13} \\
\omega^x_3 & = dx^{03} + dx^{12}, \qquad \omega^y_3 = dy^{03} + dy^{12}.
\end{split}
\end{equation}
\end{comment}
The group ${\mathrm{Spin} }(7)$ consists of all linear transformations of ${\mathbb R}^8$ that preserve $\Psi_0$ under pullback. It is a simply-connected, simple, Lie subgroup of ${\mathrm{SO} }(8)$ of dimension 21 (see {\it e.g.} Walpuski and Salamon \cite{walpuskimasters}, \S 9).

The Lie algebra of ${\mathrm{Spin} }(7)$ corresponds to the subspace of 2-forms $\eta \in \Lambda^2 {\mathbb R}^8$
satisfying
\begin{equation}\label{spin7def}
\eta + \ast \left( \Psi_0 \wedge \eta \right) = 0.
\end{equation}
Let $\Lambda^{2+}_{d} \subset \Lambda^2 {\mathbb R}^8$ be the subalgebra spanned by the three elements
$$\omega_1^x - \omega_1^y, \qquad \omega_2^x - \omega_2^y, \qquad \omega_3^x + \omega_3^y.$$
\begin{comment}
\begin{equation*
 \left( \begin{array}{cc}
I_1 &  0 \\
0 & - I_1 \end{array} \right), \quad \left( \begin{array}{cc}
I_2 &  0 \\
0 & - I_2 \end{array} \right), \quad \left( \begin{array}{cc}
I_3 &  0 \\
0 & I_3 \end{array} \right).
\end{equation*}
\end{comment}
Also denote the subspace
\begin{equation}
\mathfrak{F}_{x,y} = \left\{L_{ij} dy^i \wedge dx^j \mid L \in \mathfrak{F} \right\} \subset \Lambda^2 {\mathbb R}^8.
\end{equation}
Then we have the following decomposition:
\begin{equation}\label{spin7splitting}
{\mathrm{Lie}}({\mathrm{Spin} }(7)) = \Lambda^{2-}_{x} \oplus \Lambda^{2-}_{y} \oplus \Lambda^{2+}_{d} \oplus \mathfrak{F}_{x,y} .
\end{equation}
One readily checks that each factor of (\ref{spin7splitting}) satisfies (\ref{spin7def}).
\begin{comment}
The standard complex structure on ${\mathbb R}^8 = {\mathbb C}^4 = \H^2$ is given by
\begin{equation*
\left( \begin{array}{cc}
I_1 &  0 \\
0 & I_1 \end{array} \right).
\end{equation*}
\end{comment}

Take complex coordinates
$$z^1 = x^0 + i x^1, \quad z^2 = x^2 + i x^3, \quad z^3 = y^0 + i y^1, \quad z^4 = y^2 + i y^3$$
for ${\mathbb R}^8 \cong {\mathbb C}^4.$
We have the standard holomorphic volume form and K\"ahler form
\begin{equation}\label{stdkahlerforms}
\Omega = dz^1 \wedge dz^2 \wedge dz^3 \wedge dz^4, \qquad {\bf \omega} = \frac{i}{2} \sum_{i = 1}^4 dz^i \wedge d\bar{z}^i = \omega_1^x + \omega_1^y.
\end{equation}
One may verify that
\begin{equation}\label{psi0kahler}
\Psi_0 =  \frac{1}{2} \omega \wedge \omega + \Re \Omega.
\end{equation}
The 15-dimensional group ${\mathrm{SU} }(4),$ with its standard presentation, is therefore a subgroup of ${\mathrm{Spin} }(7).$ In particular, the 10-dimensional group ${\mathrm{Sp} }(2)$ of orthogonal quaternionic matrices, linear over $I_1, I_2,$ and $I_3,$ is also a subgroup.

The 14-dimensional Lie group ${\mathbb G}_2$ is the subgroup of ${\mathrm{Spin} }(7)$ stabilizing a point on $S^7.$ Equivalently, ${\mathbb G}_2$ is the subgroup of $\mathrm{GL}(7)$ that preserves the model 3-form
\begin{equation}\label{phi0}
\phi_0 = \frac{\partial}{\partial x^0} \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \Psi_0 = dx^{123} + dx^1 \wedge \omega_1^y + dx^2 \wedge \omega_2^y - dx^3 \wedge \omega_3^y.
\end{equation}
Denote the dual 4-form by
\begin{equation}\label{psi0}
\psi_0 = \ast_{{\mathbb R}^7} \phi_0 = dy^{1234} + dx^{23} \wedge \omega_1^y - dx^{13} \wedge \omega_2^y - dx^{12} \wedge \omega_3^y.
\end{equation}
The Lie algebra ${\mathrm{Lie}}({\mathbb G}_2)$ corresponds to the subspace of 2-forms $\xi \in \Lambda^2 {\mathbb R}^7$ satisfying
\begin{equation*}
\xi + * \left( \phi_0 \wedge \xi \right) = 0
\end{equation*}
or equivalently
\begin{equation*}
\psi_0 \wedge \xi = 0.
\end{equation*}

\begin{rmk}\label{rmk:convention}
The (standard) presentation of ${\mathrm{Spin} }(7),$ in which the diagonal ${\mathrm{Sp} }(1)_r$ is a subgroup, can be obtained from (\ref{psidef}) by the coordinate change
\begin{equation*}
\left( \begin{array}{cc}
\mathbf{1} &  0 \\
0 & I_3 \end{array} \right).
\end{equation*}
However, Lemma \ref{lemma:instapullback} below fails in that convention.
In fact, there is no convention in which both ${\mathrm{Sp} }(2)$ (giving a transitive group of automorphisms) and a commuting copy of ${\mathrm{Sp} }(1)$ (giving the Hopf fibration) are subgroups of ${\mathrm{Spin} }(7);$
this is an essential difficulty.

\end{rmk}

\subsection{Nearly parallel ${\mathbb G}_2$-structures}\label{ss:npg2} Let $M$ be an oriented 7-manifold. Recall that a \emph{${\mathbb G}_2$-structure} on $M$ is defined by a global 3-form $\phi$ that is \emph{positive}, in the sense that
\begin{equation}
G_\phi(v) = \left( v \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \phi \right) \wedge \left( v \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \phi \right) \wedge \phi > 0
\end{equation}
for all $x \in M$ and $v \neq 0 \in T_x M.$ Any such $\phi$ is pointwise equivalent to the model 3-form $\phi_0$ given by (\ref{phi0}) above (see \cite{walpuskimasters}, Theorem 3.2).

A positive 3-form defines a unique Riemannian metric $g_\phi$ on $M$ by the requirement
\begin{equation}\label{g2metric}
6 g_\phi(v,v)  Vol_{g_\phi} = G_\phi(v) \,\, \forall \,\, v \in TM.
\end{equation}
To any $\phi,$ we also associate the dual 4-form
\begin{equation}
\psi = \ast_{g_{\phi}} \phi.
\end{equation}
Recall that a ${\mathbb G}_2$-structure is said to be \emph{closed} if $d \phi = 0,$ and \emph{coclosed} if $d \psi = 0.$

We are concerned with ${\mathbb G}_2$-structures satisfying (\ref{nearlypar}), where we assume
\begin{equation}\label{pm4}
\tau_0 = \pm 4.
\end{equation}
The basic reference for nearly parallel structures is Friedrich, Kath, Moroianu, and Semmelmann \cite{friedrichkathmoroianusemmelmann}. With the normalization (\ref{pm4}), nearly parallel ${\mathbb G}_2$-manifolds are Einstein, with
\begin{equation}
\mathrm{Ric}_g = 6 g.
\end{equation}
There are three further equivalent formulations of the nearly parallel condition (\ref{nearlypar}).
The first is that the induced ${\mathrm{Spin} }(7)$-structure on the cone over $M$ be torsion-free. The second and most frequently used condition
is that $M$ possess a nonzero Killing spinor.
The third is as follows:
\begin{lemma}\label{lemma:gradphi} A ${\mathbb G}_2$-structure $\phi$ is nearly parallel if and only if
\begin{equation*}
\nabla \phi = \frac{\tau_0}{4} \psi.
\end{equation*}
Here $\nabla$ is the Levi-Civita connection associated to the metric $g_\phi$ defined by (\ref{g2metric}).
\end{lemma}
\begin{proof} See Karigiannis \cite{karigiannisflowsofg2}, Theorem 2.27.
\end{proof}

\begin{example} Define the \emph{standard ${\mathbb G}_2$-structure on $S^7$} by
\begin{equation}\label{phistdzerothdef}
\phi_{std} = \left. \vec{r} \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \Psi_0 \right|_{S^7}
\end{equation}
where
$$\vec{r} = x^i \frac{\partial}{\partial x^i} + y^i \frac{\partial}{\partial y^i}$$
is the coordinate vector field on ${\mathbb R}^8.$
The group of global automorphisms of $\phi_{std}$ is ${\mathrm{Spin} }(7).$

To obtain a more explicit expression for $\phi_{std},$ we define the 3-form
\begin{equation*}
\nu^x = \vec{r} \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \mathrm{Vol}_{{\mathbb R}^4_x}= x^0 dx^{123} - x^1 dx^{023} + x^2 dx^{013} - x^3 dx^{012}
\end{equation*}
on ${\mathbb R}^4_x,$ and the 1-forms
\begin{equation}\label{zetaidef}
\begin{split}
\zeta_1^x &= \vec{r}\mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \omega_1^x = x^0 dx^1 - x^1 dx^0 + x^2 dx^3 - x^3 dx^2 \\
\zeta_2^x &= \vec{r} \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \omega_2^x = x^0 dx^2 - x^2 dx^0 - x^1 dx^3 + x^3 dx^1 \\
\zeta_3^x & = \vec{r} \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \omega_3^x = x^0 dx^3 - x^3 dx^0 + x^1 dx^2 - x^2 dx^1.
\end{split}
\end{equation}
Define $\nu^y$ and $\zeta_i^y$ similarly. We then have
\begin{equation}\label{phistdfirstdef}
\begin{split}
\phi_{std}
& = \nu^x + \nu^y + \zeta_1^x \wedge \omega_1^y + \zeta_1^y \wedge \omega_1^x + \zeta_2^x \wedge \omega_2^y + \zeta_2^y \wedge \omega_2^x - \zeta_3^x \wedge \omega_3^y - \zeta_3^y \wedge \omega_3^x.
\end{split}
\end{equation}
It is easy to check, using the ${\mathrm{Spin} }(7)$-invariance, that $\phi_{std}$ defines the round metric
on $S^7.$

Notice that $d \nu^x = 4 {\mathrm{Vol}}_{{\mathbb R}^4_x}$ and $d \zeta_i^x = 2 \omega_i^x,$ and similarly for $y,$ so
\begin{equation}\label{phiPsi0}
d \phi_{std} = \left. 4 \Psi_0 \right|_{S^7}.
\end{equation}
Also note that
\begin{equation}\label{psistd}
\psi_{std} = \ast_{std} \phi_{std} = \left. \ast_{{\mathbb R}^8} \Psi_0 \right|_{S^7} = \left. \Psi_0 \right|_{S^7}.
\end{equation}
It follows from (\ref{phiPsi0}-\ref{psistd}) that $\phi_{std}$ is a nearly parallel ${\mathbb G}_2$-structure.

\begin{comment}
It will be useful to rewrite $\phi_{std}$ using global ${\mathrm{Sp} }(2)$-invariant coframes. Let
\begin{equation*
\begin{split}
\zeta_i & = \zeta_i^x + \zeta_i^y, \qquad \omega_i^\circ = \left. \omega^x_i + \omega^y_i \right|_{S^7} \\
\bar{\omega}_i  & = \omega_i^\circ - \frac{1}{2}\epsilon_{ijk} \zeta_j \wedge \zeta_k, \qquad i = 1,2,3 \\
\nu & = \zeta_1 \wedge \zeta_2 \wedge \zeta_3.
\end{split}
\end{equation*}
From (\ref{phistdfirstdef}) and the ${\mathrm{Sp} }(2)$-invariance, we have
\begin{equation
\begin{split}
\phi_{std}
& = \nu + \zeta_1 \wedge \bar{\omega}_1 - \zeta_2 \wedge \bar{\omega}_2 + \zeta_3 \wedge \bar{\omega}_3.
\end{split}
\end{equation}
\end{comment}
\end{example}
\begin{example}\label{ex:squashed} Define the \emph{squashed ${\mathbb G}_2$-structure}
\begin{equation*}
\phi_{sq} = \frac{27}{25} \left( \begin{split} \frac{1}{5} (\nu_x + \nu_y) + \frac{16}{5} \left( \begin{split}\zeta_1^x \wedge \zeta_2^x \wedge \zeta_3^y + \zeta_1^x \wedge \zeta_2^y \wedge \zeta_3^x + \zeta_1^y \wedge \zeta_2^x \wedge \zeta_3^x \\ + \zeta_1^x \wedge \zeta_2^y \wedge \zeta_3^y + \zeta_1^y \wedge \zeta_2^x \wedge \zeta_3^y + \zeta_1^y \wedge \zeta_2^y \wedge \zeta_3^x  \end{split} \right) \\
- \zeta^x_1 \wedge \omega^y_1 - \zeta_1^y \wedge \omega_1^x - \zeta^x_2 \wedge \omega^y_2 - \zeta_2^y \wedge \omega_2^x - \zeta^x_3 \wedge \omega^y_3 - \zeta_3^y \wedge \omega_3^x \end{split}\right).
\end{equation*}
The squashed ${\mathbb G}_2$-structure was discovered by Awada, Duff, and Pope \cite{awadaduffpope}, and has automorphism group ${\mathrm{Sp} }(2) {\mathrm{Sp} }(1).$ Appendix \ref{appendix:squashed} includes a proof that $\phi_{sq}$ is nearly parallel. This is a special case of the theorem of Friedrich {\it et al.} \cite{friedrichkathmoroianusemmelmann}, which constructs a strictly-nearly-parallel ${\mathbb G}_2$-structure by shrinking the fibers of any 3-Sasakian 7-manifold.
\end{example}

\begin{rmk}
Alexandrov and Semmelmann \cite{alexandrovsemmelmannnearlyparallel} have shown that both the standard and the squashed structures are rigid among nearly parallel ${\mathbb G}_2$-structures. These remain the only known nearly parallel structures on the 7-sphere, with any of its differentiable structures.
We also note that the (non-)existence of a \emph{closed}
${\mathbb G}_2$-structure on the 7-sphere is a well-known open problem.
\end{rmk}

\subsection{Instantons on nearly parallel ${\mathbb G}_2$-manifolds} Recall that a connection $A$ is called a \emph{${\mathbb G}_2$-instanton} if its curvature satisfies (\ref{instantondef1}),
or equivalently
\begin{equation}\label{instantondef2}
\psi \wedge F_A = 0.
\end{equation}
If the ${\mathbb G}_2$-structure $\phi$ is nearly parallel, then from (\ref{instantondef1}) and (\ref{nearlypar}), we have
\begin{equation*}
\begin{split}
 0 & = D_A^*F_A - * D_A \left( \phi \wedge F_A \right) \\
 & = D_A^*F_A - * \left( \tau_0 \psi \wedge F_A -\phi \wedge D_A F_A \right) \\
 & = D_A^*F_A.
 \end{split}
 \end{equation*}
We have used (\ref{instantondef2}) and the Bianchi identity in the last line. Hence, in the nearly-parallel case, any ${\mathbb G}_2$-instanton is Yang-Mills.

The linearization of (\ref{instantondef2}) is
$$\psi \wedge D_A \alpha = 0$$
for $\alpha \in \Omega^1\left( {\mathfrak g}_E \right),$ and an infinitesimal gauge transformation  $u \in \Omega^0 \left( {\mathfrak g}_E \right)$ acts by $u \mapsto D_A u.$ The infinitesimal deformations of a ${\mathbb G}_2$-instanton $A,$ modulo gauge, therefore correspond to the first cohomology group of the following self-dual elliptic complex:
\begin{equation}\label{deformationcomplex}
\Omega^0 \left( {\mathfrak g}_E \right) \stackrel{D_A}{\longrightarrow} \Omega^1 \left( {\mathfrak g}_E \right) \stackrel{ \psi \wedge D_A }{\longrightarrow} \Omega^6 \left( {\mathfrak g}_E \right) \stackrel{D_A}{\longrightarrow} \Omega^7 \left( {\mathfrak g}_E \right).
\end{equation}
Folding (\ref{deformationcomplex}), and writing $d = D_A,$ we obtain the \emph{deformation operator}
\begin{equation}\label{deformationoperator}
\begin{split}
{\mathscr L}_A & :  \Omega^0\left( {\mathfrak g}_E \right) \oplus \Omega^1 \left( {\mathfrak g}_E \right) \to \Omega^0\left( {\mathfrak g}_E \right) \oplus \Omega^1 \left( {\mathfrak g}_E \right) \\
& \begin{pmatrix}
u \\
\alpha
\end{pmatrix}
\mapsto
\begin{pmatrix}
d^*\alpha \\
du + * \left( \psi \wedge d\alpha \right)
\end{pmatrix}.
\end{split}
\end{equation}
This is a first-order, self-adjoint, elliptic operator.

Squaring (\ref{deformationoperator}), we obtain
\begin{equation}\label{lausquared}
\begin{split}
{\mathscr L}_A^2  \begin{pmatrix}
u \\
\alpha
\end{pmatrix} & =
\begin{pmatrix}
d^* d u - *\left( d \psi \wedge d \alpha + \psi \wedge F_A \wedge \alpha \right) \\
d d^* \alpha + * \left( \psi \wedge d \left( * \left( \psi \wedge d \alpha \right) + F_A \wedge u \right) \right)
\end{pmatrix} \\
& = \begin{pmatrix}
d^* d u \\
d d^*\alpha + * \left( \psi \wedge d * \left( \psi \wedge d\alpha \right) \right)
\end{pmatrix}.
\end{split}
\end{equation}
Over a compact manifold, integration by parts implies
\begin{equation*}
\ker \mathscr{L}_A = \ker \mathscr{L}_A^2.
\end{equation*}
\begin{comment}
\begin{equation*}
\mathscr{L}_A \begin{pmatrix} u \\ \alpha \end{pmatrix} = 0 \Leftrightarrow \mathscr{L}_A^2 \begin{pmatrix} u \\ \alpha \end{pmatrix} = 0.
\end{equation*}
\end{comment}
Hence, $du \equiv 0$ for any infinitesimal deformation on a compact nearly parallel ${\mathbb G}_2$-manifold, and $u \equiv 0$ if $A$ is irreducible.

We shall use the following interior product notation:
\begin{equation}\label{2formaction}
\begin{split}
 dx^{01} \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \frac{\partial}{\partial x^1} = dx^0, \qquad dx^{01} \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \frac{\partial}{\partial x^0}  = - dx^1, \quad  \text{{\it etc.}}
\end{split}
\end{equation}
\begin{comment}
\begin{equation}\label{2formaction}
\begin{split}
e_0 \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} dx^{01} & = e_1,  \qquad dx^{01} \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} e_1 = e_0\\
e_1 \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} dx^{01} & = - e_0, \qquad dx^{01} \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} e_0  = - e_1, \quad  \text{{\it etc.}}
\end{split}
\end{equation}
\end{comment}
We also take interior products between differential forms, {\it e.g.}
$$dx^{01} \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} dx^1 = dx^0, \qquad dx^{01} \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} dx^{01} = 1$$
and similarly in general using the metric.
In particular, for a 2-form $b,$ we have
$$* \left( \psi \wedge b \right) = \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} b.$$
For a ${\mathfrak g}_E$-valued 1-form $\alpha,$ we shall write
\begin{equation*}
{\mathscr L}_A(\alpha) = {\mathscr L}_A \begin{pmatrix}
0 \\
\alpha
\end{pmatrix} = \begin{pmatrix}
d^*\alpha \\
\phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d\alpha
\end{pmatrix}.
\end{equation*}
Then (\ref{lausquared}) becomes
\begin{equation}\label{lasquaredfirst}
{\mathscr L}_A^2(\alpha) = d d^*\alpha + \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d \left( \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d \alpha \right).\end{equation}

\begin{comment}
\begin{proof} This is an algebraic identity, and can be checked for the standard forms $\phi_0$ and $\psi_0$ above (\ref{phi0}-\ref{psi0}).  The result is immediate from (\ref{phi0}-\ref{psi0}).
\end{proof}
\end{comment}

\begin{lemma}[\cite{bryantlaplacianremarks}, (2.7-8)]\label{lemma:phiidents} The following identities hold between any positive 3-form $\phi,$ the associated metric $g = g_\phi,$ and the dual 4-form $\psi = *_{\phi} \phi:$
\begin{equation*}
g^{pq} \phi_{p ij} \phi_{q k \ell} = g_{i k} g_{j\ell} - g_{i\ell} g_{j k} + \psi_{i j k \ell}
\end{equation*}
\begin{equation*}
g^{pq} g^{\ell m} \phi_{p\ell i} \psi_{qm jk} = 2 \phi_{ijk}.
\end{equation*}
\end{lemma}
\begin{proof} Since these are zeroth-order identities, it suffices to check them for the standard 3- and 4-form, given by (\ref{phi0}-\ref{psi0}), and the standard metric. This is easily accomplished using the fact that ${\mathbb G}_2$ acts transitively on orthonormal pairs of vectors.
\end{proof}

\begin{lemma}[{\it Cf.} \cite{yuanqideformations}, Lemma 7.1]\label{lemma:dbstar}
For a nearly parallel $G_2$-structure and any 2-form $b,$ there holds
\begin{equation}\label{dbstar:maineq}
d\left( b \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \phi \right) \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \phi = d^*b - db \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \psi + \frac{\tau_0}{2} b \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \phi.
\end{equation}
\end{lemma}
\begin{proof}
Write $b = \frac12 b_{ij} dx^i \wedge dx^j$ in normal coordinates. We then have
\begin{equation*}
\begin{split}
\left( d \left( b \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \phi \right) \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \phi \right)_k & = \frac12 \left( \frac12 \left( \nabla_m (b_{ij} \phi_{ij\ell}) - \nabla_\ell (b_{ij} \phi_{ijm}) \right) \right) \phi_{m \ell k} \\
& = \frac14 \left( \nabla_m b_{ij} \phi_{ij \ell} + b_{ij} \nabla_m \phi_{ij\ell} - \nabla_\ell b_{ij} \phi_{ijm} - b_{ij} \nabla_{\ell} \phi_{ijm} \right) \phi_{m \ell k} \\
& = - \frac12 \nabla_m b_{ij} \phi_{\ell ij} \phi_{\ell mk} + \frac{\tau_0}{8} b_{ij} \psi_{m ij \ell} \phi_{m \ell k}
\end{split}
\end{equation*}
where we have used Lemma \ref{lemma:gradphi} in the last line.
By Lemma \ref{lemma:phiidents}, this becomes
\begin{equation*}
\begin{split}
\left( d \left( b \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \phi \right) \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \phi \right)_k & = - \frac12 \nabla_m b_{ij} \left( g_{im} g_{jk} - g_{ik} g_{jm} + \psi_{ijmk} \right) + \frac{\tau_0}{4} b_{ij} \phi_{ijk} \\
& = - \nabla_i b_{ik} - \frac12 \nabla_m b_{ij} \psi_{mijk} + \frac{\tau_0}{4} b_{ij} \phi_{ijk}
\end{split}
\end{equation*}
which agrees with the expression (\ref{dbstar:maineq}).
\end{proof}

\begin{prop}\label{prop:weitz} For a ${\mathbb G}_2$-instanton with respect to a nearly parallel ${\mathbb G}_2$-structure $\phi,$ we have
\begin{equation}\label{weitwithsa}
\begin{split}
\mathscr{L}_A^2(\alpha)
& = \frac{\tau_0}{2} \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d \alpha + \mathscr{S}_A \left( \alpha \right)
\end{split}
\end{equation}
where
$$\mathscr{S}_A\left( \alpha \right) = \nabla_A^*\nabla_A \alpha + {\mathrm{Ric}}(\alpha) - 2 \left[ F_A \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \alpha \right]$$
is the Yang-Mills stability operator (see Bourguignon-Lawson \cite{bourguignonlawson}).
In the case of the round $7$-sphere, we have
\begin{equation}\label{LAsquared}
\begin{split}
\mathscr{L}_A^2(\alpha ) & = 2 \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d \alpha + \nabla_A^*\nabla_A \alpha + 6 \alpha - 2 \left[ F_A \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \alpha \right] .
\end{split}
\end{equation}
\end{prop}
\begin{proof} From (\ref{lasquaredfirst}) and Lemma \ref{lemma:dbstar}, we have
\begin{equation}\label{weitzmess}
\begin{split}
{\mathscr L}_A^2(\alpha) & = d d^*\alpha + \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d \left( \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d\alpha \right) \\
& = d d^*\alpha + d^* d \alpha - d^2 \alpha \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \psi + \frac{\tau_0}{2} \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d \alpha \\
& = \left( d d^* + d^* d \right) \alpha - \left( F_A \wedge \alpha \right) \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \psi + \frac{\tau_0}{2} \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d \alpha \\
& = \frac{\tau_0}{2} \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d \alpha + \left( d d^* + d^* d \right) \alpha + \left( F_A \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \psi \right) \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \alpha.
\end{split}
\end{equation}
But since $A$ is an instanton, we have
$$F_A \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \psi = * \left( F_A \wedge \phi \right) = - F_A.$$
Substituting into (\ref{weitzmess}) and applying the Bochner formula yields (\ref{weitwithsa}). Then (\ref{LAsquared}) is obtained by substituting $\tau_0 = 4$ and ${\mathrm{Ric}}_g = 6g$ on the round 7-sphere.
\end{proof}

\begin{rmk} Instantons on compact manifolds with holonomy ${\mathbb G}_2$ (other than the Levi-Civit{\`a} connection) were first constructed by Walpuski \cite{walpuskig2onkummer}, and have subsequently been produced in a few different contexts.
For the state of the art, see \cite{walpuskiarithmeticinstantons}.

Instantons on nearly parallel ${\mathbb G}_2$-manifolds have been studied by Harland and N{\"o}lle \cite{harlandnolle}, and by Ball and Oliveira \cite{balloliveira} in the case of Aloff-Wallach spaces. Ragini Singhal \cite{raginisinghal} also studies instantons on homogeneous nearly parallel ${\mathbb G}_2$-manifolds, using methods similar to those of Charbonneau and Harland \cite{charbharland} in the context of nearly K{\"a}hler 6-manifolds.
\end{rmk}

\section{Hopf fibration and standard instantons}\label{sec:quatinst}

In this section, we give an explicit description of the standard (A)SD instanton and its ${\mathbb G}_2$ relative. We shall use the following variant of Atiyah's quaternionic notation \cite{atiyahgeometry}:
\begin{equation*}
\begin{split}
x & = x^0 \mathbf{1} - x^1 {\mathbf i} - x^2 {\mathbf j} - x^3 {\mathbf k}, \qquad \qquad \,\,\,\, \bar{x} = x^0 \mathbf{1} + x^1 {\mathbf i} + x^2{\mathbf j} + x^3 {\mathbf k}  \\
 dx & = dx^0 \mathbf{1} - dx^1 {\mathbf i} - dx^2 {\mathbf j} - dx^3 {\mathbf k} , \qquad d\bar{x}  = dx^0 \mathbf{1} + dx^1 {\mathbf i} + dx^2 {\mathbf j} + dx^3 {\mathbf k}.
\end{split}
\end{equation*}
Here $x$ and $\bar{x}$ are $\H$-valued functions, and $dx$ and $d\bar{x}$ are $\H$-valued differential forms on ${\mathbb R}^4,$ with signs determined by the convention (\ref{quaternionbasis}). We define $y$ and $\bar{y}$ similarly, and will identify
\begin{equation*}
{\mathbb R}^8 = \H_x \oplus \H_y
\end{equation*}
as above.

\subsection{The quaternionic Hopf fibration(s)}\label{ss:hopf}
The (right) Hopf fibration
\begin{equation}\label{hopf}
\pi: \H^2 \setminus \{ (0,0) \} \rightarrow \H \P^1
\end{equation}
is the quotient projection under right-multiplication by $\H^{\times}.$ The restriction of (\ref{hopf}) to the 7-sphere and to multiplication by ${\mathrm{Sp} }(1)_r$ gives the fibration (\ref{quathopf}).

To see the identification $\H \P^1 \cong S^4$
explicitly, observe that ${\mathrm{Sp} }(2)$ acts on $S^7$ by isometries commuting with ${\mathrm{Sp} }(1)_r,$ where the stabilizer of a fiber is the subgroup ${\mathrm{Sp} }(1)_l \times {\mathrm{Sp} }(1)_l \subset {\mathrm{Sp} }(2).$ The group ${\mathrm{Sp} }(2)$ has a 5-dimensional irreducible representation $W,$ given by the conjugation action on the space of  $2\times 2$ traceless self-adjoint quaternionic matrices:
\begin{equation}\label{wdef}
W = \left\{ \left( \begin{array}{cc}
a  & \bar{z} \\
z & -a  \end{array} \right) \mid a \in {\mathbb R}, z \in \H \right\}.
\end{equation}
This action sets up a map
\begin{equation*}\label{quotientid}
\H \P^1 = {\mathrm{Sp} }(2)/ {\mathrm{Sp} }(1)_l \times {\mathrm{Sp} }(1)_l \,\, \tilde{\longrightarrow} \,\, {\mathrm{SO} }(5) / {\mathrm{SO} }(4) = S^4
 \end{equation*}
which is an isometry, up to a factor of $1/2.$

\begin{comment}

We claim that the tautological bundle over $\H \P^1$ is the positive spin bundle $S^+ \to S^4,$ and its orthogonal complement is $S^-.$
To check this, it suffices to write down the Clifford multiplication $\gamma : TS^4 \to \mathrm{Hom} \left( S^+ , S^- \right),$ satisfying
\begin{equation}\label{cliffordrule}
\gamma(v)^* \gamma(v) = |v|^2.
\end{equation}
But a tangent vector $v \in T_x\H \P^1$ is exactly an $\H_r$-linear map from the tautological bundle to its complement, {\it i.e.}, $\gamma(v): S^+_x \to S^-_x.$ Such a map is necessarily given by multiplication by a quaternion on the left, which automatically satisfies (\ref{cliffordrule}).
With the orientation convention on $\H \P^1$ coming from (\ref{quaternionbasis}), one checks that $\gamma$ sets up an isomorphism
\begin{equation*
\Lambda^2_+(T^*S^4) \tilde{\longrightarrow} {\text{End}}_0 \, S^+
\end{equation*}
as required.

\end{comment}

We write $P^+$ for the principal ${\mathrm{Sp} }(1)$-bundle associated to the right Hopf fibration. The \emph{standard self-dual instanton} is defined to be the connection on $P^+$ induced by the round metric on the total space. The corresponding connection form on $P^+$ is the $\mathfrak{su}(2)$-valued 1-form
\begin{equation*
\frac{ \Im \left[ \bar{x} \, d x + \bar{y} \, d y \right] }{ |x|^2 + |y|^2 }.
\end{equation*}
Pulling back to ${\mathbb R}^4$ by the map $x \mapsto (x,1),$
we obtain the well-known connection matrix
\begin{comment}  instanton on ${\mathbb R}^4:$
\begin{equation*}\label{standardsdinst}
\frac{\Im  \bar{x} dx }{1 + |x|^2}.
\end{equation*}
Its curvature is the $\mathfrak{su}(2)$-valued self-dual 2-form
\begin{equation*}
\frac{ d\bar{x} \wedge d x}{\left( 1 + |x|^2 \right)^2 }.
\end{equation*}
\end{comment}
\begin{equation*
\frac{\Im  \bar{x} d x }{1 + |x|^2}
\end{equation*}
whose curvature is the $\mathfrak{su}(2)$-valued self-dual 2-form
\begin{equation*
\frac{ d \bar{x} \wedge d x }{\left( 1 + |x|^2 \right)^2 }.
\end{equation*}
See Atiyah \cite{atiyahgeometry}, \S 1, for these formulae, as well as a generalization giving the complete ADHM construction.

Similarly, we define the \emph{standard anti-self-dual (ASD) instanton} $P^-$ to be the principal bundle associated to the left Hopf fibration, with connection form
\begin{equation}\label{stdasdconnform}
\frac{ \Im \left[ w\, d \bar{w} + z \, d\bar{z} \right] }{ |w|^2 + |z|^2 }.
\end{equation}
In the stereographic chart on ${\mathbb R}^4,$ this has a connection matrix
\begin{equation}\label{stdinstaconnmatrix}
B_0(x) = \frac{\Im x d \bar{x} }{1 + |x|^2}
\end{equation}
and curvature the $\Im \H$-valued self-dual 2-form
\begin{equation}\label{stdinstacurvature}
F_{B_0}(x) = \frac{ d x \wedge d \bar{x} }{\left( 1 + |x|^2 \right)^2 }.
\end{equation}

The basic link between instantons on $S^7$ and $S^4$ is as follows; a more general statement appears in Proposition \ref{prop:hym} below.

\begin{comment}
 It follows from (\ref{quotientid}) that the tautological bundle over $\H \P^1_r$ is the positive spin bundle $S^+ \to S^4,$ and its orthogonal complement is $S^-.$
To check this, it suffices to write down the Clifford multiplication $\zeta : TS^4 \to \mathrm{Hom} \left( S^+ , S^- \right),$ satisfying
\begin{equation}\label{cliffordrule}
\zeta(v)^* \zeta(v) = |v|^2
\end{equation}
for all tangent vectors $v.$ But a tangent vector $v$ to $\H \P^1_r$ at a point $x$ is just an $\H_r$-linear map from the tautological bundle to its complement ({\it i.e.} $\zeta(v): S^+_x \to S^-_x$); such a map is necessarily given by multiplication by a quaternion on the left, which satisfies (\ref{cliffordrule}).
\end{comment}

\begin{comment}
Explicitly, given a point $(a,b)^{T} \in S^7$ and a tangent vector $(x,y)^{T},$ both considered as vectors in $\H^2,$ with
$$ \bar{x}a +  \bar{y} b = 0$$ the Clifford multiplication on the total spin bundle $S^+ \oplus S^- \simeq \underline{{\mathbb R}^8}$ is given by left-multiplication by
\begin{equation}
\zeta_{\left( \begin{array}{c} a \\ b \end{array} \right) }\left(  \begin{array}{c} x \\ y \end{array} \right) =  \left( \begin{array}{c} a \\ b \end{array} \right) \left(  \begin{array}{cc} \bar{x} & \bar{y} \end{array} \right) - \left(  \begin{array}{c} x \\ y \end{array} \right)\left( \begin{array}{cc} \bar{a} & \bar{b} \end{array} \right) = \left( \begin{array}{cc}
a \bar{x} - x \bar{a} & x \bar{b} - a \bar{y} \\
y \bar{a} - b \bar{x} & y \bar{b} - b \bar{y} \end{array} \right).
\end{equation}
This interchanges $S^+$ and $S^-,$ and squares to $- \left( |x|^2 + |y|^2 \right),$ as required.
\end{comment}
\begin{comment}
With the orientation convention on $\H \P^1_r$ coming from (\ref{quaternionbasis}), one can also check that $\zeta$ sets up an isomorphism
\begin{equation}\label{selfdualS+}
\Lambda^2_+(T^*S^4) \tilde{\longrightarrow} {\text{End}}_0 \, S^+
\end{equation}
as required.
\end{comment}

\begin{lemma}\label{lemma:instapullback} Let $B$ be a connection on a principal bundle over $S^4.$ Then $B$ is an ASD instanton if and only if the pullback $A = \pi^*B$ by the (right) Hopf fibration is a ${\mathbb G}_2$-instanton, for either the standard or the squashed nearly parallel ${\mathbb G}_2$-structure on $S^7.$
\end{lemma}
\begin{proof}
By ${\mathrm{Sp} }(2)$-invariance both of $\phi_{std}$ and of the fibration, it suffices to consider the point $(1,0).$
From (\ref{phistdfirstdef}), we have
\begin{equation}\label{phis30}
 \phi (x,0)  = \nu^x + \zeta_1^x \wedge \omega_1^y + \zeta_2^x \wedge \omega_2^y - \zeta_3^x \wedge \omega_3^y.
\end{equation}
and
\begin{equation*}
\phi(1,0) = dx^{123} + dx^1 \wedge \omega_1^y + dx^2 \wedge \omega_2^y - dx^3 \wedge \omega_3^y.
\end{equation*}
Then it is clear that $\phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} F_A = \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \pi^*F_B$ vanishes if and only if $B$ is ASD.

The same argument applies on $\phi_{sq}.$
\end{proof}

\subsection{The standard ${\mathbb G}_2$-instanton}\label{ss:stdg2inst}
Let
\begin{equation}
P = P^-\times_{S^4} P^+ \stackrel{\pi_2}{\longrightarrow} S^7
\end{equation}
be the fiber product over $S^4$ of $P^-$ with $P^+,$ considered as a principal ${\mathrm{Sp} }(1)$-bundle via the projection to $P^+ = S^7.$
According to \S \ref{ss:hopf} and Lemma \ref{lemma:instapullback}, the pullback of the standard ASD instanton on $P^-$ is a ${\mathbb G}_2$-instanton on $P \to S^7,$ which we call the \emph{standard ${\mathbb G}_2$-instanton}, $A_0.$

We may obtain a connection matrix for $A_0$ by pulling back the connection form of the standard ASD instanton (\ref{stdasdconnform})
by the fiber-preserving map $\H^{\times 2}_r \to \H^{\times 2}_l $ given by
\begin{equation*}
\begin{split}
w & = y^{-1} \\
z & = x^{-1}.
\end{split}
\end{equation*}
\begin{comment}
\begin{equation*}
\begin{split}
\begin{pmatrix}
x \\
y
\end{pmatrix} & \mapsto  \begin{pmatrix}
y^{-1} \\
x^{-1}
\end{pmatrix}.
\end{split}
\end{equation*}
\end{comment}
This gives
\begin{equation*}
\begin{split}
A_0(x,y) & = \frac{1}{ |x|^{-2} + |y|^{-2} } \Im \left[ x^{-1} d \left( \bar{x}^{-1} \right) + y^{-1} d \left( \bar{y}^{-1} \right) \right] \\
& = \frac{|x|^2|y|^2}{ |x|^{2} + |y|^{2} } \Im \left[ - x^{-1} \bar{x}^{-1} d \bar{x} \bar{x}^{-1} - y^{-1} \bar{y}^{-1} d\bar{y} \bar{y}^{-1}  \right] \\
& = \frac{-1}{|x|^2 + |y|^2} \Im \left[ |y|^2 d\bar{x} \, \bar{x}^{-1} + |x|^2 d\bar{y} \, \bar{y}^{-1} \right] \\
& = \frac{1}{|x|^2 + |y|^2} \Im \left[ |y|^2  x^{-1} \, dx + |x|^2 y^{-1} \, dy \right]
\end{split}
\end{equation*}
which is (\ref{standardg2instanton}) above.
The singularity along the $x$-axis can be removed by applying the gauge transformation $g(x,y) = y/|y|:$ noting that
$dg g^{-1} = - \Im y d\bar{y} / |y|^2,$ we obtain
\begin{equation*}
\begin{split}
 g(A_0) = gA_0 \bar{g} - dg \bar{g} & = \frac{1}{|x|^2 + |y|^2} \Im \left[  y  x^{-1} dx \bar{y} + \frac{|x|^2}{|y|^2} dy \bar{y} + \frac{|x|^2 + |y|^2}{|y|^2} y d\bar{y} \right] \\
 & = \frac{1}{|x|^2 + |y|^2} \Im \left[  y  x^{-1} dx \bar{y} + \frac{|x|^2}{|y|^2} dy \bar{y} - \frac{|x|^2 }{|y|^2} dy \bar{y} + y d\bar{y} \right] \\
 & = \frac{1}{|x|^2 + |y|^2} \Im \left[ y x^{-1} dx \bar{y} + y d\bar{y} \right].
\end{split}
\end{equation*}
 The singularity along the $y$-axis can be removed similarly.
However,
they cannot be removed simultaneously, since the bundle $P \to S^7$ is nontrivial;
recall that topological ${\mathrm{SU} }(2)$-bundles on $S^7$ are classified by $\pi_6(S^3) = {\mathbb Z}_{12}.$

\begin{prop}[Crowley and Goette \cite{crowleygoettekreck}, (1.18)]
For an $SU(2)$-bundle on $S^4$ with $c_2(E) = \kappa,$ the pullback bundle
on $S^7$ has homotopy class
\begin{equation*}
\frac{\kappa \left( \kappa + 1 \right)}{2} \in {\mathbb Z}_{12}.
\end{equation*}
\end{prop}

\subsection{Curvature calculation}\label{ss:curvcalc} We check directly that $A_0$ is a ${\mathbb G}_2$-instanton on $S^7.$ By (\ref{phistdzerothdef}), this is equivalent to showing that (\ref{standardg2instanton}) is a ${\mathrm{Spin} }(7)$-instanton on ${\mathbb R}^8.$ We calculate
\begin{equation*}
\begin{split}
d \left( x^{-1} dx \right)
& = - x^{-1} dx  \wedge  x^{-1} dx
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
d \left( \frac{ |y|^2}{|x|^2 + |y|^2} \right)
& = \frac{ |x|^2|y|^2}{\left( |x|^2 + |y|^2 \right)^2} 2 \Re \left[ y^{-1} dy - x^{-1} dx \right] \\
& = - \, d \left( \frac{|x|^2}{|x|^2 + |y|^2} \right).
\end{split}
\end{equation*}
This gives
\begin{equation}\label{dacalc}
\begin{split}
dA_0 & = \frac{1}{\left( |x|^2 + |y|^2 \right)^2} \Im \left[ \begin{split}
&  x^{-1} dx \wedge \left(-2 |x|^2|y|^2 \Re \left[ y^{-1} dy - x^{-1} dx \right] - \left( |y|^4 + |x|^2 |y|^2 \right) x^{-1} dx  \right)   \\
& + y^{-1} dy \wedge \left(-2 |x|^2 |y|^2 \Re \left[ x^{-1} dx  - y^{-1} dy \right] - \left( |x|^4 + |x|^2 |y|^2 \right) y^{-1} dy \right) \end{split} \right] \\
& = \frac{1}{\left( |x|^2 + |y|^2 \right)^2} \Im \left[ \begin{split}
& x^{-1} dx \wedge \left(-2 |x|^2|y|^2 \Re y^{-1} dy  + |y|^2 d\bar{x} x  -  |y|^4 x^{-1} dx \right)  \\
& + y^{-1} dy \wedge \left( - 2 |x|^2 |y|^2 \Re  x^{-1} dx + |x|^2 d\bar{y} y  -  |x|^4 y^{-1} dy \right) \end{split} \right] \\
& = \frac{1}{\left( |x|^2 + |y|^2 \right)^2} \Im \left[ \begin{split}
&  |y|^2 x^{-1} dx  \wedge d\bar{x} x +  |x|^2 y^{-1} dy \wedge d\bar{y} y - \bar{x} dx \wedge d \bar{y}  y  - \bar{y} dy  \wedge d\bar{x} x \\
& - 2 |x|^2 |y|^2  y^{-1} dy \wedge x^{-1} dx -  |x|^4 y^{-1} dy \wedge y^{-1} dy - |y|^4 x^{-1}  dx \wedge x^{-1} dx \end{split} \right].
\end{split}
\end{equation}
On the other hand, we have
\begin{equation}\label{asquaredcalc}
A_0 \wedge A_0 = \frac{1}{\left( |x|^2 + |y|^2 \right)^2} \Im \left[ |y|^4 x^{-1} dx \wedge x^{-1} dx + |x|^4 y^{-1} dy \wedge y^{-1} dy + 2 |x|^2 |y|^2 x^{-1} dx \wedge y^{-1} dy  \right].
\end{equation}
Adding (\ref{dacalc}) and (\ref{asquaredcalc}), we obtain the curvature form
\begin{equation}\label{stdg2instcurv}
\begin{split}
F_{A_0} (x,y) & = dA_0 + A_0 \wedge A_0 \\
& = \frac{1}{\left( |x|^2 + |y|^2 \right)^2} \Im \left[
 |y|^2 x^{-1} dx \wedge d\bar{x} x +  |x|^2 y^{-1} dy \wedge d\bar{y} y - 2 \bar{x} dx \wedge d \bar{y} y  \right] \\
& = \frac{1}{ \left(|x|^2 + |y|^2 \right)^2 |x|^2 |y|^2 } \Im \left[ \overline{\left( d\bar{x} x |y|^2 - d \bar{y} y |x|^2 \right)} \wedge \left(d\bar{x} x |y|^2 - d \bar{y} y |x|^2 \right) \right].
\end{split}
\end{equation}
The 2-forms $dx \wedge d\bar{x},$ $dy \wedge d\bar{y},$ and $dx \wedge d \bar{y}$ are each invariant under ${\mathrm{Sp} }(1)_r.$ Therefore, the 2-form part of $F_{A_0}$ lies in ${\mathrm{Lie}}({\mathrm{Sp} }(2)) \subset {\mathrm{Lie}}({\mathrm{Spin} }(7)),$ as claimed.

\begin{comment}
\begin{rmk}
If we apply the gauge transformation $g(x,y) = \bar{y}/|y|,$ as before, we get
\begin{equation}\label{stdg2curvatureingauge}
\begin{split}
 F_{g(A_0)} = g F_{A_0} \bar{g} (x,y) & = \frac{1}{\left( |x|^2 + |y|^2 \right)^2} \Im \left[
\bar{y} x d\bar{x} \wedge dx x^{-1} y  +  |x|^2 d\bar{y} \wedge dy - 2 d \bar{y} \wedge dx \bar{x} y \right].
\end{split}
\end{equation}
Notice that in this gauge, the curvature along the $x$-axis is given by
\begin{equation}
\begin{split}
 F_{g(A_0)} (x,0) & = \frac{
d\bar{y} \wedge dy }{ |x|^2 }.
\end{split}
\end{equation}
Lemma \ref{lemma:eigenvalues} and the Weitzenbock formula (\ref{LAsquared}) can be used to show that all deformations of the standard instanton are of the form $ F_{A_0} \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} X$ for vector fields $X$ on $S^7.$
\end{rmk}
\end{comment}

\subsection{Linear deformations}\label{ss:lineardefos}

Let $A$ be a conical instanton on ${\mathbb R}^8 \setminus \{0\}$
whose curvature $F_A$ takes values in ${\mathrm{Lie}}({\mathrm{Sp} }(2)) \otimes {\mathfrak g}_E.$ Suppose that $F_A(x,y)$ spans ${\mathrm{Lie}}({\mathrm{Sp} }(2))$ as $(x,y)$ varies over ${\mathbb R}^8.$ From (\ref{stdg2instcurv}), this is seen to be the case for $A_0.$

Given any $8\times 8$ matrix $M,$ we associate the vector field
\begin{equation*}
X_M = M \cdot \vec{r} = M^i{}_{j} x^j \frac{\partial}{\partial x^i}
\end{equation*}
on ${\mathbb R}^8,$ as well as the ${\mathfrak g}_E$-valued 1-form
\begin{equation}\label{lineardeformation}
\alpha_M = X_M \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} F_{A} \in \Omega^1 \left( {\mathfrak g}_E \right).
\end{equation}
Notice that (\ref{lineardeformation}) corresponds to pushforward by the diffeomorphism generated by $X_M,$ with its horizontal lift to the bundle. For, working in a local gauge where $A_{X_M} = 0,$ we have
$$\frac{d}{dt} \exp(-tX_M)^* A = X_M \left( A \right) = F(X_M, -) = \alpha_M.$$ The action on the curvature is given by
\begin{equation}\label{deformationexpthing}
\frac{d}{dt} \exp(-tX_M)^* F_A = \frac{d}{dt} F_{\exp^*(-t X_M) A}  = D_{A} \alpha_M.
\end{equation}

\begin{lemma}\label{lemma:coulomb} We have
\begin{equation}
\left( D_{A}^{S^7} \right)^* \alpha_M = 0
\end{equation}
if and only if $M \in {\mathrm{Lie}}({\mathrm{Sp} }(2))^\perp \subset{\mathbb R}^{8\times 8}.$
\end{lemma}
\begin{proof} Let $\alpha = \alpha_M$ and $F = F_A.$ In coordinates on ${\mathbb R}^8,$ we have
\begin{equation}\label{R8coulomb}
\left( D_A^{{\mathbb R}^8} \right)^*\alpha = \nabla^i X^j F_{ij} + X^j \nabla^i F_{ij} = M_{ij} F_{ij}.
\end{equation}
Since the curvature spans ${\mathrm{Lie}}({\mathrm{Sp} }(2)),$ the expression (\ref{R8coulomb}) vanishes identically if and only if $M$ lies in the orthogonal complement of ${\mathrm{Lie}}({\mathrm{Sp} }(2)).$
The result then follows from the formula
$$\left( D_A^{{\mathbb R}^8} \right)^*\alpha = \left( D_A^{S^7} \right)^*\alpha - \langle \hat{r}, \nabla_{\hat{r}} \left( \alpha \left( \hat{r} \right) \right) \rangle$$
and the fact that $\alpha(\hat{r}) \equiv 0$ for a conical instanton.
\end{proof}

\begin{prop}\label{prop:15deformations} Let $W$ be the 5-dimensional space (\ref{wdef}).
Then $\ker {\mathscr L}_A$ contains the 15-dimensional space
\begin{equation}\label{15deformations}
\{ \alpha_M \mid M \in W \oplus W I_1 \oplus W I_2 \}
\end{equation}
where $\alpha_M$ is defined by (\ref{lineardeformation}).
\end{prop}
\begin{proof}
According to (\ref{deformationexpthing}), the space (\ref{15deformations}) must be contained in $\ker {\mathscr L}_A.$ For, the subspace $\{\alpha_M \mid M \in W\}$ corresponds to pushforward by elements of ${\mathrm{GL} }(2, \H),$ which preserve the algebra ${\mathrm{Lie}}({\mathrm{Sp} }(2)).$
Meanwhile, per the decomposition (\ref{spin7splitting}), we have
$$W I_1 \oplus W I_2 \subset \Lambda^2_d \oplus \mathfrak{F}_{x,y} \subset {\mathrm{Lie}} \left( {\mathrm{Spin} }(7) \right).$$

Since the curvature $F_A$ spans ${\mathrm{Lie}}({\mathrm{Sp} }(2))$ pointwise, these elements are all nonzero. By Lemma \ref{lemma:coulomb}, the space (\ref{15deformations}) is in Coulomb gauge.
\end{proof}

\begin{comment}
\begin{proof}[Proof of Proposition \ref{prop:15deformations}] By the construction of \S \ref{ss:stdg2inst}, we know that $A_0$ is invariant modulo gauge under ${\mathrm{Sp} }(2).$ Hence, the action of ${\mathrm{Sp} }(2) \subset {\mathrm{Spin} }(7)$ on vector fields gives an action on the space $\{ X \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} F_{A_0} \} \subset \Omega^1\left( {\mathfrak g}_E \right),$ which preserves the kernel of the deformation operator ${\mathscr L}_{A_0}.$

Let $R = {\mathrm{Lie}}({\mathrm{Sp} }(2)) \subset \Lambda^2 {\mathbb R}^8$ be the adjoint representation of ${\mathrm{Sp} }(2),$ and $W$ as above. We may decompose ${\mathbb R}^8 \otimes {\mathbb R}^8 = \Lambda^2 {\mathbb R}^8 \oplus {\mathrm{Sym} }^2{\mathbb R}^8$ into irreducible ${\mathrm{Sp} }(2)$-representations as
\begin{equation}\label{15deformations:so8}
\begin{split}
\Lambda^2 {\mathbb R}^8 = R \oplus W I_1 \oplus W I_2 \oplus W I_3\oplus \langle I_1 \rangle \oplus \langle I_2 \rangle \oplus \langle I_3 \rangle
\end{split}
\end{equation}
and
\begin{equation}\label{15deformations:sym8}
{\mathrm{Sym} }^2{\mathbb R}^8 = W \oplus R I_1 \oplus R I_2 \oplus R I_3 \oplus \langle \mathrm{Id} \rangle.
\end{equation}

We first check the factors of (\ref{15deformations:so8}). According to (\ref{spin7splitting}), $W I_1 \oplus W I_2$ belongs to ${\mathrm{Lie}}({\mathrm{Spin} }(7)),$ and by Lemma \ref{lemma:coulomb}, the subspace $\{ \alpha_M \mid M \in W I_1 \oplus W I_2\}$ is in Coulomb gauge. Hence this subspace is contained in $\ker {\mathscr L}_{A_0}.$ Also by Lemma \ref{lemma:coulomb}, the subspace $\{\alpha_M \mid M \in R \}$ is not in Coulomb gauge (and indeed is trivial modulo gauge).
The three trivial representations in (\ref{15deformations:so8}) yield zero when substituted into $F = F_{A_0}.$

It remains to rule out the factor $W I_3,$ for which it suffices to check that the vector field
\begin{equation}
X = \left( \begin{array}{cc}
I_3 &  0 \\
0 & -I_3 \end{array} \right) \vec{r} = \left( \begin{array}{c}
-x \cdot {\mathbf k} \\
y \cdot {\mathbf k} \end{array} \right)
\end{equation}
does not yield an element of $\ker {\mathscr L}_{A_0},$ as follows.

As remarked above, these correspond to known deformations of $A_0,$ which are in Coulomb gauge by Lemma \ref{lemma:coulomb}. The given space (\ref{15deformations}) therefore lies within $\ker {\mathscr L}_A.$
\end{proof}

\end{comment}

\begin{comment}

\begin{thm}\label{thm:stddefs} The space of infinitesimal deformations of $A_0$ consists of the 15-dimensional space of linear deformations (\ref{15deformations}).
\end{thm}
\begin{proof} This is a special case of Theorem \ref{thm:generaldefos} below, for which an independent proof is given in the Appendix.
\end{proof}

\end{comment}

\section{Infinitesimal deformations}
\label{sec:generaldefos}

In this section, we calculate the infinitesimal deformations, as a ${\mathbb G}_2$-instanton, of the pullback to $S^7$ of a general irreducible ASD instanton on $S^4.$ We write
$$\nabla = \nabla^{S^7} = \pi_{S^7} \circ \nabla^{{\mathbb R}^8}$$
for the Levi-Civita connection on $S^7_{std},$ which we shall couple to the connection on any auxiliary bundle. We shall also write
\begin{equation}\label{nablavh}
\nabla^v = \nabla_{\pi_v}, \qquad \nabla^h = \nabla_{\pi_h}.
\end{equation}
Here $\pi_v$ is the orthogonal projection to the the vertical tangent space of the Hopf fibration, and $\pi_h$ is the complementary projection.

\subsection{Vertical and horizontal components}
Let $\Omega^1_h$ be the annihilator of vertical vector fields along the Hopf fibration, and $\Omega^1_v$ its orthogonal complement. We have
\begin{equation*}
\Omega^1_{S^7} = \Omega^1_v \oplus \Omega^1_h.
\end{equation*}
Letting
$$\Omega^{(p,q)} = \Lambda^p \Omega^1_v \otimes \Lambda^q \Omega^1_h \subset \Omega^{p + q}_{S^7}$$
we have a decomposition
$$\Omega^{k}_{S^7} = \bigoplus_{p + q = k} \Omega^{(p,q)}.$$
An element of $\Omega^{(p,q)}$ will be referred to as a $(p,q)$-form.

Let $\nu$ denote the $(3,0)$ volume form of the Hopf fibration, and let
$$\bar{\nu} = * \nu.$$
The $(0,2)$-forms split as
$$\Omega^{(0,2)} = \Omega^{2+}_h \oplus \Omega^{2-}_h$$
where $\Omega^{2\pm}_h$ are the (anti-)self-dual components
with respect to the $(0,4)$ volume form $\bar{\nu}.$

For a $(0,1)$-form $b,$ we shall write $d^v b$ for the $(1,1)$ part of $db$ and $d^h b$ for the $(0,2)$ part. A similar notation will be used for $(1,0)$-forms (see Lemma \ref{lemma:dalpha} below).

\begin{defn}\label{defn:coframes} Let $\omega_i^x, \omega_i^y, \zeta_i^x,$ and $\zeta_i^y$ be as in (\ref{2formbases}) and (\ref{zetaidef}). For $i = 1,2,3,$ define the global ${\mathrm{Sp} }(2)$-invariant forms on $S^7$:
\begin{equation*}
\begin{split}
\zeta_i & = \vec{r} \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \left( \omega^x_i + \omega^y_i \right) \\
\omega_i^\circ & = \left. \omega^x_i + \omega^y_i \right|_{S^7} \\
\bar{\omega}_i  & = \omega_i^\circ - \frac{1}{2}\epsilon_{ijk} \zeta_j \wedge \zeta_k \\
\end{split}
\end{equation*}
Notice that $\{\zeta_i\}$ and $\{ \bar{\omega}_i \}$ are global frames for $\Omega^1_v$ and $\Omega^{2+}_h,$ respectively. The vertical volume form is given by
$$\nu = \zeta_1 \wedge \zeta_2 \wedge \zeta_3.$$
From (\ref{phistdfirstdef}) and the ${\mathrm{Sp} }(2)$-invariance, we may re-express $\phi_{std}$ as follows:
\begin{equation}\label{phistdsp2def}
\begin{split}
\phi_{std}
& = \stackrel{ (3,0) }{\overbrace{ \nu} } + \stackrel{ (1,2) }{\overbrace{ \zeta_1 \wedge \bar{\omega}_1 + \zeta_2 \wedge \bar{\omega}_2 - \zeta_3 \wedge \bar{\omega}_3 }}.
\end{split}
\end{equation}
\end{defn}
\noindent We now derive the basic properties of these frames, and use them to decompose the Laplace operator into horizontal and vertical parts.

\begin{lemma}\label{lemma:gradgammai} The frame $\{\zeta_i\}$ is coclosed, and satisfies
\begin{equation}\label{gradgammai:gradvh}
\nabla^v \zeta_i = \frac{1}{2} \epsilon_{ijk} \zeta_j \wedge \zeta_k, \qquad \nabla^h \zeta_i = \bar{\omega}_i
\end{equation}
\begin{equation}\label{gradgammai:laplace}
\nabla^* \nabla^v \zeta_i = 2 \zeta_i, \qquad \nabla^* \nabla^h \zeta_i = 4 \zeta_i.
\end{equation}
Here $\nabla^v, \nabla^h$ are defined by (\ref{nablavh}) above.
\end{lemma}
\begin{proof} From the Definition \ref{defn:coframes} and (\ref{zetaidef}), for $X, Y \in T S^7,$ we have
\begin{equation}\label{gradgammai:grad}
\left( \nabla_X \zeta_i \right)  (Y) = \left( \nabla^{{\mathbb R}^8}_X \zeta_i \right)(Y) = \left( \omega^x_i + \omega^y_i \right)(X,Y) = \omega^\circ_i (X, Y).
\end{equation}
Coclosedness of $\zeta_i$ and (\ref{gradgammai:gradvh}) follow directly from (\ref{gradgammai:grad}).

Since $\zeta_i$ is coclosed and equal to the restriction of a linear form, we have
\begin{equation}\label{gradgammai:spherelaplace}
\nabla^*\nabla_{S^3} \zeta_i = 2 \zeta_i, \qquad \nabla^*\nabla_{S^7} \zeta_i = 6 \zeta_i
\end{equation}
as can be verified directly from (\ref{gradgammai:grad}). Then (\ref{gradgammai:laplace}) follows from (\ref{gradgammai:spherelaplace}) and the fact that $\nabla = \nabla^v + \nabla^h.$
\end{proof}

\begin{lemma}\label{lemma:dalpha}
Let $\alpha = a + b$ be a 1-form on $S^7,$ with $a = f_i \zeta_i \in \Omega^1_v $ and $b \in  \Omega^1_h .$ Then
\begin{equation*}
\begin{split}
d \alpha = \,\, \stackrel{ (2,0) }{\overbrace{ d^v a }} & + \stackrel{ (1,1)}{ \overbrace{d^h \! f_i \wedge \zeta_i } } + \stackrel{ (0,2) }{\overbrace{2 f_i \bar{\omega}_i }} \\
& + d^v b \qquad + \, d^h b. \end{split}
\end{equation*}
\end{lemma}
\begin{proof} This is a standard decomposition result, which can be seen directly as follows. By Lemma \ref{lemma:gradgammai}, we have
\begin{equation*}
d \zeta_i = 2 \omega_i^\circ, \quad d^v \zeta_i = \epsilon_{ijk} \zeta_j \wedge \zeta_k, \quad i = 1,2,3.
\end{equation*}
Therefore
\begin{equation}\label{dgammai}
d \zeta_i = \stackrel{ (2,0) }{\overbrace{ \epsilon_{ijk} \zeta_j \wedge \zeta_k }} + \stackrel{ (0,2) }{\overbrace{ 2 \bar{\omega}_i. }}
\end{equation}
The result follows directly from (\ref{dgammai}).
\end{proof}

\begin{lemma}\label{lemma:coclosed} If $a = f_i \zeta_i$ is (co)closed on each $S^3$ fiber, then $\left( \nabla^*\nabla^h  f_i \right) \zeta_i$ is again fiberwise (co)closed.
\end{lemma}
\begin{proof} Let $U_j$ be the dual vector field of $\zeta_j,$ for $j = 1,2,3.$ The Killing vector fields $U_j$ commute with the operators $\nabla^*\nabla$ and $\nabla^*\nabla^v,$ hence also with $\nabla^* \nabla^h = \nabla^*\nabla - \nabla^*\nabla^v.$ Since $d^v a$ and $(d^v)^* a = d^*a$ are determined by $U_j (f_i),$ we conclude that $\nabla^* \nabla^h$ preserves (co)closedness on the fibers.
\end{proof}

\begin{comment}
Let $\{e_j\}_{j = 1}^7$ be a basis of vector fields which is orthonormal and satisfies $\nabla_{e_j} e_k (p)= 0$ at a given point $p,$ with
\begin{equation}
e_j(p) \in T_v S^7, \quad j = 1,2,3.
\end{equation}
\end{comment}

\begin{prop}\label{prop:laplacev} Let $\alpha = a + b$ be a 1-form as above, where $a = f_i \zeta_i$ and $b \in \Omega^1_h.$ The vertical component of the Laplacian on $S^7$ is given by
\begin{equation}\label{laplacev:maineq}
\left( \nabla^* \nabla \alpha \right)^v =  \nabla^* \nabla^v a + 4a + \left(\nabla^* \nabla^h \! f_i  + 2\langle d^h b, \bar{\omega}_i  \rangle \right)\zeta_i.
\end{equation}
Here $\nabla^* \nabla^v$ denotes the Laplacian on the $S^3$ fiber.
\end{prop}
\begin{comment}
\begin{rmk}
Here the inner product as 2-tensors is intended:
$$\langle \omega_i , \nabla b \rangle = \langle \omega_i(e_j, e_k), \nabla_{e_j} b (e_k ) \rangle.$$
\end{rmk}
\end{comment}
\begin{proof}
We have
\begin{equation*}
\left( \nabla^*\nabla \alpha \right)^v = \left( \nabla^*\nabla a \right)^v + \left( \nabla^*\nabla b \right)^v.
\end{equation*}
For the first term, we write
\begin{equation}\label{laplacev:avfirst}
\left( \nabla^*\nabla a \right)^v = \nabla^*\nabla^v a + \left( \nabla^* \nabla^h a \right)^v
\end{equation}
and
\begin{equation*}
\begin{split}
\left( \nabla^* \nabla^h a \right)^v & = \left( \nabla^* \left( \nabla^h \! f_i \zeta_i + f_i \nabla^h \zeta_i \right) \right)^v \\
& = \left( \nabla^* \nabla^h f_i \right) \zeta_i + \left( \nabla^h f_i \nabla^*\zeta_i - \nabla f_i \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \bar{\omega}_i \right)^v + f_i \nabla^* \nabla^h \zeta_i \\
& = \left( \nabla^* \nabla^h f_i \right) \zeta_i + 4 f_i \zeta_i
\end{split}
\end{equation*}
where we have used Lemma \ref{lemma:gradgammai}. Then (\ref{laplacev:avfirst}) yields
\begin{equation}\label{laplacev:av}
\left( \nabla^*\nabla a \right)^v = \nabla^*\nabla^v a + 4 a + \nabla^* \nabla^h f_i \zeta_i
\end{equation}
which gives the case $b = 0$ of the formula (\ref{laplacev:maineq}).

Assuming now that $Y$ is horizontal and $b$ is of type $(0,1),$ we compute
\begin{equation*}
\begin{split}
0 \equiv \nabla_Y \langle \zeta_i , b \rangle & = \langle \nabla_Y \zeta_i , b \rangle + \langle \zeta_i, \nabla_X b \rangle \\
\langle \zeta_i, \nabla_Y b \rangle & = - \langle \bar{\omega}_i(Y, -) , b \rangle.
\end{split}
\end{equation*}
For $U$ vertical, we have
\begin{equation*}
\begin{split}
0 \equiv \nabla_U \langle \zeta_i , b \rangle & = \langle \nabla_U \zeta_i , b \rangle + \langle \zeta_i, \nabla_U b \rangle
\end{split}
\end{equation*}
and
\begin{equation*}
\langle \zeta_i, \nabla_U b \rangle = 0.
\end{equation*}
Hence, for $X \in TS^7,$ we have
\begin{equation*}
\left( \nabla_X b \right)^v = - \zeta_i \langle \bar{\omega}_i(X, -) , b \rangle.
\end{equation*}
Next, let $\{e_j\}_{j = 1}^7$ be an orthonormal basis of vector fields that satisfies $\nabla_{e_j} e_k = 0$ at a given point. We compute
\begin{equation*}
\begin{split}
0 & \equiv \nabla_{e_j} \nabla_{e_j} \langle \zeta_i , b \rangle \\
& = \langle \nabla_{e_j} \nabla_{e_j} \zeta_i , b \rangle + 2 \langle \omega_i^\circ \left(e_j, - \right), \nabla_{e_j} b \rangle + \langle \zeta_i, \nabla_{e_j} \nabla_{e_j} b \rangle \\
& = - 6 \langle \zeta_i , b \rangle + 2 \langle \omega_i^\circ \left(e_j, - \right), \nabla_{e_j} b \rangle + \langle \zeta_i, \nabla_{e_j} \nabla_{e_j} b \rangle.
\end{split}
\end{equation*}
Since $\langle \zeta_i, b \rangle \equiv 0,$ we are left with
\begin{equation*}
\langle \zeta_i, \nabla_{e_j} \nabla_{e_j} b \rangle = -2 \langle \omega_i^\circ \left(e_j, - \right), \nabla_{e_j} b \rangle
\end{equation*}
which may be rewritten as
\begin{equation}\label{laplacev:bv}
\left( \nabla^* \nabla b \right)^v = 2 \zeta_i \langle \bar{\omega}_i , d b \rangle.
\end{equation}
Combining (\ref{laplacev:av}) and (\ref{laplacev:bv}) yields (\ref{laplacev:maineq}).
\end{proof}

\begin{rmk} The previous results carry over when $\nabla$ is coupled to a connection that is trivial along the fibers of the Hopf fibration.
\end{rmk}

\subsection{Vanishing of the vertical component} This subsection proves our vanishing theorem for the vertical component of an infinitesimal deformation in Coulomb gauge.

\begin{lemma}\label{lemma:horizproperties} Let $B$ be an ASD instanton on $S^4,$ and put $A = \pi^*B.$ Assume that $f$ is a section of $\pi^* {\mathfrak g}_E \to S^7$ satisfying
\begin{equation}\label{horizproperties:assumption}
\nabla_A^h f = 0.
\end{equation}
Then $f = \pi^*h$ is the pullback of a section of ${\mathfrak g}_E \to S^4,$ with
\begin{equation}\label{horizproperties:conclusion}
\nabla_B h = 0.
\end{equation}
\end{lemma}
\begin{proof} We have
\begin{equation}\label{horizproperties:mixedpartials}
0 = \nabla^v \nabla^h f = \nabla^h \nabla^v f
\end{equation}
since $f$ is a $0$-form and the curvature $F_A$ is purely horizontal. By (\ref{horizproperties:assumption}), we have
$$D_A f = D_A^v f.$$
Applying $D_A$ to both sides, by Proposition \ref{lemma:dalpha}, we obtain
\begin{equation*
\begin{split}
D_A^2 f = \left[ F_A , f \right] & = D_A D_A^v f \\
& = (D_A^v)^2 f + D_A^h D_A^v f + 2 D^v f (U_i) \bar{\omega}_i.
\end{split}
\end{equation*}
By (\ref{horizproperties:mixedpartials}) and the fact that $\left( D_A^v \right)^2 = 0$ for a pullback connection, the first two terms vanish, yielding
\begin{equation}\label{horizproperties:curvaturecomparison}
\left[ F_A , f \right] = 2 \left( \nabla_{U_i} f \right) \bar{\omega}_i.
\end{equation}
Each side of (\ref{horizproperties:curvaturecomparison}) is of type $(0,2);$ however, the LHS is anti-self-dual and the RHS is self-dual. Therefore both sides of (\ref{horizproperties:curvaturecomparison}) must vanish, giving
$$\nabla^v f = 0.$$
Hence, $f$ is constant on the fibers, so $f = \pi^*h$ for a section $h$ of ${\mathfrak g}_E.$ Then (\ref{horizproperties:assumption}) implies (\ref{horizproperties:conclusion}), as desired.
\end{proof}

\begin{thm}\label{thm:vertvanishing}\label{thm:vanishing} Let $B$ be an irreducible ASD instanton on a bundle $E \to S^4,$ and $A = \pi^* B$
its pullback under the Hopf fibration.
If $\alpha \in \Omega^1 \left( \pi^*{\mathfrak g}_E \right)$ satisfies ${\mathscr L}_A \alpha = 0,$ then the vertical part of $\alpha$ vanishes.
\end{thm}
\begin{proof} We write $\nabla = \nabla_A$ and $d = D_A$ throughout the proof.

Decompose $\alpha = a + b$ into vertical and horizontal parts as above. Let $a = f_i \zeta_i,$ and write the self-dual part of $d^h b$ as
\begin{equation*}
\left( d^h b \right)^+ = c_i \bar{\omega}_i .
\end{equation*}
From Proposition \ref{lemma:dalpha} and (\ref{phistdsp2def}), our assumption ${\mathscr L}_A \alpha = 0$ implies
\begin{equation}\label{phidalpha}
\begin{split}
0 = \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d \alpha
& = \stackrel{(1,0)}{ \overbrace{ \quad \qquad \nu \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d^v a \qquad \quad } } + \stackrel{(0,1)}{ \overbrace{ \phi_{1,2} \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \left( d^h a + d^v b \right) }} \\
& \,\, + \phi_{1,2} \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \left( 2 f_i \bar{\omega}_i + d^h b \right).
\end{split}
\end{equation}
Noting that $\bar{\omega}_i \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \bar{\omega}_i = 2,$ per (\ref{phistdsp2def}), the $(1,0)$-part of (\ref{phidalpha}) comes out to
\begin{equation*}
\begin{split}
0 & = \zeta_1 \left( *d^v a(U_1) + 4 f_1 + 2 c_1 \right)  \\
& + \zeta_2  \left( *d^v a(U_2) + 4 f_2 + 2 c_2 \right) \\
& + \zeta_3 \left( *d^v a(U_3) - 4 f_3 - 2 c_3 \right)
\end{split}
\end{equation*}
where $U_j$ is the dual vector field to $\zeta_j,$ for $j = 1,2,3,$ as above. We conclude that
\begin{equation}\label{ciformulafirst}
\begin{split}
c_1 & = -2 f_1 - \frac12 *d^v a (U_1) \\
c_2 & = -2 f_2 - \frac12 *d^v a (U_2) \\
c_3 & = -2 f_3 + \frac12 *d^v a (U_3).
\end{split}
\end{equation}
We rewrite this as
\begin{equation}\label{ciformula}
c_i \zeta_i = -2a + \frac{1}{2} \mu(a)
\end{equation}
where $\mu: \Omega^1_v \to \Omega^1_v,$ defined by (\ref{ciformulafirst}), obeys
\begin{equation}\label{munorm}
|\mu (a)| = |*d^v a | = |d^v a |.
\end{equation}
Now, the decomposition of the Laplacian (\ref{laplacev:maineq}) reads
\begin{equation}\label{laplacedecompfordef}
\begin{split}
\left( \nabla^* \nabla \alpha \right)^v & = \nabla^* \nabla^v a + 4 a + 4 c_i\zeta_i  + \left( \nabla^* \nabla^h \! f_i \right) \zeta_i \\
& = \nabla^* \nabla^v a - 4 a + 2 \mu(a)  + \left( \nabla^* \nabla^h \! f_i \right) \zeta_i
\end{split}
\end{equation}
where we have used (\ref{ciformula}). Returning to (\ref{LAsquared}), we have
\begin{equation*}
\begin{split}
0 = \mathscr{L}_A^2 \alpha & = 2 \phi_{std} \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \alpha + \nabla^*\nabla \alpha - 2 \left[ F_A \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \alpha \right] + 6 \alpha \\
& = \nabla^*\nabla \alpha - 2 \left[ F_A \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} b \right] + 6 \alpha.
\end{split}
\end{equation*}
Taking the vertical part and inserting (\ref{laplacedecompfordef}), since $F_A$ is purely horizontal, we obtain
\begin{equation}\label{decomp0}
\begin{split}
0 = \left( \mathscr{L}_A^2 \alpha \right)^v = \nabla^* \nabla^v a + 2a + 2 \mu(a)  + \left( \nabla^* \nabla^h \! f_i \right) \zeta_i.
\end{split}
\end{equation}
Applying the Bochner formula on $S^3$ yields
\begin{equation}\label{decomp1}
0 = \left( d^* d^v + d^v d^* \right) a + 2 \mu(a)  + \left( \nabla^* \nabla^h \! f_i \right) \zeta_i.
\end{equation}
We now split $a$ into fiberwise closed and coclosed parts
$$a = a_{cl} + a_{cocl}$$
and write
$$a_{cocl} = g_i \zeta_i.$$
Taking an inner product with $a_{cocl}$ in (\ref{decomp1}), and integrating over $S^7,$ yields
\begin{equation}\label{decomp2}
0 = \| d^v a \|^2 + 2 (a_{cocl}, \mu(a))  + \left( g_i, \nabla^* \nabla^h \! g_i  \right).
\end{equation}
Here we have used the orthogonality between fiberwise closed and coclosed vertical 1-forms, the fact that $d^*a_{cocl} = (d^v)^*a_{cocl} = 0,$ and Lemma \ref{lemma:coclosed}.

Note that the first eigenvalue of the Hodge Laplacian on coclosed 1-forms on $S^3$ is 4, so
$$\| d^v a \| \geq 2 \| a_{cocl} \|.$$
Inserting this into (\ref{decomp2}), using Cauchy-Schwartz and (\ref{munorm}), we obtain
\begin{equation*}
\begin{split}
0 &\geq 2 \| a_{cocl} \| \| d^v a \| - 2 \| a_{cocl} \| \| \mu(a) \|  + \sum_{i = 1}^3 \| \nabla^h \! g_i \|^2 \\
& \geq 2 \| a_{cocl} \| \| d^v a \| - 2 \| a_{cocl} \| \| d^v a \| +\sum_{i = 1}^3 \| \nabla^h \! g_i \|^2 \\
& \geq \sum_{i = 1}^3 \| \nabla^h \! g_i \|^2.
\end{split}
\end{equation*}
This yields
$$\nabla^h g_i \equiv 0$$
for $i = 1,2,3.$
Since $B$ is assumed irreducible, Lemma \ref{lemma:horizproperties} implies that $g_i \equiv 0,$ and hence $a_{cocl} \equiv 0.$

But then $a = a_{cl}$ is fiberwise closed, so $\mu(a) = 0$ by (\ref{munorm}). Since the RHS in (\ref{decomp0}) is positive,
we conclude that $a \equiv 0,$ as desired.
\end{proof}

\subsection{Space of infinitesimal deformations} This subsection proves our main theorem, which calculates the deformations of a pulled-back instanton on $S^7$ in terms of the ASD deformations on $S^4.$

Consider the operator
$$\varphi = \left( \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d(-) \right)^h : \Omega^1_h \to \Omega^1_h$$
given by the restriction of $\phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d(-)$ to the space of horizontal 1-forms. For $b \in \Omega^1_h,$ we have
\begin{equation*
\varphi(b) = \left( \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d b \right)^h = \left( \phi^{(1,2)} \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d^v b \right)^h.
\end{equation*}
Let $S^3_0 \subset \{ (x, 0) \} \subset \H^2$ be the fiber of the Hopf fibration contained in the $x$-axis, and let
$$\varphi_0= \left. \varphi \right|_{S^3_0}.$$
Notice that
\begin{equation*}
\begin{split}
dy^j \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \zeta_i & = \left( \zeta_i^y \right)^j = \omega_{ijk} y^k.
\end{split}
\end{equation*}
We may therefore define a local frame for $\Omega^1_h$ near $S^3_0$ by
\begin{equation*}
\bar{e}^j = dy^j - \zeta_i \omega_{ijk} y^k, \qquad j = 0, \ldots, 3.
\end{equation*}
We calculate
\begin{equation*}
\begin{split}
\left. d \bar{e}^j \right|_{S_0^3} & = \omega_{ijk} \zeta_i \wedge \bar{e}^k \\
& = \left. d^v \bar{e}^j \right|_{S_0^3}.
\end{split}
\end{equation*}
Then
\begin{equation*}
\begin{split}
\varphi_0 \left( \bar{e}^j \right) = \left( \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d^v \bar{e}^j \right)^h & = - \left( \omega_{1jk} \omega_{1 k \ell} + \omega_{2jk} \omega_{2 k \ell} - \omega_{3jk} \omega_{3 k \ell}  \right) \bar{e}^\ell \\
\end{split}
\end{equation*}
and
\begin{equation}\label{phi0ebar}
\varphi_0 \left( \bar{e}^j \right) = \bar{e}^j.
\end{equation}

\begin{lemma}\label{lemma:horiz} The kernel of $\varphi_0$ consists of sections of the form
\begin{equation*}
L_{ij} x^i \bar{e}^j
\end{equation*}
where $L \in \mathfrak{F}$ is a Fueter map of ${\mathbb R}^4,$ per (\ref{fueter}) above.
\end{lemma}
\begin{proof}
Let $\alpha = \alpha_j \bar{e}^j$ be an arbitrary section of $\Omega^1_h$ near $S^3_0.$ Per (\ref{phi0ebar}) and (\ref{phis30}), we have
\begin{equation}\label{phi0fueter}
\begin{split}
\varphi_0 \left( \alpha \right) & = \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \left( U_i \left( \alpha_j \right) \zeta^i \wedge \bar{e}^j  \right) + \alpha_j \bar{e}^j \\
& = \left( U_1 \left( \alpha_j \right) \omega_{1 j \ell} + U_2 \left( \alpha_j \right) \omega_{2 j \ell} - U_3 \left( \alpha_j \right) \omega_{3 j \ell} + \alpha_\ell \right) \bar{e}^\ell \\
& =: \beta_\ell \bar{e}^\ell.
\end{split}
\end{equation}
Further, we calculate
\begin{equation*}
\begin{split}
\varphi_0^2 \left( \alpha \right) \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \bar{e}^i & = U_1 \left( \beta_\ell \right) \omega_{1 \ell i} + U_2 \left( \beta_\ell \right) \omega_{2 \ell i} - U_3 \left( \beta_\ell \right) \omega_{3 \ell i} + \beta_i \\
& = \omega_{1 \ell i} U_1 \left(U_1 \alpha_j \omega_{1j\ell} + U_2 \alpha_j \omega_{2j\ell} - U_3 \alpha_j \omega_{3j\ell} + \alpha_\ell \right) \\
& \quad + \omega_{2 \ell i}  U_2 \left( U_1 \alpha_j \omega_{1j\ell} + U_2 \alpha_j \omega_{2j\ell} - U_3 \alpha_j \omega_{3j\ell} + \alpha_\ell \right) \\
& \quad - \omega_{3 \ell i} U_3 \left( U_1 \alpha_j \omega_{1j\ell} + U_2 \alpha_j \omega_{2j\ell} - U_3 \alpha_j \omega_{3j\ell} + \alpha_\ell \right) \\
& \quad + \beta_i \\
& = - \left( U_1^2 + U_2^2 + U_3^2 \right) \alpha_i \\
& \quad + \omega_{2j\ell} \omega_{1 \ell i} U_1 U_2\left( \alpha_j \right) + \omega_{1 j \ell} \omega_{2 \ell i} U_2 U_1 \left( \alpha_j \right) \\
& \quad - \omega_{3j\ell} \omega_{1 \ell i} U_1 U_3 \left( \alpha_j \right) - \omega_{1 j \ell} \omega_{3 \ell i} U_3 U_1 \left( \alpha_j \right) \\
& \quad - \omega_{3j\ell} \omega_{2 \ell i} U_2 U_3 \left( \alpha_j \right) - \omega_{2 j \ell} \omega_{3 \ell i} U_3 U_2 \left( \alpha_j \right) \\
& \quad + \left( \beta_i - \alpha_i \right) + \beta_i \\
& = - \left( U_1^2 + U_2^2 + U_3^2 \right) \alpha_i  + \omega_{3 j i} \left[ U_1, U_2 \right] \alpha_j + \omega_{2 j i} \left[ U_1, U_3 \right] \alpha_j - \omega_{1 j i} \left[ U_2, U_3 \right] \alpha_j  \\
& \quad + 2\beta_i - \alpha_i  \\
& = - \left( U_1^2 + U_2^2 + U_3^2 \right) \alpha_i  + 2 \left( - \omega_{3 ji } U_3 + \omega_{2ji} U_2 + \omega_{1ji} U_1 \right) \alpha_j \\
& \quad + 2\beta_i - \alpha_i.
\end{split}
\end{equation*}
Again applying (\ref{phi0fueter}), we obtain
\begin{equation}\label{horiz:phi02}
\varphi_0^2 \left( \alpha \right) \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} \bar{e}^i = - \left( U_1^2 + U_2^2 + U_3^2 \right) \alpha_i  + 4\beta_i - 3 \alpha_i.
\end{equation}
Notice from (\ref{gradgammai:gradvh}) that $\nabla_{U_i} U_i = 0.$
\begin{comment} that
\begin{equation*}
\begin{split}
\nabla_{U_1} U_1 & = \pi_{S^3} \left( U_1 \left( x^j \right) \omega_{1\ell j} \frac{\partial }{\partial x^\ell} \right) \\
& = \pi_{S^3} \left( x^k \omega_{1kj} \omega_{1\ell j} \frac{\partial }{\partial x^\ell} \right) \\
& = \pi_{S^3} \left( x^\ell \frac{\partial }{\partial x^\ell} \right) = 0
\end{split}
\end{equation*}
and similarly for $U_2$ and $U_3.$
\end{comment}
Hence the first term on the RHS of (\ref{horiz:phi02}) is the Laplace-Beltrami operator on $S^3_0,$ applied component-wise to $\alpha.$
We conclude that
\begin{equation*}
\varphi_0^2 \left( \alpha \right) = \Delta_{S^3} \alpha + 4 \varphi_0 \left( \alpha \right) - 3 \alpha.
\end{equation*}
In particular, if $\varphi_0(\alpha) = 0,$ we have
\begin{equation*}
\Delta_{S^3} \alpha_j = 3 \alpha_j.
\end{equation*}
Hence, $\alpha_j$ lies in the first nonzero eigenspace, and is the restriction of a linear function on ${\mathbb R}^4:$
$$\alpha_j = x^i L_{ij}.$$
Substituting back into (\ref{phi0fueter}), we have
\begin{equation*}
\varphi_0(\alpha) = x^i \left( L_{ij} + \omega_{1ik} L_{k\ell}\omega_{1 \ell j} + \omega_{2 i k} L_{k\ell}\omega_{2 \ell j} - \omega_{3 i k} L_{k \ell}\omega_{3 \ell j} \right) \bar{e}^j = 0
\end{equation*}
which recovers (\ref{fueter}), as desired.
\end{proof}

\begin{prop}\label{prop:kerphi} The kernel of $\varphi$ is the
subspace of $\Omega^1_h$ given by
\begin{equation}\label{3subsheaves}
\pi^* \Omega^1_{S^4} \oplus  I_1 \left( \pi^* \Omega^1_{S^4} \right) \oplus I_2 \left( \pi^* \Omega^1_{S^4} \right).
\end{equation}
Here the complex structures $I_1$ and $I_2$ act pointwise on $\Omega^1_h \subset \Omega^1_{{\mathbb R}^8}.$
\end{prop}
\begin{proof} Over the fiber $S^3_0,$ this follows directly from Lemma \ref{lemma:horiz} and the description (\ref{fueterfactors}) of $\mathfrak{F}.$ Since ${\mathrm{Sp} }(2)$ preserves $\pi^* \Omega^1_{S^4}$ and commutes with $I_1$ and $I_2,$ the subspace (\ref{3subsheaves}) is also invariant under ${\mathrm{Sp} }(2).$ Hence, it agrees with $\ker \varphi$ globally.
\end{proof}

\begin{thm}\label{thm:generaldefos} Let $A = \pi^*B$ be the pullback of an irreducible ASD instanton on $S^4,$ and let
$$V = \ker \left( D_B^+ \oplus D_B^* \right) \subset \Omega^1_{S^4} \left( {\mathfrak g}_E \right)$$
denote the space of infinitesimal deformations of $B.$ The space of infinitesimal deformations of $A,$ as a ${\mathbb G}_2$-instanton, is given by
\begin{equation}\label{generaldefos:maineq}
\ker {\mathscr L}_A = \pi^* V \oplus  I_1 \left( \pi^* V \right) \oplus I_2 \left( \pi^* V \right) \subset \Omega^1_{S^7} \left( {\mathfrak g}_{\pi^*E} \right).
\end{equation}
For structure group ${\mathrm{SU} }(2),$ this has dimension $3 \left( 8\kappa - 3 \right).$
\end{thm}
\begin{proof} Let $\alpha \in \ker {\mathscr L}_A.$ According to Theorem \ref{thm:vertvanishing}, the vertical component of $\alpha$ vanishes, so it is a global section of $\Omega^1_h ({\mathfrak g}_E).$ The image $\varphi(\alpha) \in \Omega^1_h$ under the horizontal component of the deformation operator must therefore vanish; by Proposition \ref{prop:kerphi}, we have
$$\alpha = \alpha_{0} + \alpha_{1} + \alpha_{2}$$
according to (\ref{3subsheaves}).

It remains to show that ${\mathscr L}_A(\alpha) = 0$ if and only if $d^*\alpha_{i} = 0$ and $(d^h\alpha_{i})^+ = 0,$ for $i = 0,1,2.$ Clearly $d^*\alpha = 0$ if and only if $d^*\alpha_{i} = 0 \, \forall \, i.$
We have
\begin{equation}\label{dhalphaplus}
(d^h \alpha)^+ = \sum_{i = 1}^3 (d^h \alpha_{i} )^+
\end{equation}
where $(d^h \alpha_{i} )^+$
are linearly independent.
But the map $\phi \, \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} $ is a bundle isomorphism from $\Omega^{2+}_h$ to $\Omega^1_v,$ so
$$\left(\phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} d \alpha \right)^v = \phi \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} (d^h \alpha)^+ = 0$$
if and only if $(d^h \alpha)^+ = 0.$ By (\ref{dhalphaplus}), this implies that $(d^h \alpha_{i} )^+ = 0$ for $i = 0,1,2,$ as desired.

The dimension formula for structure group ${\mathrm{SU} }(2)$ follows from the Atiyah-Hitchin-Singer Theorem \cite{ahs}.
\end{proof}

\section{Global picture}\label{sec:global}

In this section, we discuss the global structure of the components of the moduli space obtained by pullback. We first prove Theorem \ref{thm:grassmann} on the structure of the $\kappa = 1$ component. For higher charge, the picture necessarily involves Hermitian-Yang-Mills connections on the twistor space ${\mathbb C} \P^3 \to S^4.$

\subsection{Proof of Theorem \ref{thm:grassmann}}
Let $W$ be given by (\ref{wdef}). Taking $A_0$ in the gauge (\ref{standardg2instanton}), we may define the smooth 5-dimensional family of connections
\begin{equation*}
V = \{\exp(w)^* A_0 \mid w \in W \} \subset {\mathscr A}_E.
\end{equation*}
By the construction of \S \ref{ss:hopf}, this family is equal to the pullback by the Hopf fibration of the 5-dimension family of unit-charge ASD instantons on $S^4.$
Further define a smooth map
\begin{equation}\label{firstg2map}
\begin{split}
{\mathrm{Spin} }(7) \times V & \to {\mathscr A}_E \\
\left(\sigma , A \right) & \mapsto \sigma^*A .
\end{split}
\end{equation}
By construction, the image of (\ref{firstg2map}) consists of ${\mathbb G}_2$-instantons.

Denote the principal bundle
$$Q = {\mathrm{Spin} }(7) \to {\mathrm{Spin} }(7) / {\mathrm{Sp} }(2) \times {\mathrm U}(1) .$$
Note that ${\mathrm{Sp} }(2) \times {\mathrm U}(1)$ acts on $V,$ modulo gauge, by the 5-dimensional representation of ${\mathrm{Sp} }(2)$ and the trivial representation of ${\mathrm U}(1).$ Taking the quotient by the gauge group ${\mathscr G}_E,$ the map (\ref{firstg2map}) descends to a smooth map from the associated bundle
$$X = Q \times_{{\mathrm{Sp} }(2) \times {\mathrm U}(1)} V$$
to the space of connections modulo gauge:
\begin{equation}\label{secondg2map}
X
\to {\mathscr A}_E / {\mathscr G}_E.
\end{equation}
Notice that
$${\mathrm{Spin} }(7)/ {\mathrm{Sp} }(2) \times {\mathrm U}(1) \cong {\mathrm{SO} }(7) / {\mathrm{SO} }(5) \times {\mathrm{SO} }(2) = {\mathbb G}^{or}(5,7).$$
Hence, the domain $X$ in (\ref{secondg2map}) is equal to the vector bundle over ${\mathbb G}^{or}(5,7)$ associated to the standard representation of ${\mathrm{SO} }(5),$ {\it i.e.}, the tautological 5-plane bundle.

We claim that the map (\ref{secondg2map}) is a proper embedding.
By Proposition \ref{prop:15deformations}, the image of the differential of (\ref{secondg2map}) has rank at least 15 at each point. But $X$ is 15-dimensional, so the map is an embedding.
The base space ${\mathbb G}^{or}(5,7)$ is compact, and as $x \in X$ tends to infinity in a fiber $V,$ the curvature of the ASD instanton blows up, as does the curvature of the pullback. This implies that the connection tends to infinity in ${\mathscr A}_E / {\mathscr G}_E,$ so (\ref{secondg2map}) is a proper map.

By Theorem \ref{thm:generaldefos}, every element of the image of (\ref{secondg2map}), being equivalent modulo ${\mathrm{Spin} }(7)$ to a pullback from $S^4,$ has a 15-dimensional space of infinitesimal deformations. Since $\dim(X) = 15,$
we conclude that (\ref{secondg2map}) is a diffeomorphism onto a connected component of the ${\mathbb G}_2$-instanton moduli space.
\qed

\begin{rmk} By the same construction, we may compactify $X$ fiberwise to a $\bar{B}^5$-bundle,
where a boundary point records bubbling along an associative great sphere.
\end{rmk}

\subsection{Chern-Simons functional and Hermitian-Yang-Mills connections}

Given two connections $A$ and $A_1$ on a bundle $E,$ let $a = A - A_1$ and define the relative Chern-Simons 3-form:
\begin{equation}
cs(A, A_1) = - Tr \left( a \wedge \left( F_{A_1} + \frac{1}{2} d_{A_1}a + \frac{1}{3} a \wedge a \right) \right).
\end{equation}
This satisfies
$$d cs(A, A_1) = Tr \left( F_A \wedge F_A - F_{A_1} \wedge F_{A_1} \right).$$
On a 7-manifold $M$ with ${\mathbb G}_2$-structure, we may define the global relative \emph{Chern-Simons functional}
\begin{equation}
CS_{\psi}(A, A_1) = \frac{-1}{4 \pi^4}\int_M \psi \wedge cs(A, A_1).
\end{equation}
Let
\begin{equation}\label{bigpi}
\Pi : S_{std}^7 \to {\mathbb C} \P^3
\end{equation}
be the natural projection for the standard complex structure $I_1.$

\begin{prop}\label{prop:hym} Let $B$ be a connection on an ${\mathrm{SU} }(n)$-bundle $E \to {\mathbb C}\P^3.$ Then $A = \Pi^*B$ is a ${\mathbb G}_2$-instanton on $S^7_{std}$ if and only if $B$ is Hermitian-Yang-Mills.

Given any two such connections $(E,B)$ and $(E_1,B_1)$ for which $\Pi^* E \cong \Pi^*E_1,$ we have
\begin{equation*}
CS_\psi \left(A, A_1 \right) = \langle \left[ \omega_{FS} \right] \cup \left( c_2(E) - c_2(E_1) \right), \left[ {\mathbb C}\P^3 \right] \rangle.
\end{equation*}
\end{prop}
\begin{proof} By ${\mathrm{SU} }(4)$-invariance, it suffices to consider the point $p = (1, 0, \ldots, 0) \in S^7.$ We shall write
\begin{equation}
T_p S^7 = {\mathbb R}_{x_1} \oplus {\mathbb C}^3.
\end{equation}
Then, from the expression (\ref{psi0kahler}), we have
\begin{equation}
\phi_{std}(p) = \frac{\partial}{\partial x_0} \mathbin{\raisebox{.7\depth}{\scalebox{1}[-1]{$\lnot$}} \,} \Psi_0 = dx^1 \wedge \omega + \Re dz_2 \wedge dz_3 \wedge dz_4
\end{equation}
where $\omega$ is the standard K{\"a}hler form on ${\mathbb C}^3,$ given by (\ref{stdkahlerforms}). Letting
$$F = F_A(p) = \left. \Pi^*F_B \right|_{T_p S^7} = \left. F_B \right|_{{\mathbb C}^3}$$
we have
\begin{equation}
\phi_0 \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} F = dx^1 \left( \omega . F \right) + \Re \, dz_2 \wedge dz_3 \wedge dz_4 \mathbin{\raisebox{.7\depth}{\scalebox{-1}[-1]{$\lnot$}}} F^{2,0} .
\end{equation}
Therefore, $A$ is a ${\mathbb G}_2$-instanton if and only if
$$\omega .F = 0, \quad F^{2,0} = 0$$
which is to say, $B$ is Hermitian-Yang-Mills on ${\mathbb C} \P^3.$

Next, we have
\begin{equation}\label{thingtointegrate}
\begin{split}
 - 4 \pi^4 CS_\psi (A, A_1) & = \int_{S^7} \psi \wedge cs(A, A_1) \\
& = \frac{1}{4} \int_{S^7} d \phi \wedge cs(A, A_1) \\
& = - \frac{1}{4} \int_{S^7} \phi \wedge d cs(A, A_1) \\
& = - \frac{1}{4} \int_{S^7} \phi \wedge Tr \left( F_A \wedge F_A  - F_{A_1} \wedge F_{A_1} \right).
\end{split}
\end{equation}
Assume that both $A$ and $A_1$ are pullbacks of Hermitian-Yang-Mills connections on ${\mathbb C} \P^3.$ Computing at $p$ as above, we have $F = F_A^{1,1}$ and are left with
\begin{equation}\label{thisexpression}
\phi_0 \wedge Tr \left( F \wedge F \right) = dx^1 \wedge \omega \wedge Tr \left( F \wedge F \right).
\end{equation}
Since the Fubini-Study metric on ${\mathbb C}\P^3$ satisfies
$$\Pi^*\omega_{FS} = \frac{\omega}{\pi}$$
on $S^7,$ the expression (\ref{thisexpression}) agrees globally with
$$\pi \zeta_1 \wedge \Pi^* \left( \omega_{FS} \wedge Tr \left( F_B \wedge F_B \right) \right).$$
Integrating (\ref{thingtointegrate}) over the fibers of $\Pi,$ we obtain
\begin{equation*}
\begin{split}
-4 \pi^4 CS_\psi (A, A_1) & = - \frac{\pi}{4} \int_{S^7} \zeta_1 \wedge \Pi^* \left( \omega_{FS} \wedge Tr \left( F_B \wedge F_B - F_{B_1} \wedge F_{B_1} \right) \right) \\
& = - \frac{\pi^2}{2} \int_{{\mathbb C}\P^3} \omega_{FS} \wedge Tr \left( F_B \wedge F_B - F_{B_1} \wedge F_{B_1} \right) \\
& = - \frac{\pi^2}{2} \langle \left[ \omega_{FS} \right] \cup 8 \pi^2 \left(c_2(E) - c_2(E_1) \right) , \left[ {\mathbb C} \P^3 \right] \rangle \\
& = -4 \pi^4 \langle \left[ \omega_{FS} \right] \cup \left(c_2(E) - c_2(E_1) \right) , \left[ {\mathbb C} \P^3 \right] \rangle
\end{split}
\end{equation*}
as claimed.
\end{proof}

\subsection{Discussion}\label{ss:discussion}
Theorem \ref{thm:generaldefos} can be explained in light of Proposition \ref{prop:hym}, as follows.
Owing to the Ward correspondence, the space of infinitesimal deformations of the pullback of an ASD instanton to the twistor space ${\mathbb C} \P^3 \to S^4,$ as a Hermitian-Yang-Mills connection, is the complexification of the space of ASD deformations.
In fact, there is an $S^1$ family of complex structures
\begin{equation*
\cos (\theta) I_1 + \sin (\theta) I_2
\end{equation*}
such that the quotient map $\Pi_\theta$
satisfies Proposition \ref{prop:hym} and factorizes the given Hopf fibration:
\[
\xymatrix{
S^7 \ar@{->}[r]^{\Pi_\theta} \ar@{->}[dr]_{\pi} & {\mathbb C}\P^3 \ar@{->}[d] \\
 & S^4.}
\]
The span of the pullbacks by $\Pi_\theta$ of the deformation space over ${\mathbb C} \P^3,$ for $\theta \in S^1,$ gives the larger space (\ref{generaldefos:maineq}).

According to this argument, the fibration to ${\mathbb C}\P^3$
accounts for all of the infinitesimal deformations identified by Theorem \ref{thm:generaldefos}.
As such, we expect that only the deformations coming from (\ref{bigpi}), or its 6-dimensional family of ${\mathrm{Spin} }(7)$ conjugates,
are integrable. So for $\kappa > 1,$ the generic dimension of the pulled-back component of the ${\mathbb G}_2$-instanton moduli space should be
\begin{equation}\label{higherchargedimformula}
2\left( 8 \kappa - 3 \right) + 6 = 16 \kappa.
\end{equation}
These components appear to be singular along the subset of instantons coming from $S^4$ ({\it i.e.}, instanton bundles on ${\mathbb C}\P^3$ satisfying a reality condition),
which are themselves manifolds of dimension
$$ 8 \kappa - 3 + 10 = 8 \kappa + 7.$$

More broadly, we would like to know where the instantons obtained via pullback
fit within the full moduli space of ${\mathbb G}_2$-instantons on $S^7.$ While
examples that are unrelated to any fibration
may exist,
the following guess is appropriate based on \cite{yuanqicyproduct} and the present work.
For the statement, fix a reference Hermitian-Yang-Mills connection $B_1$ on an ${\mathrm{SU} }(n)$-bundle $E_1 \to {\mathbb C} \P^3,$ and let $E = \Pi^* E_1 \to S^7$ and $A_1 = \Pi^* B_1.$

\begin{conj}[Donaldson]\label{conj:donaldson} Let $A$ be a ${\mathbb G}_2$-instanton on $E \to S^7_{std},$ for which
$$CS_{\psi}(A, A_1) \in {\mathbb Z}.$$
Then $A$ is equivalent, modulo the action of ${\mathrm{Spin} }(7)$ and ${\mathscr G}_E,$ to the pullback of a Hermitian-Yang-Mills connection on ${\mathbb C} \P^3.$
\end{conj}