$\displaystyle F\text{ is an }I\text{-primitive HYM connection}$
$\displaystyle\iff F=F_{I}+F_{W}$ $\displaystyle F\text{ is an
}\mathrm{Sp}(n)\text{-instanton}$ $\displaystyle\iff F=F_{W}$

Now, for
$\mathbf{\lambda}=(\lambda_{1},\lambda_{2},\lambda_{3})\in\mathbb{R}^{3}$
satisfying the conditions

$\displaystyle\lambda_{1}+\lambda_{2}+\lambda_{3}$ $\displaystyle=0$
$\displaystyle\lambda_{1}-2\lambda_{2}-2\lambda_{3}$ $\displaystyle>0,$ (7.4)

define a $(4n-4)$-form $\Phi_{\lambda}\in\Omega^{4n-4}(X)$ by

$\Phi_{\lambda}=\textstyle\frac{1}{(2n-2)!}\left(\lambda_{1}\omega_{I}^{2n-2}+\lambda_{2}\omega_{J}^{2n-2}+\lambda_{3}\omega_{K}^{2n-2}\right)\\!,$

and define constants $a_{I},a_{J},a_{K}$ and $b_{I},b_{J},b_{K}$ by

$\displaystyle\begin{bmatrix}a_{I}\\\ a_{J}\\\ a_{K}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}1&-2&-2\\\ -2&1&-2\\\
-2&-2&1\end{bmatrix}\begin{bmatrix}\lambda_{1}\\\ \lambda_{2}\\\
\lambda_{3}\end{bmatrix}$ $\displaystyle\begin{bmatrix}b_{I}\\\ b_{J}\\\
b_{K}\end{bmatrix}=\begin{bmatrix}1-2n&-2&-2\\\ -2&1-2n&-2\\\
-2&-2&1-2n\end{bmatrix}\begin{bmatrix}\lambda_{1}\\\ \lambda_{2}\\\
\lambda_{3}\end{bmatrix}\\!.$ (7.5)

Note that $a_{I}>0$ by assumption.

######  Lemma 7.5.

For each $\lambda\in\mathbb{R}^{3}$ satisfying (7.4), we have

$\sum_{L\in\\{I,J,K\\}}a_{L}\left\|F_{L}\right\|^{2}+b_{L}\left\|\mu_{L}\omega_{L}\right\|^{2}=\int_{X}\mathrm{tr}(F\wedge
F)\wedge\Phi_{\lambda}.$

(Note that there is no $F_{W}$ term on the left side.) Moreover, the integral
on the right side depends only on the topology of the bundle $P\to X$.

###### Proof.

Fix $\lambda\in\mathbb{R}^{3}$ satisfying (7.4). For $L\in\\{I,J,K\\}$, we set
$\Theta_{L}:=\frac{\omega_{L}^{2n-2}}{(2n-2)!}$. By the proof of Lemma 7.3, we
have

$\displaystyle\mathrm{tr}(F\wedge F)\wedge\Theta_{I}$
$\displaystyle=\left(-2|F^{2,0}_{I}|^{2}+|F^{1,1}_{I,0}|^{2}+\left(1-2n\right)|\mu_{I}\omega_{I}|^{2}\right)\mathrm{vol}_{X}$
$\displaystyle\mathrm{tr}(F\wedge F)\wedge\Theta_{J}$
$\displaystyle=\left(-2|F^{2,0}_{J}|^{2}+|F^{1,1}_{J,0}|^{2}+\left(1-2n\right)|\mu_{J}\omega_{J}|^{2}\right)\mathrm{vol}_{X}$
(7.6) $\displaystyle\mathrm{tr}(F\wedge F)\wedge\Theta_{K}$
$\displaystyle=\left(-2|F^{2,0}_{K}|^{2}+|F^{1,1}_{K,0}|^{2}+\left(1-2n\right)|\mu_{K}\omega_{K}|^{2}\right)\mathrm{vol}_{X}.$

We now decompose the first two terms on the right hand sides with respect the
splitting (7.3). The result is

$\displaystyle|F^{2,0}_{I}|^{2}$
$\displaystyle=|F_{J}|^{2}+|F_{K}|^{2}+|\mu_{J}\omega_{J}|^{2}+|\mu_{K}\omega_{K}|^{2}$
$\displaystyle|F^{2,0}_{J}|^{2}$
$\displaystyle=|F_{I}|^{2}+|F_{K}|^{2}+|\mu_{I}\omega_{I}|^{2}+|\mu_{K}\omega_{K}|^{2}$
(7.7) $\displaystyle|F^{2,0}_{K}|^{2}$
$\displaystyle=|F_{I}|^{2}+|F_{J}|^{2}+|\mu_{I}\omega_{I}|^{2}+|\mu_{J}\omega_{J}|^{2}$

and

$\displaystyle|F^{1,1}_{0,L}|^{2}$ $\displaystyle=|F_{L}|^{2}+|F_{W}|^{2}.$
(7.8)

For brevity, we introduce the following shorthand: For constants
$p_{I},p_{J},p_{K},r,q_{I},q_{J},q_{K}\in\mathbb{R}$, we let

$\displaystyle[p_{I},p_{J},p_{K};\ r;\ q_{I},q_{J},q_{K}]$
$\displaystyle:=\sum_{L\in\\{I,J,K\\}}p_{L}|F_{L}|^{2}+r|F_{W}|^{2}+\sum_{L\in\\{I,J,K\\}}q_{L}|\mu_{L}\omega_{L}|^{2}.$

Then using (7.1.2) and (7.8), we can write equations (7.1.2) as follows:

$\displaystyle\mathrm{tr}(F\wedge F)\wedge\Theta_{I}$
$\displaystyle=\left[1,-2,-2;\ 1;\ 1-2n,-2,-2\right]\mathrm{vol}_{X}$
$\displaystyle\mathrm{tr}(F\wedge F)\wedge\Theta_{J}$
$\displaystyle=\left[-2,1,-2;\ 1;\ -2,1-2n,-2\right]\mathrm{vol}_{X}$ (7.9)
$\displaystyle\mathrm{tr}(F\wedge F)\wedge\Theta_{K}$
$\displaystyle=\left[-2,-2,1;\ 1;\ -2,-2,1-2n\right]\mathrm{vol}_{X}$

Therefore, since $\lambda_{1}+\lambda_{2}+\lambda_{3}=0$, we have that

$\mathrm{tr}(F\wedge F)\wedge\Phi_{\lambda}=[a_{I},a_{J},a_{K};\ 0;\
b_{I},b_{J},b_{K}]\,\mathrm{vol}_{X}.$

∎

######  Corollary 7.6.

If $A$ is an $I$-primitive HYM connection, then $\|F_{I}\|$ depends only on
the topology of the bundle $P\to X$.

###### Proof.

If $A$ is an $I$-primitive HYM connection, then $F_{J}=F_{K}=0$ and
$\mu_{I}=\mu_{J}=\mu_{K}=0$. Therefore, by Lemma 7.5, the quantity

$a_{I}\left\|F_{I}\right\|^{2}=\int_{X}\mathrm{tr}(F\wedge
F)\wedge\Phi_{\lambda}$

is topological, and $a_{I}$ is a positive constant. ∎

We are now in a position to prove Theorem 7.1. Suppose that $A$ is a primitive
Hermitian Yang-Mills connection with respect to $I$, so that its curvature
$2$-form decomposes as $F=F_{I}+F_{W}$. By assumption, $P$ admits an
$\mathrm{Sp}(n)$-instanton. Let $\widetilde{A}$ denote such an
$\mathrm{Sp}(n)$-instanton, and let $\widetilde{F}$ denote its curvature
$2$-form. Since both $A$ and $\widetilde{A}$ are $I$-primitive HYM
connections, Corollary 7.6 implies that

$\|F_{I}\|=\|\widetilde{F}_{I}\|.$

But since $\widetilde{A}$ is an $\mathrm{Sp}(n)$-instanton, we have
$\widetilde{F}=\widetilde{F}_{W}$, meaning that $\widetilde{F}_{I}=0$.
Therefore, $F_{I}=0$, so that $F=F_{W}$, and hence $A$ is an
$\mathrm{Sp}(n)$-instanton. This completes the proof.

###  7.2 The Asymptotically Conical Case

This section is aimed at stating and proving Theorem 7.15, a Lewis-type
theorem for instantons on asymptotically conical (AC) hyperkähler manifolds
$X$. In essence, the idea is to adapt Lemma 7.5 and Corollary 7.6 to the AC
setting. This is accomplished in Proposition 7.14, which, by applying Stokes’
Theorem to truncations of $X$ at radius $r>0$, features a Chern-Simons type
functional $\mathrm{CS}_{\lambda}[A]$ that measures the failure of a pHYM
connection $A$ from being an $\mathrm{Sp}(n)$-instanton. By estimating
$\mathrm{CS}_{\lambda}[A]$ at infinity, we show that pHYM connections with
sufficiently fast decay will satisfy $\mathrm{CS}_{\lambda}[A]=0$ for large
values of $r$, which implies the desired Lewis theorem.
Our approach follows that of Papoulias [37], who obtains an analogous result
for AC $\mathrm{Spin}(7)$-manifolds. Preparation of this section was also
aided by helpful discussions in [9], [15], and [20].

####  7.2.1 Asymptotically conical Riemannian manifolds

######  Definition 7.7.

Let $(X^{N},h_{X})$ be a Riemannian $N$-manifold, $N\geq 2$. We say that $X$
is _asymptotically conical (AC)_ with _rate $\nu<0$_ if there exists a compact
set $K\subset X$, a constant $R>0$, a compact connected Riemannian
$(N-1)$-manifold $(\Sigma^{N-1},g_{\Sigma})$, and a diffeomorphism

$\Psi\colon(R,\infty)\times\Sigma\to X-K$

such that

$\left|\nabla^{k}_{g_{C}}(\Psi^{*}h_{X}-g_{C})\right|_{g_{C}}=O(r^{\nu-k})\ \
\ \text{as }r\to\infty$

for all $k\in\mathbb{Z}_{\geq 0}$, where
$(C,g_{C})=((R,\infty)\times\Sigma,dr^{2}+r^{2}g_{\Sigma})$ is the metric cone
of $\Sigma$. We then call $\Sigma$ the _asymptotic link_ of $X$.

Let $h_{X}^{-1}$ and $g_{C}^{-1}$ denote the dual metrics on $1$-forms, and
let $\mathrm{vol}_{X}$ and $\mathrm{vol}_{C}$ denote the corresponding volume
forms. It can be shown that the AC condition implies the estimates

$\displaystyle\left|\nabla^{k}_{g_{C}}(\Psi^{*}h_{X}^{-1}-g_{C}^{-1})\right|_{g_{C}}$
$\displaystyle=O(r^{\nu-k})$ (7.10)
$\displaystyle\left|\nabla^{k}_{g_{C}}(\Psi^{*}\mathrm{vol}_{X}-\mathrm{vol}_{C})\right|_{g_{C}}$
$\displaystyle=O(r^{\nu-k}).$

See [9, $\S$3.1] for a proof.

We now set notation. Let $r\colon C\to(R,\infty)$ denote the natural radius
function, and let $\rho=r\circ\Psi^{-1}\colon X-K\to(R,\infty)$ denote the
corresponding function on $X-K$. For $a>R$, we set

$\displaystyle\Sigma_{a}$ $\displaystyle=\\{a\\}\times\Sigma=\\{p\in C\colon
r(p)=a\\}$ $\displaystyle S_{a}$ $\displaystyle=\\{x\in X\colon\rho(x)=a\\}.$

Let $\iota_{a}\colon\Sigma\to\Sigma_{a}$ denote the identification
$\iota_{a}(p)=(a,p)$, and let
$\Psi_{a}:=\left.\Psi\right|_{\Sigma_{a}}\colon\Sigma_{a}\to S_{a}$. We equip
each $\Sigma_{a}\subset C$ with its induced metric $g_{a}$, and similarly
equip each $S_{a}\subset X$ with its induced metric $h_{a}$. The following
diagram summarizes the manifolds under consideration, where vertical arrows
denote the obvious inclusions.

${{(C,g_{C})=(R,\infty)\times\Sigma}}$${(X-K,h_{X})}$${(\Sigma,g_{\Sigma})}$${(\Sigma_{r},g_{r})=\\{r\\}\times\Sigma}$${(S_{r},h_{r})}$$\scriptstyle{\Psi}$$\scriptstyle{\iota_{r}}$$\scriptstyle{\Psi_{r}}$

To streamline notation, we will often suppress the identification maps
$\iota_{r}$ (as well as their pushforwards and pullbacks) from the notation.
Let $\\{e_{1},\ldots,e_{N-1}\\}$ be a $g_{\Sigma}$-orthonormal basis of
$T_{p}\Sigma$, and let $\\{e^{1},\ldots,e^{N-1}\\}$ be the dual
$g_{\Sigma}^{-1}$-orthonormal basis of $T_{p}^{*}\Sigma$. Note that
$\\{\frac{1}{r}e_{j}\\}$ is a $g_{r}$-orthonormal basis of $T_{p}\Sigma_{r}$,
and $\\{re^{j}\\}$ is a $g_{r}^{-1}$-orthonormal basis of
$T_{p}^{*}\Sigma_{r}$. In particular,

$\mathrm{vol}_{\Sigma_{r}}=r^{N-1}\,e^{1}\wedge\cdots\wedge
e^{N-1}=r^{N-1}\,\mathrm{vol}_{\Sigma}.$

In this way, we are able to relate the metric $g_{r}$ to $g_{\Sigma}$.
However, we also will need to relate $\Psi_{r}^{*}h_{r}$ to $g_{\Sigma}$. This
is accomplished by the following lemma.

######  Lemma 7.8.

We have

$\displaystyle\mathrm{vol}_{\Psi_{r}^{*}h_{r}}$
$\displaystyle=O\\!\left(r^{N-1}\right)\mathrm{vol}_{\Sigma}$ (7.11)
$\displaystyle\Psi_{r}^{*}h_{r}$
$\displaystyle=O\\!\left(r^{2}\right)g_{\Sigma}$ (7.12)
$\displaystyle\Psi_{r}^{*}h_{r}^{-1}$
$\displaystyle=O\\!\left(r^{-2}\right)g_{\Sigma}^{-1}$ (7.13)

###### Proof.

Let us fix $p\in\Sigma$ and suppress the identification map
$\iota_{r}\colon\Sigma\to\Sigma_{r}$ throughout. Since $\Psi_{r}^{*}h_{r}$ is
a symmetric bilinear form, the spectral theorem yields a
$g_{\Sigma}$-orthonormal basis $\\{e_{1},\ldots,e_{N-1}\\}$ of $T_{p}\Sigma$
with respect to which $\Psi_{r}^{*}h_{r}$ is diagonal. That is, there exist
$\tau_{j}=\tau_{j}(r)$ for which

$\displaystyle\Psi_{r}^{*}h_{r}=r^{2}\sum\tau_{j}^{2}(e^{j})^{2}$

where in this sum (and all those throughout the proof) the index range is
$j=1,\ldots,N-1$. In particular,

$\displaystyle\Psi_{r}^{*}h_{r}^{-1}$
$\displaystyle=\frac{1}{r^{2}}\sum\frac{1}{\tau_{j}^{2}}e_{j}^{2}$
$\displaystyle\mathrm{vol}_{\Psi_{r}^{*}h_{r}}$
$\displaystyle=r^{N-1}\tau_{1}\cdots\tau_{N-1}\,e^{1}\wedge\cdots\wedge
e^{N-1}.$

Now, since $h_{X}$ is asymptotically conical, we have
$\left|\Psi^{*}h_{X}-g_{C}\right|_{g_{C}}=O(r^{\nu})$. Thus, for each
$j=1,\ldots,N-1$,

$\displaystyle|\tau_{j}^{2}-1|\leq\left(\sum\left|\tau_{j}^{2}-1\right|^{2}\right)^{1/2}$
$\displaystyle=\left(\sum\left|\tau_{j}^{2}(\Psi^{*}h_{X})\\!\left(\frac{1}{\tau_{j}r}e_{j},\frac{1}{\tau_{j}r}e_{j}\right)-g_{C}\\!\left(\frac{1}{r}e_{j},\frac{1}{r}e_{j}\right)\right|^{2}\right)^{1/2}=O(r^{\nu})$

and so

$\tau_{j}^{2}=1+O(r^{\nu})=O(1).$

Therefore,

$r^{N-1}\tau_{1}\cdots\tau_{N-1}=O(r^{N-1})$

proves (7.11) and

$\Psi_{r}^{*}h_{r}=r^{2}\sum\tau_{j}^{2}(e^{j})^{2}=O(r^{2})\sum(e^{j})^{2}=O(r^{2})g_{\Sigma}$

proves (7.12). Next, since $h_{X}$ is asymptotically conical, we have
$\left|\Psi^{*}h_{X}^{-1}-g_{C}^{-1}\right|_{g_{C}}=O(r^{\nu})$ from (7.10).
Thus, for each $j=1,\ldots,N-1$,

$\displaystyle\left|\frac{1}{\tau_{j}^{2}}-1\right|\leq\left(\sum\left|\frac{1}{\tau_{j}^{2}}-1\right|^{2}\right)^{1/2}$
$\displaystyle=\left(\sum\left|\frac{1}{\tau_{j}^{2}}(\Psi^{*}h_{X}^{-1})\\!\left(\tau_{j}re^{j},\tau_{j}re^{j}\right)-g_{C}^{-1}\\!\left(re^{j},re^{j}\right)\right|^{2}\right)^{1/2}=O(r^{\nu})$

and so

$\frac{1}{\tau_{j}^{2}}=1+O(r^{\nu})=O(1).$

Therefore,

$\Psi_{r}^{*}h_{r}^{-1}=\frac{1}{r^{2}}\sum\frac{1}{\tau_{j}^{2}}e_{j}^{2}=O(r^{-2})\sum
e_{j}^{2}=O(r^{-2})g_{\Sigma}^{-1},$

which proves (7.13). ∎

Before continuing, we establish a few more estimates.

######  Lemma 7.9.

Let $E\to X$ be a Hermitian vector bundle.

  1. (a)

Let $\gamma\in\Lambda^{k}(T^{*}X)\otimes E$. Then
$\left|\Psi_{r}^{*}\gamma\right|_{g_{\Sigma}}\leq
r^{k}\left|\Psi^{*}\gamma\right|_{g_{C}}$, suppressing pullbacks via the
inclusions $S_{r}\hookrightarrow X-K$ from the notation.

  2. (b)

Let $\gamma\in\Omega^{k}(X)$. Then
$\left|\Psi^{*}\gamma\right|_{g_{C}}=O(1)\left|\gamma\right|_{h_{X}}$.

###### Proof.

(a) For $\theta\in\Lambda^{k}(T^{*}C)\otimes\Psi^{*}E$, we may write
$\theta=\theta_{H}+dr\wedge\theta_{V}$ where
$\partial_{r}\,\lrcorner\,\theta_{H}=0$. Then

$\left|\iota_{r}^{*}\theta\right|_{g_{\Sigma}}=r^{k}\left|\theta_{H}\right|_{g_{C}}\leq
r^{k}\left|\theta\right|_{g_{C}}.$

Taking $\theta=\Psi^{*}\gamma$ gives the result.

(b) Let $\\{e_{1},\ldots,e_{N}\\}$ be a $g_{C}$-orthonormal basis of
$T_{(r,p)}C$ with respect to which $\Psi^{*}h_{X}$ is diagonal. This means
that there exist $\mu_{j}=\mu_{j}(r)$ for which

$\displaystyle g_{C}$ $\displaystyle=(e^{1})^{2}+\cdots+(e^{N})^{2},$
$\displaystyle\Psi^{*}h_{X}$
$\displaystyle=\mu_{1}^{2}(e^{1})^{2}+\cdots+\mu_{N}^{2}(e^{N})^{2}.$

Repeating the argument in the proof of Lemma 7.8 shows that each
$\mu_{j}=O(1)$. Now, in terms of the basis $\\{e_{j}\\}$, we may write
$\Psi^{*}\gamma=\sum_{I}A_{I}e^{I}$, say, where we sum over multi-indices
$I=(i_{1},\ldots,i_{k})$ and write $e^{I}=e^{i_{1}}\wedge\cdots\wedge
e^{i_{k}}$. Therefore,

$\displaystyle\left|\Psi^{*}\gamma\right|_{g_{C}}^{2}=\sum_{I}A_{I}^{2}=\sum_{I}\frac{1}{\mu_{I}^{2}}A_{I}^{2}\mu_{I}^{2}=O(1)\sum_{I}\frac{1}{\mu_{I}^{2}}A_{I}^{2}=O(1)\left|\Psi^{*}\gamma\right|^{2}_{\Psi^{*}h_{X}}=O(1)\left|\gamma\right|^{2}_{h_{X}}.$

∎

####  7.2.2 Asymptotically conical hyperkähler manifolds

######  Definition 7.10.

Let $(X^{4n+4},h_{X},(I,J,K),(\omega_{I},\omega_{J},\omega_{K}))$ be a
hyperkähler $(4n+4)$-manifold, $n\geq 1$. We say that $X$ is _asymptotically
conical (AC)_ with _rate $\nu<0$_ if there exists a compact set $K\subset X$,
a constant $R>0$, a compact connected $3$-Sasakian $(4n+3)$-manifold
$(\Sigma^{4n+3},g_{\Sigma})$, and a diffeomorphism

$\Psi\colon(R,\infty)\times\Sigma\to X-K$

such that for each $L\in\\{I,J,K\\}$ and $k\in\mathbb{Z}_{\geq 0}$,

$\displaystyle\left|\nabla^{k}_{g_{C}}(\Psi^{*}\omega_{L}-\beta_{L})\right|_{g_{C}}$
$\displaystyle=O(r^{\nu-k})\ \ \ \text{as }r\to\infty$
$\displaystyle\left|\nabla^{k}_{g_{C}}(\Psi^{*}L-L_{C})\right|_{g_{C}}$
$\displaystyle=O(r^{\nu-k})\ \ \ \text{as }r\to\infty$

where $C=\mathrm{C}(\Sigma)=(R,\infty)\times\Sigma$ is the metric cone over
$\Sigma$, and $(g_{C}$, $(I_{C},J_{C},K_{C})$,
$(\beta_{I},\beta_{J},\beta_{K}))$ is the natural conical hyperkähler
structure on $C$. We call $\Sigma$ the _asymptotic link_ of $X$.

It is well-known that AC hyperkähler $(4n+4)$-manifolds of rate $\nu$ are AC
Riemannian $(4n+4)$-manifolds of rate $\nu$, meaning that we have the estimate

$\left|\nabla^{k}_{g_{C}}(\Psi^{*}h_{X}-g_{C})\right|_{g_{C}}=O(r^{\nu-k})\ \
\ \text{as }r\to\infty$

See [9, $\S$3.3] for a proof. From now on, we assume that $X$ is an AC
hyperkähler manifold of rate $\nu$.

As in $\S$7.1.2, for
$\lambda=(\lambda_{1},\lambda_{2},\lambda_{3})\in\mathbb{R}^{3}$ satisfying
both $\lambda_{1}+\lambda_{2}+\lambda_{3}=0$ and
$\lambda_{1}-2\lambda_{2}-2\lambda_{3}>0$, we define a $4n$-form
$\Phi_{\lambda}\in\Omega^{4n}(X)$ by

$\Phi_{\lambda}=\textstyle\frac{1}{(2n)!}(\lambda_{1}\omega_{I}^{2n}+\lambda_{2}\omega_{J}^{2n}+\lambda_{3}\omega_{K}^{2n}).$

The following algebraic fact will be used repeatedly in the sequel. To state
it, recall the decomposition

$\Lambda^{2}(T^{*}X)=U_{I}\oplus U_{J}\oplus U_{K}\oplus
W\oplus\mathbb{R}\omega_{I}\oplus\mathbb{R}\omega_{J}\oplus\mathbb{R}\omega_{J}$

of Proposition 3.2. For a $2$-form $F\in\Lambda^{2}(T^{*}X)$, we let $F_{I}$
and $F_{W}$ denote its components in $U_{I}$ and $W$, respectively.

######  Lemma 7.11.

Let $F\in\Lambda^{2}(T^{*}X)$. Then:

  1. (a)

$F_{W}\wedge\Phi_{\lambda}=0$.

  2. (b)

$F_{I}\wedge\Phi_{\lambda}=(-\lambda_{1}+\lambda_{2}+\lambda_{3})\ast\\!F_{I}$.

###### Proof.

(a) Note that $F_{W}\wedge\frac{1}{(2n)!}\omega_{L}^{2n}=-\ast\\!F_{W}$ for
each $L\in\\{I,J,K\\}$. Therefore,

$\displaystyle F_{W}\wedge\Phi_{\lambda}$
$\displaystyle=-(\lambda_{1}+\lambda_{2}+\lambda_{3})\ast\\!F_{W}=0.$

(b) Note that $F_{I}\wedge\omega_{I}^{2n}=-\ast\\!F_{I}$, whereas
$F_{I}\wedge\omega_{J}^{2n}=F_{K}\wedge\omega_{J}^{2n}=\ast F_{I}$. Therefore,

$\displaystyle F_{I}\wedge\Phi_{\lambda}$
$\displaystyle=\textstyle\frac{1}{(2n)!}\left(\lambda_{1}\,F_{I}\wedge\omega_{I}^{2n}+\lambda_{2}\,F_{I}\wedge\omega_{J}^{2n}+\lambda_{3}\,F_{I}\wedge\omega_{K}^{2n}\right)$
$\displaystyle=(-\lambda_{1}+\lambda_{2}+\lambda_{3})\ast\\!F_{I}.$

∎

####  7.2.3 The Chern-Simons functional

Let $X^{4n+4}$ be an AC hyperkähler manifold of rate $\nu<0$. To study gauge-
theoretic objects over $X$, we consider a class of bundles that respects the
asymptotic geometry of $X$.

######  Definition 7.12.

A principal bundle $P\to X^{4n+4}$ is _asymptotically framed_ if there exists
a principal bundle $Q\to\Sigma$ (called an _asymptotic framing_) such that
$\Psi^{*}P\cong\pi^{*}Q$, where $\pi\colon C=(R,\infty)\times\Sigma\to\Sigma$
is the projection onto the second factor.

${Q}$${\pi^{*}Q\cong\Psi^{*}P}$${P}$${\Sigma}$${C}$${X-K}$$\scriptstyle{\Psi}$$\scriptstyle{\pi}$

Fix a connection $A_{\infty}$ on $Q\to\Sigma$. A connection $A$ on $P\to X$ is
called _asymptotically conical (AC)_ of _rate $\alpha$_ _with respect to
$A_{\infty}$_ provided that

$\left|\nabla^{k}_{g_{C}}(\Psi^{*}A-\pi^{*}A_{\infty})\right|_{g_{C}}=O(r^{\alpha-k})\
\ \text{ as }r\to\infty$

for all $k\geq 0$. We let

$\displaystyle\mathcal{A}_{\alpha}(P,A_{\infty})$
$\displaystyle=\left\\{\text{AC connections on }P\text{ of rate }\alpha\text{
with respect to }A_{\infty}\right\\}$
$\displaystyle\mathcal{M}_{\alpha}(P,A_{\infty})$
$\displaystyle=\left\\{A\in\mathcal{A}_{\alpha}(P,A_{\infty})\colon A\text{ is
an }\mathrm{Sp}(n+1)\text{-instanton}\right\\}\\!.$

We now seek AC analogues of Lemma 7.5 and Corollary 7.6, which as stated apply
only to _compact_ hyperkähler manifolds. To this end, by using the truncations
$X_{r}=\\{x\in X\colon\rho(x)\leq r\\}$ and their boundaries $S_{r}=\\{x\in
X\colon\rho(x)=r\\}$, we introduce the following functional.

######  Definition 7.13.

Let $P\to X^{4n+4}$ be an asymptotically framed principal bundle, and let
$A_{0}$ be an $\mathrm{Sp}(n+1)$-instanton on $P$. For a connection $A$ on
$P$, its _Chern-Simons functional_ is

$\displaystyle\mathrm{CS}_{\lambda}[A]\colon[R,\infty)$
$\displaystyle\to[0,\infty)$ $\displaystyle\mathrm{CS}_{\lambda}[A](r)$
$\displaystyle:=\int_{S_{r}}\mathrm{tr}\\!\left(F_{A}\wedge(A-A_{0})-\frac{1}{3}(A-A_{0})^{3}\right)\wedge\Phi_{\lambda}.$

The key properties of the CS functional are summarized in the following
proposition. To state it, recall once more that if $A$ is a connection on
$P\to X$, then its curvature $2$-form $F:=F_{A}$ decomposes as

$F=F_{I}+F_{J}+F_{K}+F_{W}+\mu_{I}\omega_{I}+\mu_{J}\omega_{J}+\mu_{K}\omega_{K}.$

In particular, $A$ is $I$-primitive HYM if and only if $F=F_{I}+F_{W}$, and
$A$ is an $\mathrm{Sp}(n+1)$-instanton if and only if $F_{I}=0$.

######  Proposition 7.14.

Let $A$ be a connection on $P$.

  1. (a)

We have

$\mathrm{CS}_{\lambda}[A](r)=\sum_{L\in\\{I,J,K\\}}\int_{X_{r}}\left(a_{L}|F_{L}|^{2}+b_{L}|\mu_{L}\omega_{L}|^{2}\right)\mathrm{vol}_{X_{r}},$

where the constants $a_{L}$ and $b_{L}$ are as in (7.5). In particular,
$\mathrm{CS}_{\lambda}[A]$ is independent of the choice of
$\mathrm{Sp}(n+1)$-instanton $A_{0}$.

  2. (b)

If $A$ is $I$-primitive HYM, then

$\mathrm{CS}_{\lambda}[A](r)=a_{I}\int_{X_{r}}|F_{I}|^{2}\,\mathrm{vol}_{X_{r}}.$

Thus, $A$ is an $\mathrm{Sp}(n+1)$-instanton if and only if
$\mathrm{CS}_{\lambda}[A](r)=0$ for all sufficiently large $r\geq R$.

###### Proof.

(a) Let $A$ be a connection on $P$, let $A_{0}$ be an
$\mathrm{Sp}(n+1)$-instanton on $P$, and let $F$ and $F_{0}$ be their
respective curvature forms. It is well-known that

$d\left[\mathrm{tr}\\!\left((F+F_{0})\wedge(A-A_{0})-\frac{1}{3}(A-A_{0})^{3}\right)\right]=\mathrm{tr}(F^{2})-\mathrm{tr}(F_{0}^{2}).$

Wedging with $\Phi_{\lambda}$, integrating over $X_{r}$, and then applying
Stokes’ Theorem gives:

$\displaystyle\int_{S_{r}}\mathrm{tr}\\!\left((F+F_{0})\wedge(A-A_{0})-\frac{1}{3}(A-A_{0})^{3}\right)\wedge\Phi_{\lambda}$
$\displaystyle=\int_{X_{r}}\mathrm{tr}(F^{2})\wedge\Phi_{\lambda}-\int_{X_{r}}\mathrm{tr}(F_{0}^{2})\wedge\Phi_{\lambda}$
$\displaystyle=\sum_{L\in\\{I,J,K\\}}\int_{X_{r}}\left(a_{L}|F_{L}|^{2}+b_{L}|\mu_{L}\omega_{L}|^{2}\right)\mathrm{vol}_{X_{r}},$

where in the last step we applied Lemma 7.5 to both integrals. To simplify the
left side, note that since $A_{0}$ is an $\mathrm{Sp}(n+1)$-instanton, we have
$F_{0}\in\Gamma(W\otimes\mathrm{ad}_{P})$. Therefore, Lemma 7.11(a) gives
$F_{0}\wedge\Phi_{\lambda}=0$, and so

$\mathrm{tr}(F_{0}\wedge(A-A_{0}))\wedge\Phi_{\lambda}=\mathrm{tr}(F_{0}\wedge(A-A_{0})\wedge\Phi_{\lambda})=0.$

Therefore,

$\int_{S_{r}}\mathrm{tr}\\!\left(F\wedge(A-A_{0})-\frac{1}{3}(A-A_{0})^{3}\right)\wedge\Phi_{\lambda}=\sum_{L\in\\{I,J,K\\}}\int_{X_{r}}\left(a_{L}|F_{L}|^{2}+b_{L}|\mu_{L}\omega_{L}|^{2}\right)\mathrm{vol}_{X_{r}}.$

(b) This follows immediately from part (a). ∎

####  7.2.4 The asymptotically conical Lewis theorem

We are finally in a position to state and prove the AC analogue of Theorem
7.1.

######  Theorem 7.15.

Let $P\to X^{4n+4}$ be an asymptotically framed principal $G$-bundle over an
asymptotically conical hyperkähler manifold $X$ of rate $\nu<0$, where $G$ is
a compact Lie group. Let $\Sigma^{4n+3}$ be the $3$-Sasakian asymptotic link
of $X$, let $Q\to\Sigma$ be an asymptotic framing, and fix a tri-contact
instanton $A_{\infty}$ on $Q\to\Sigma$.
Suppose that $\nu\leq-2-\frac{4}{3}n$, and that
$\mathcal{M}_{\alpha}(P,A_{\infty})$ is non-empty for some $\alpha<\nu+1$. If
$A\in\mathcal{A}_{\alpha}(P,A_{\infty})$ is an asymptotically conical
$I$-primitive HYM connection satisfying

$\displaystyle\left|\partial_{r}\,\lrcorner\,\Psi^{*}F_{I}\right|_{g_{C}}$
$\displaystyle=O(r^{\beta})\ \ \text{ where }\beta<-4n-\nu-4,$ (7.14)

where $F=F_{A}$ is the curvature of $A$, then $A$ is an
$\mathrm{Sp}(n+1)$-instanton.

###### Proof.

Let $A\in\mathcal{A}_{\alpha}(P,A_{\infty})$ be an asymptotically conical
$I$-primitive HYM connection satisfying (7.14). Since
$\mathcal{M}_{\alpha}(P,A_{\infty})$ is non-empty, there exists an
$\mathrm{Sp}(n+1)$-instanton $A_{0}$ satisfying

$|\Psi^{*}A_{0}-\pi^{*}A_{\infty}|_{g_{C}}=O(r^{\alpha}).$

Consequently,

$\left|\Psi^{*}(A-A_{0})\right|_{g_{C}}\leq\left|\Psi^{*}A-\pi^{*}A_{\infty}\right|_{g_{C}}+\left|\Psi^{*}A_{0}-\pi^{*}A_{\infty}\right|_{g_{C}}=O(r^{\alpha}).$

In particular, suppressing pullbacks via the inclusions $S_{r}\hookrightarrow
X-K$ from the notation, Lemma 7.9(a) gives

$\displaystyle\left|\Psi_{r}^{*}(A-A_{0})\right|_{g_{\Sigma}}$
$\displaystyle\leq r\left|\Psi^{*}(A-A_{0})\right|_{g_{C}}=O(r^{\alpha+1})$
(7.15)
$\displaystyle\left|\partial_{r}\,\lrcorner\,\Psi_{r}^{*}F_{I}\right|_{g_{\Sigma}}$
$\displaystyle\leq
r\left|\partial_{r}\,\lrcorner\,\Psi^{*}F_{I}\right|_{g_{C}}=O(r^{\beta+1}).$
(7.16)

Now, following the technique of [37], we split

$\displaystyle\mathrm{CS}_{\lambda}[A](r)$
$\displaystyle=\frac{1}{2}\int_{S_{r}}\mathrm{tr}\\!\left(F\wedge(A-A_{0})-\frac{1}{3}(A-A_{0})^{3}\right)\wedge\Phi_{\lambda}\leq\left|I_{1}(r)\right|+\left|I_{2}(r)\right|\\!,$

where

$\displaystyle I_{1}(r)$
$\displaystyle=\frac{1}{2}\int_{S_{r}}\mathrm{tr}\\!\left(F\wedge(A-A_{0})\right)\wedge\Phi_{\lambda},$
$\displaystyle I_{2}(r)$
$\displaystyle=\frac{1}{6}\int_{S_{r}}\mathrm{tr}\\!\left((A-A_{0})^{3}\right)\wedge\Phi_{\lambda}.$

We aim to establish the following two estimates:

$\displaystyle\left|I_{1}(r)\right|$
$\displaystyle=O\\!\left(r^{4n+3+\alpha+\beta}\right)$ (7.17)
$\displaystyle\left|I_{2}(r)\right|$
$\displaystyle=O\\!\left(r^{4n+3+3\alpha}\right)\\!.$ (7.18)

Note that these two bounds imply the result. To see this, observe that our
assumptions imply that $4n+3+\alpha+\beta<0$ and $4n+3+3\alpha<0$. Therefore,
the Monotone Convergence Theorem, followed by Proposition 7.14(b), followed by
the above estimates gives

$a_{I}\int_{X}|F_{I}|^{2}\,\mathrm{vol}=\lim_{r\to\infty}a_{I}\int_{X_{r}}|F_{I}|^{2}\,\mathrm{vol}_{X_{r}}=\lim_{r\to\infty}\mathrm{CS}_{\lambda}[A](r)\leq\lim_{r\to\infty}(|I_{1}(r)|+|I_{2}(r)|)=0,$

whence $F_{I}=0$, and $A$ is an $\mathrm{Sp}(n+1)$-instanton.
We begin by proving (7.17). Set $\tau:=-\lambda_{1}+\lambda_{2}+\lambda_{3}$.
Since $A$ is $I$-primitive HYM, Lemma 7.11 implies that
$F\wedge\Phi_{\lambda}=\tau\ast_{X}\\!F_{I}$, and hence

$\left|I_{1}(r)\right|=\left|\frac{1}{2}\int_{S_{r}}\mathrm{tr}\\!\left(F\wedge(A-A_{0})\right)\wedge\Phi_{\lambda}\right|=\left|\frac{\tau}{2}\int_{S_{r}}\mathrm{tr}((A-A_{0})\wedge\ast_{X}F_{I})\right|.$

Now, setting
$\widehat{\partial}_{\rho}:=\frac{1}{\left|\partial_{\rho}\right|}\partial_{\rho}$,
we have

$\displaystyle\mathrm{tr}((A-A_{0})\wedge\ast_{X}F_{I})$
$\displaystyle=\mathrm{tr}((A-A_{0})\wedge\ast_{S_{r}}(\widehat{\partial}_{\rho}\,\lrcorner\,F_{I}))=-\frac{1}{|\partial_{\rho}|}\left\langle
A-A_{0},\,\partial_{\rho}\,\lrcorner\,F_{I}\right\rangle_{h_{r}}\mathrm{vol}_{S_{r}}.$

Moreover, since $X$ is asymptotically conical of rate $\nu$, we have

$\left|h_{X}(\partial_{\rho},\partial_{\rho})-1\right|=\left|(\Psi^{*}h_{X})(\partial_{r},\partial_{r})-g_{C}(\partial_{r},\partial_{r})\right|=O(r^{\nu}),$

which implies that $\frac{1}{|\partial_{\rho}|}=O(1)$. Therefore,

$\displaystyle\left|I_{1}(r)\right|$
$\displaystyle=\left|\frac{\tau}{2}\int_{S_{r}}\frac{1}{|\partial_{\rho}|}\left\langle
A-A_{0},\,\partial_{\rho}\,\lrcorner\,F_{I}\right\rangle_{h_{r}}\mathrm{vol}_{S_{r}}\right|$
$\displaystyle=O(1)\left|\int_{\Sigma}\Psi_{r}^{*}\\!\left(\left\langle
A-A_{0},\,\partial_{\rho}\,\lrcorner\,F_{I}\right\rangle_{h_{r}}\mathrm{vol}_{S_{r}}\right)\right|$
$\displaystyle=O(1)\left|\int_{\Sigma}\left\langle\Psi_{r}^{*}(A-A_{0}),\,\Psi_{r}^{*}(\partial_{\rho}\,\lrcorner\,F_{I})\right\rangle_{\Psi_{r}^{*}h_{r}}\mathrm{vol}_{\Psi_{r}^{*}h_{r}}\right|$
$\displaystyle=O(r^{4n+1})\left|\int_{\Sigma}\left\langle\Psi_{r}^{*}(A-A_{0}),\,\partial_{r}\,\lrcorner\,\Psi_{r}^{*}F_{I}\right\rangle_{g_{\Sigma}}\mathrm{vol}_{\Sigma}\right|$

where in the last step we used (7.11) and (7.13). Finally, from the bounds
(7.15) and (7.16), we obtain

$\left|\left\langle\Psi_{r}^{*}(A-A_{0}),\,\partial_{r}\,\lrcorner\,\Psi_{r}^{*}F_{I}\right\rangle_{g_{\Sigma}}\right|\leq\left|\Psi_{r}^{*}(A-A_{0})\right|_{g_{\Sigma}}\left|\partial_{r}\,\lrcorner\,\Psi_{r}^{*}F_{I}\right|_{g_{\Sigma}}=O(r^{\alpha+\beta+2}),$

which gives (7.17).
It remains to prove (7.18). For this, we estimate

$\displaystyle\left|I_{2}(r)\right|=\left|\frac{1}{6}\int_{S_{r}}\mathrm{tr}\\!\left[(A-A_{0})^{3}\wedge\Phi_{\lambda}\right]\right|$
$\displaystyle=\left|\frac{1}{6}\int_{\Sigma}\mathrm{tr}\\!\left[(\Psi_{r}^{*}(A-A_{0}))^{3}\wedge\Psi_{r}^{*}\Phi_{\lambda}\right]\right|$
$\displaystyle\leq\frac{1}{6}\int_{\Sigma}\left|\Psi_{r}^{*}(A-A_{0})\right|_{g_{\Sigma}}^{3}\left|\Psi_{r}^{*}\Phi_{\lambda}\right|_{g_{\Sigma}}\,\mathrm{vol}_{\Sigma}.$

Now, the bound (7.15) gives
$\left|\Psi_{r}^{*}(A-A_{0})\right|_{g_{\Sigma}}^{3}=O(r^{3\alpha+3})$.
Moreover, Lemma 7.9(a) and (b) give

$\left|\Psi_{r}^{*}\Phi_{\lambda}\right|_{g_{\Sigma}}\leq
r^{4n}\left|\Psi^{*}\Phi_{\lambda}\right|_{g_{C}}=O(r^{4n})\left|\Phi_{\lambda}\right|_{h_{X}}.$

Finally, since $\nabla\Phi_{\lambda}=0$, we have
$\left|\Phi_{\lambda}\right|_{h_{X}}=\text{constant}$. Thus, we conclude that
$\left|I_{2}(r)\right|=O(r^{4n+3\alpha+3})$ as claimed. ∎

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