Examples of Deformed Spin(7)-Instantons/Donaldson–Thomas Connections

Abstract

We construct examples of deformed Hermitian Yang–Mills connections and deformed $\textrm{Spin}(7)$-instantons (also called $\textrm{Spin}(7)$ deformed Donaldson–Thomas connections) on the cotangent bundle of $\mathbb {C}\mathbb {P}^2$ endowed with the Calabi hyperKähler structure. Deformed $\textrm{Spin}(7)$-instantons on cones over 3-Sasakian 7-manifolds are also constructed. We show that these can be used to distinguish between isometric structures and also between $\textrm{Sp}(2)$ and $\textrm{Spin}(7)$ holonomy cones. To the best of our knowledge, these are the first non-trivial examples of deformed $\textrm{Spin}(7)$-instantons.

1 Introduction

In this paper we construct cohomogeneity one examples of deformed $\textrm{Spin}(7)$-instantons (also called $\textrm{Spin}(7)$ deformed Donaldson–Thomas connections [23, 24]) on $T^*\mathbb {C}\mathbb {P}^2$ endowed with its $\textrm{SU}(3)$-invariant Calabi hyperKähler structure [6]. We study their relations with the deformed Hermitian Yang–Mills connections associated to the underlying hyperKähler 2-forms. Furthermore, we show that the deformed $\textrm{Spin}(7)$ instantons on the hyperKähler cone are different to those on the $\textrm{Spin}(7)$ cone obtained when the 3-Sasakian link is equipped with the squashed metric. We also construct deformed Hermitian Yang–Mills connections on $T^*\mathbb {C}\mathbb {P}^1$ and the flag manifold $\textrm{SU}(3)/\mathbb {T}^2$ to illustrate certain similarities and differences with our examples on $T^*\mathbb {C}\mathbb {P}^2$. Before stating our results more precisely we first recall a few basic notions and elaborate on the motivation for this work.

Given a Kähler manifold $(M^{2n},g,J,\omega )$ and a Hermitian line bundle $L\rightarrow M$, a connection 1-form A on L is called a deformed Hermitian Yang–Mills (dHYM) connection with phase $e^{i\theta }$ if its curvature $F_A$, viewed as a real 2-form, satisfies

$$\begin{aligned} F_A^{0,2}&=0, \end{aligned}$$

(1.1)

$$\begin{aligned} \textrm{Im}((\omega +iF_A)^n)&=\tan (\theta )\textrm{Re}((\omega +iF_A)^n), \end{aligned}$$

(1.2)

for some constant $\theta$. The dHYM equations (1.1)-(1.2) were introduced in [27, 30] where it was shown that these arise naturally in the context of the SYZ conjecture for mirror symmetry [33]. Roughly, the SYZ conjecture asserts that mirror pairs of Calabi-Yau 3-folds admit special Lagrangian torus fibrations in certain limits. Solutions to the dHYM equations arise as mirrors to special Lagrangian sections of these fibrations via a real Fourier-Mukai transform cf. [9, Section 1.2]. Thus, the SYZ conjecture suggests an intricate relation between the calibrated geometry and gauge theory of mirror pairs of Calabi-Yau 3-folds. The SYZ conjecture also has higher dimensional analogues in the context of $\textrm{G}_2$ and $\textrm{Spin}(7)$ geometry [14], and a similar argument as in the Calabi-Yau case leads to the notion of deformed $\textrm{G}_2$- and deformed $\textrm{Spin}(7)$-instantons as mirrors to (co)-associatives and Cayley submanifolds respectively via the Fourier-Mukai transform cf. [21, 23, 24, 26].

In recent years, dHYM connections have received considerable interest, see for instance [8, 9, 18, 19] and the references therein, and there are by now many known examples on compact Kähler manifolds; especially provided by the work of Chen in [7] where their existence is related to a notion of stability. By contrast, very little is known in the context of $\textrm{G}_2$ and $\textrm{Spin}(7)$ geometry. The first non-trivial examples of deformed $\textrm{G}_2$-instantons were constructed by Lotay-Oliveira in [29] over certain nearly parallel $\textrm{G}_2$ manifolds (these are special classes of Einstein manifolds whose cone metrics have holonomy contained in $\textrm{Spin}(7)$). Here ‘non-trivial’ means that these connections are not flat and do not arise via pullback from lower-dimensional constructions. Subsequently in [12], we constructed non-trivial examples of deformed $\textrm{G}_2$-instantons over the spinor bundle of $S^3$ endowed with the complete Brandhuber et al. $\textrm{G}_2$ holonomy metric [3] and also the conical Bryant-Salamon metric [5]. These currently exhaust all the known non-trivial examples of deformed $\textrm{G}_2$-instantons. To the best of our knowledge, there are at present no known non-trivial examples of deformed $\textrm{Spin}(7)$-instantons and it is precisely this problem that motivated this work.

Given a $\textrm{Spin}(7)$ manifold $(M^8,\Phi )$ and a Hermitian line bundle $L \rightarrow M$, a connection A on L is called a deformed $\textrm{Spin}(7)$-instanton if $F_A$ satisfies:

$$\begin{aligned}{} & {} \pi ^4_7(F_A\wedge F_A) =0, \end{aligned}$$

(1.3)

$$\begin{aligned}{} & {} \pi ^2_7(F_A-\frac{1}{6}* F_A^3)=0, \end{aligned}$$

(1.4)

where $\pi ^2_7$ and $\pi ^4_7$ are projection maps determined by the $\textrm{Spin}(7)$ 4-form $\Phi$, see Sect. 2. If M is a Calabi-Yau 4-fold then one can view (1.3), (1.4) as a generalisation of (1.1)-(1.2) in the case when $\theta =0$ cf. [23, Theorem 7.5] and [24, Lemma 7.2]. In this paper we consider the deformed $\textrm{Spin}(7)$-instanton equations (1.3), (1.4) on $T^*\mathbb {C}\mathbb {P}^2$ endowed with its well-known hyperKähler metric arising from the Calabi ansatz [6]. This set up is of particular interest for several reasons: (1) Since the Calabi hyperKähler structure is invariant under a cohomogeneity one action of $\textrm{SU}(3)$, working in this invariant set up we have that the complicated systems of PDEs given by (1.1), (1.2) and (1.3), (1.4) reduce to more tractable systems of ODEs. (2) As $T^*\mathbb {C}\mathbb {P}^2$ is a hyperKähler manifold, it admits a triple of orthogonal Kähler forms $\omega _1,\omega _2,\omega _3$. Hence one can consider the dHYM equations with respect to each Kähler form and compare how the solutions vary. (3) We can also view $T^*\mathbb {C}\mathbb {P}^2$ as a $\textrm{Spin}(7)$ manifold with respect to different $\textrm{Spin}(7)$ 4-forms, defined by (2.27), and again we can compare the deformed $\textrm{Spin}(7)$-instantons in each case. (4) Closely related to $T^*\mathbb {C}\mathbb {P}^2$ is the spinor bundle of $\mathbb {C}\mathbb {P}^2$ which only exists as an orbifold. The latter admits a $\textrm{Spin}(7)$ holonomy metric due to the Bryant-Salamon construction in [5] and moreover, it is also invariant under the cohomogeneity one action of $\textrm{SU}(3)$. As such we can study the deformed $\textrm{Spin}(7)$-instanton equations in this set up as well. Note that the Bryant–Salamon and Calabi metrics are both asymptotic to a conical metric over a 3-Sasakian 7-manifold; however, in the latter case the metric has holonomy $\textrm{Sp}(2)$ and in the former case $\textrm{Spin}(7)$. Thus, we can compare the deformed $\textrm{Spin}(7)$-instantons over each cone. With this in mind we can now elaborate on our main results.

Statement of main results Let $\omega _1,\omega _2,\omega _3$ denote the hyperKähler triple on $T^*\mathbb {C}\mathbb {P}^2$ as given in Proposition 5.1 and let $\Phi _1,\Phi _2, \Phi _3$ denote the $\textrm{Spin}(7)$ 4-forms defined by (2.27). Parametrising the space of complex line bundles on $T^*\mathbb {C}\mathbb {P}^2$ by $k \in \mathbb {Z} \cong H^2(T^*\mathbb {C}\mathbb {P}^2,\mathbb {Z})$, we have that:

Theorem 1.1

For generic values of k and $\theta$, there are 2 distinct $\textrm{SU}(3)$-invariant dHYM connections on $T^*\mathbb {C}\mathbb {P}^2$ with respect to $\omega _1$. On the other hand, for generic values of k and $\theta$, there are 4 distinct $\textrm{SU}(3)$-invariant dHYM connections on $T^*\mathbb {C}\mathbb {P}^2$ with respect to $\omega _2$ and $\omega _3$. Moreover, when $\theta =0$, there is a unique hyper-holomorphic connection which occurs as a dHYM connection for $\omega _1,\omega _2$ and $\omega _3$.

An explicit description of these solutions is given in Theorems 5.8 and 5.10, see also Remarks 5.9 and 5.11. Thus, the moduli spaces of $\textrm{SU}(3)$-invariant dHYM connections with respect to the Kähler forms $\omega _1,\omega _2,\omega _3$ are all distinct and they intersect in exactly one point, namely, the hyper-holomorphic connection. It is also worth pointing out that the Kähler form $\omega _1$ is distinguished from $\omega _2$ and $\omega _3$ by the fact that it is non-trivial in cohomology. Since $\omega _1,\omega _2,\omega _3$ are all compatible with the Calabi metric $g_{Ca}$, this shows that the dHYM connections distinguish between isometric Kähler structures. A similar result was demonstrated by Lotay-Oliveira in [29] whereby they showed that deformed $\textrm{G}_2$-instantons can be used to distinguish between isometric $\textrm{G}_2$-structures on 3-Sasakian 7-manifolds. In the deformed $\textrm{Spin}(7)$ case for each fixed k we show that:

Theorem 1.2

The moduli spaces of deformed $\textrm{Spin}(7)$-instantons with respect to $\Phi _1,\Phi _2,\Phi _3$ all consist of a 0-dimensional component comprising of the hyper-holomorphic connection and all the other components are 1-dimensional. For $\Phi _1$, there are 2 one dimensional components while for $\Phi _2$ and $\Phi _3$ there are 4 one dimensional components.

An explicit description of these solutions is given in Theorems 5.12 and 5.14. As before we see that the moduli spaces of deformed $\textrm{Spin}(7)$-instantons distinguish between isometric $\textrm{Spin}(7)$-structures. It turns out that all the above deformed $\textrm{Spin}(7)$-instantons also occur as dHYM connections with respect to certain Kähler forms lying in the 2-sphere generated by $\langle \omega _1,\omega _2,\omega _3\rangle$ and also for different phase angle $\theta$, see Remarks 5.13 and 5.15. In particular, this shows that deformed $\textrm{Spin}(7)$-instantons are not equivalent to dHYM connections with phase 1 when M has holonomy contained in $\textrm{SU}(4)$ but is non-compact; this is rather different from the compact situation as shown by Kawai–Yamamoto in [23, Theorem 7.5], see also Theorems 2.4 (2) below. An analogous result was shown by Papoulias in [31] where the author showed that $\textrm{Spin}(7)$-instantons and traceless HYM connections do not coincide on the Calabi-Yau manifold $T^*S^4$; this is again in contrast with the compact case as demonstrated by Lewis in [28, Theorem 3.1], see also [23, Remark 7.6]. This discrepancy is essentially due to the fact that the arguments in the compact cases in [23] and [28] both rely on certain energy estimates and these do not hold in the non-compact setting.

Consider now the Aloff–Wallach space $N_{1,1}=\textrm{SU}(3)/\textrm{U}(1)$ which admits both a 3-Sasakian structure and a strictly nearly parallel $\textrm{G}_2$-structure. The cone metric on the former structure has holonomy $\textrm{Sp}(2)\subset \textrm{Spin}(7)$ while the cone metric for latter has holonomy equal to $\textrm{Spin}(7)$. Denoting the respective $\textrm{Spin}(7)$ 4-forms by $\Phi _{ts}$ and $\Phi _{np}$ we show that:

Theorem 1.3

On the trivial line bundle over $\mathbb {R}^+ \times N_{1,1}$ there is a 2-parameter family of $\textrm{SU}(3)$-invariant deformed $\textrm{Spin}(7)$-instantons with respect to $\Phi _{ts}$ and a 3-parameter family with respect to $\Phi _{np}$.

An explicit description of these solutions is given in Theorems 6.2 and 6.3. More generally, the above also applies to any 3-Sasakian 7-manifold endowed with its squashed nearly parallel $\textrm{G}_2$-structure. Thus, this shows that the deformed $\textrm{Spin}(7)$-instantons can distinguish between the $\textrm{Sp}(2)$ and $\textrm{Spin}(7)$ holonomy cones. Curiously, if we restrict the above deformed $\textrm{Spin}(7)$-instantons to a hypersurface $\{r_0\}\times N_{1,1}$, then it turns out that these coincide, up to constant factor, with the deformed $\textrm{G}_2$-instantons of Lotay–Oliveira in [29, Corollary 4.5], see also Remark 6.4. Note that $N_{1,1}$ can be viewed as an $\textrm{SO}(3)$-bundle over $\mathbb {C}\mathbb {P}^2$ and in fact corresponds to the principal orbit of $T^*\mathbb {C}\mathbb {P}^2$ under the action of $\textrm{SU}(3)$. Thus, the underlying spaces in Theorems 1.1, 1.2 and 1.3 are all diffeomorphic on the complement of the singular orbit $\mathbb {C}\mathbb {P}^2$. Since the Calabi hyperKähler structure converges to the conical 3-Sasakian structure as the $\mathbb {C}\mathbb {P}^2$ shrinks to a point, comparing Theorems 1.2 and 1.3 shows that the dimension of the moduli space can jump under such degenerations.

Organisation of paper. In Sect. 2 we gather some basic facts about $\textrm{Sp}(2)$- and $\textrm{Spin}(7)$-structures on 8-manifolds. We describe how certain irreducible $\textrm{Spin}(7)$-modules split as $\textrm{Sp}(2)$-modules and describe the relation between various special connections in this set up; these will be useful for the computations in Sects. 5 and 6. In Sect. 3 we consider the dHYM equations on the cotangent bundle of $T^*\mathbb {C}\mathbb {P}^1$ endowed with the Eguchi-Hanson hyperKähler metric. The latter is invariant under the cohomogeneity one action of $\textrm{SU}(2)$ and hence this set up provides a simple model for our investigation on $T^*\mathbb {C}\mathbb {P}^2$. We describe all the $\textrm{SU}(2)$-invariant dHYM connections in Proposition 3.2 and Theorem 3.5. As in Theorem 1.1 we show that the number of dHYM connections depends on the Kähler form and also that there is a unique hyper-holomorphic connection arising in the intersection of all the dHYM connections. In Sect. 4 we construct dHYM connections on the flag manifold $\textrm{SU}(3)/\mathbb {T}^2$ endowed with its standard Kähler Einstein structure, see Theorem 4.3. This section serves two main purposes. Firstly it illustrates certain key differences between dHYM connections in the compact and non-compact setting, and secondly it will provide the basic set up for the cohomogeneity one $\textrm{SU}(3)$-invariant connections that we shall investigate in the subsequent sections. In Sect. 5 we consider the dHYM and deformed $\textrm{Spin}(7)$-instantons equations on $T^*\mathbb {C}\mathbb {P}^2$ endowed with the Calabi hyperKähler structure. To this end we begin by classifying the invariant connections on the Aloff-Wallach space $N_{1,1}$ which occurs as the principal orbit for the cohomogeneity one action. Using the latter together with the results of Sect. 2, we compute the dHYM and deformed $\textrm{Spin}(7)$-instantons equations to obtain a coupled system of ODEs. Surprisingly, these turn out to be completely solvable and we are able to find all the solutions explicitly, see Theorems 5.8, 5.10, 5.12 and 5.14. In Sect. 6 we consider the deformed $\textrm{Spin}(7)$-instantons equations on the orbi-spinor bundle of $\mathbb {C}\mathbb {P}^2$ endowed with the Bryant–Salamon $\textrm{Spin}(7)$-structure. The ODEs in this case turn out to be more complicated than in Sect. 5 and as such we are unable to find the general solution. However, for the conical metric and trivial line bundle we are able to construct an explicit 3-parameter family of solutions, see Theorem 6.2. By contrast, we show that for the conical Calabi structure, there is only a 2-parameter family of strict deformed $\textrm{Spin}(7)$-instantons, see Theorem 6.3.

2 Preliminaries

We begin by gathering some basic facts about $\textrm{Sp}(2)$- and $\textrm{Spin}(7)$-structures, and their relation. These will be useful in the subsequent sections to compute the deformed instanton equations.

$\textrm{Sp}(2)$-structures An $\textrm{Sp}(2)$-structure on an 8-manifold $M^8$ is determined by a triple of 2-forms $\omega _1,\omega _2,\omega _3$ which at each point can be expressed as

$$\begin{aligned} \omega _1= & {} dx_{12} + dx_{34} + dx_{56} + dx_{78}, \end{aligned}$$

(2.1)

$$\begin{aligned} \omega _2= & {} dx_{13} + dx_{42} + dx_{57} + dx_{86}, \end{aligned}$$

(2.2)

$$\begin{aligned} \omega _3= & {} dx_{14} + dx_{23} + dx_{58} + dx_{67}, \end{aligned}$$

(2.3)

for some local coordinates $\{x_i\}$. Note here and also subsequently $dx_{i...j}$ will be used as shorthand for $dx_i\wedge \cdots \wedge dx_j$. The $\omega _1,\omega _2,\omega _3$ in turn defines a Riemannian metric g and a triple of almost complex structures $J_i$. More concretely, given any vector field X on $M^8$ we can define $J_i X$ by the relation $X \lrcorner (\omega _j^2-\omega _k^2)/2= J_iX \lrcorner (\omega _j \wedge \omega _k)$, where (i, j, k) denotes a cyclic permutation of (1, 2, 3). The metric can then be defined by

$$\begin{aligned} \omega _i(\cdot ,\cdot )=g(J_i\cdot ,\cdot ). \end{aligned}$$

(2.4)

In particular, we can also define a volume form by ${{\,\textrm{vol}\,}}=\omega _i^4/24$ and hence the Hodge star operator $*$.

$(M^8,g,\omega _1,\omega _2,\omega _3)$ is called a hyperKähler manifold if the holonomy group of its Levi-Civita connection $\nabla$ is contained in $\textrm{Sp}(2)$. The latter can also be characterised by the condition that $\nabla \omega _1=\nabla \omega _2=\nabla \omega _3=0$, or equivalently, $d \omega _1=d \omega _2=d \omega _3=0$ cf. [20].

Given an $\textrm{Sp}(2)$-structure we can decompose the space of k-forms on $M^8$, denoted by $\Lambda ^k$, into irreducible $\textrm{Sp}(2)$-modules. For our purpose in this article we shall only need to consider $\Lambda ^2$ and $\Lambda ^4$. As an $\textrm{Sp}(2)$-module we have that

$$\begin{aligned} \Lambda ^2 = \langle \omega _1,\omega _2,\omega _3 \rangle \oplus E_1 \oplus E_2 \oplus E_3 \oplus \Lambda ^2_{10}, \end{aligned}$$

(2.5)

where

$$\begin{aligned} \Lambda ^2_{10}\ =\,&\{\alpha \in \Lambda ^2 \ | \ J_1(\alpha )=J_2(\alpha )=J_3(\alpha )=\alpha \} \end{aligned}$$

(2.6)

$$\begin{aligned} \ =\,&\{\alpha \in \Lambda ^2 \ | \ *(\alpha \wedge \omega _i^2)=-2\alpha \ \ \forall i \} \end{aligned}$$

(2.7)

i.e. $\Lambda ^2_{10}$ consists of 2-forms of type (1, 1) with respect to $J_1,J_2$ and $J_3$, and similarly we define

$$\begin{aligned} E_i&= \{\alpha \in \Lambda ^2 \ | \ J_i(\alpha )=\alpha ,\ J_j(\alpha )=J_k(\alpha )=-\alpha \text { and } g(\omega _i,\alpha )=0 \} \end{aligned}$$

(2.8)

$$\begin{aligned}&= \{\alpha \in \Lambda ^2 \ | \ *(\alpha \wedge \omega _i^2)=-2\alpha ,\ *(\alpha \wedge \omega _j^2)=+2\alpha ,\ *(\alpha \wedge \omega _k^2)=+2\alpha \}, \end{aligned}$$

(2.9)

where (i, j, k) denotes a cyclic permutation of (1, 2, 3). It is worth pointing out that $\Lambda ^2_{10}$ corresponds to the adjoint representation of $\textrm{Sp}(2)$ while $E_1 \cong E_2 \cong E_2\cong \mathbb {R}^5$ corresponds to the standard representation of $\textrm{SO}(5)\cong \textrm{Spin}(5)/\mathbb {Z}_2\cong \textrm{Sp}(2)/\mathbb {Z}_2$.

Before considering $\Lambda ^4$ we first observe that one can define Lefschetz type operators by wedging with $\omega _i$ to map $\Lambda ^2$ into $\Lambda ^4$, and moreover that these maps are all $\textrm{Sp}(2)$-invariant. Hence this allows us to split $\Lambda ^4$ into irreducible $\textrm{Sp}(2)$-modules as follows:

$$\begin{aligned} \Lambda ^4 = \langle \omega _1^2, \omega _2^2, \omega _3^2, \omega _1 \wedge \omega _2, \omega _1 \wedge \omega _3 , \omega _2 \wedge \omega _3\rangle \oplus \bigoplus _{i=1}^3 F_i^+ \oplus \Lambda ^{4+}_{14} \oplus \Lambda ^{4-}_5 \oplus \bigoplus _{i=1}^3 K_i^- \end{aligned}$$

(2.10)

where

$$\begin{aligned} F_i^+= & {} E_j \wedge \omega _k = E_k \wedge \omega _j, \end{aligned}$$

(2.11)

$$\begin{aligned} \Lambda ^{4+}_{14}= & {} \{\alpha \in \Lambda ^4 \ | \ \alpha \wedge \omega _i =0 \ \ \forall i \}, \end{aligned}$$

(2.12)

$$\begin{aligned} \Lambda ^{4-}_5= & {} E_1 \wedge \omega _1 = E_2 \wedge \omega _2 = E_3 \wedge \omega _3, \end{aligned}$$

(2.13)

$$\begin{aligned} K_i^-= & {} \Lambda ^2_{10} \wedge \omega _i, \end{aligned}$$

(2.14)

and again (i, j, k) denotes a cyclic permutation of (1, 2, 3) in (2.11). To emphasise on the above notation, we describe, for instance, $F^+_1$ more concretely as follows: $F_1^+$ is defined as the space of 4-forms obtained by wedging elements of the irreducible module $E_2$ with $\omega _3$, which equivalently corresponds to the space of 4-forms obtained by wedging elements of $E_3$ with $\omega _2$. The superscripts $+$ and − refer to the fact that these 4-forms are self-dual (i.e. $*\alpha =\alpha$) and anti-self dual (i.e. $*\alpha =-\alpha$) respectively. In particular, we have that $\dim (F_i^+)=\dim (\Lambda ^{4-}_5)=5$, $\dim (K^-_i)=10$ and $\dim (\Lambda ^{4+}_{14})=14$. It is also worth pointing out that $\langle \omega _1^2, \omega _2^2, \omega _3^2\rangle$, $\Lambda ^{4+}_{14}$ and $\Lambda ^{4-}_5$ consist precisely of those 4-forms which are of type (1, 1) with respect to $J_1,J_2$ and $J_3$.

Connections on hyperKähler manifolds Let E be a Hermitian line bundle over a hyperKähler manifold $M^8$ with a Hermitian connection A. Then we say that A is a Hermitian Yang–Mills (HYM) connection with respect to $\omega _i$ if its curvature form $F_A:=dA + A \wedge A$ satisfies

$$\begin{aligned} F_A^{0,2}= & {} 0, \end{aligned}$$

(2.15)

$$\begin{aligned} F_A \wedge \omega _i^3= & {} \lambda \omega ^4_i, \end{aligned}$$

(2.16)

where the (0, 2)-component of $F_A$ is taken here with respect to the complex structure $J_i$ and $\lambda$ is a constant. Note that Eq. (2.15) can also be expressed as

$$\begin{aligned} F_A \in \langle \omega _i \rangle \oplus E_i \oplus \Lambda ^2_{10} \end{aligned}$$

i.e. $J_i(F_A)=F_A$, which is also equivalent to $F_A \wedge (\omega _j+i\omega _k)^2 =0$, and Eq. (2.16) corresponds to the condition that the $\omega _i$ component of $F_A$ is constant. When $\lambda =0$ i.e. $F_A\in E_i \oplus \Lambda ^2_{10}$, such a connection A is called traceless Hermitian Yang–Mills with respect to $\omega _i$. If furthermore, we have that

$$\begin{aligned} F_A \in \Lambda ^2_{10} \end{aligned}$$

i.e. $F_A$ is of type (1, 1) with respect to $J_1$, $J_2$ and $J_3$, then we say that A is a hyper-holomorphic connection. For simplicity, we shall often called a HYM or dHYM connection with respect to the Kähler form $\omega _i$, an $\omega _i$-HYM or $\omega _i$-dHYM connection respectively. Observe that in the case of 8-manifolds, the dHYM equation (1.2) can be expressed as

$$\begin{aligned} 4(\omega ^3 \wedge F_A- \omega \wedge F_A^3) = \tan (\theta ) (\omega ^4 - 6\omega ^2 \wedge F_A^2 + F_A^4). \end{aligned}$$

(2.17)

Remark 2.1

It is not hard to see from the latter expression that if ${A_k}$ denotes a sequence of dHYM connections such that $\Vert F_{A_k}\Vert \rightarrow 0$ then $A_k$ convergences to a HYM connection (with $\lambda$ as in (2.16) determined by $\theta$); this is equivalent to the large volume limit described in [9, Section 1] and [30, Section 5.2]. An analogous result holds for deformed $\textrm{G}_2$-instantons cf. [12, Proposition 4.4] and below we shall see that this holds in the $\textrm{Spin}(7)$ case as well.

$\textrm{Spin}(7)$-structures. A $\textrm{Spin}(7)$-structure on an 8-manifold $M^8$ is determined by a 4-form $\Phi$ which at each point can be expressed as

$$\begin{aligned} + \Phi&= dx_{1234}+dx_{1256}+dx_{1278}+dx_{1357}+dx_{1386}-dx_{1458}-dx_{1467}\ \nonumber \\&\quad dx_{5678}+dx_{3478}+dx_{3456}+dx_{4286}+dx_{4257}-dx_{2367}-dx_{2358}, \end{aligned}$$

(2.18)

for some local coordinates $\{x_i\}$. Since $\textrm{Spin}(7)\subset \textrm{SO}(8)$ it follows that $\Phi$ defines both a Riemannian metric and orientation; in particular, we can again define the Hodge star operator $*$. $(M^8,\Phi )$ is called a $\textrm{Spin}(7)$-manifold if the holonomy of the associated Levi-Civita connection $\nabla$ is contained in $\textrm{Spin}(7)$. The latter can also be characterised by the condition that $\nabla \Phi =0$, or equivalently $d\Phi =0$ cf. [4, 32].

Similarly as in the $\textrm{Sp}(2)$ case, we can decompose $\Lambda ^2$ into irreducible $\textrm{Spin}(7)$ modules as:

$$\begin{aligned} \Lambda ^2 = \Lambda ^2_7 \oplus \Lambda ^2_{21}, \end{aligned}$$

(2.19)

where

$$\begin{aligned} \Lambda ^2_7&=\{\alpha \in \Lambda ^2\ |\ *(\alpha \wedge \Phi )=3\alpha \}, \end{aligned}$$

(2.20)

$$\begin{aligned} \Lambda ^2_{21}&=\{\alpha \in \Lambda ^2\ |\ *(\alpha \wedge \Phi )=-\alpha \}. \end{aligned}$$

(2.21)

Also in analogy with the $\textrm{Sp}(2)$ case, we have that $\Lambda ^2_{21}$ corresponds to the adjoint representation of $\textrm{Spin}(7)$ while $\Lambda ^2_7\cong \mathbb {R}^7$ corresponds to the standard representation of $\textrm{SO}(7)\cong \textrm{Spin}(7)/\mathbb {Z}_2$.

On the other hand, we have that $\Lambda ^4$ splits as

$$\begin{aligned} \Lambda ^4 = \langle \Phi \rangle \oplus \Lambda ^4_7 \oplus \Lambda ^4_{27} \oplus \Lambda ^4_{35}, \end{aligned}$$

(2.22)

where

$$\begin{aligned} \Lambda ^4_{35} =\{\alpha \in \Lambda ^4\ |\ * \alpha =-\alpha \} \end{aligned}$$

(2.23)

and $\Lambda ^4_7$ is defined as the span of the action of $\Lambda ^2_7\hookrightarrow \Lambda ^2 \cong \mathfrak {so}(8) \hookrightarrow \mathfrak {gl}(8)$, viewed as endomorphisms, on $\Phi$. More concretely, denoting the endomorphism action by $\diamond$, we have that for a simple 2-form $\alpha \wedge \beta$ its action on $\Phi$ is given by $(\alpha \wedge \beta ) \diamond \Phi := \alpha \wedge (\beta ^{\sharp } \lrcorner \Phi ) - \beta \wedge (\alpha ^{\sharp } \lrcorner \Phi )$. Finally $\Lambda ^4_{27}$ is simply defined as the orthogonal complement of $\langle \Phi \rangle \oplus \Lambda ^4_7\oplus \Lambda ^4_{35}$ in $\Lambda ^4$ cf. [4].

We shall denote by $\pi ^k_l$ the orthogonal projection of a k-form into the space $\Lambda ^k_l$. Of particular interest to us are the projection maps $\pi ^2_7$ and $\pi ^4_7$ since these occur in the definition of the deformed $\textrm{Spin}(7)$-instanton equations (1.3) and (1.4). In view of the above characterisation of $\Lambda ^2_7$ and $\Lambda ^2_{21}$, it follows that the projection map $\pi ^2_7$ can be expressed as

$$\begin{aligned} \pi ^2_7(\alpha ) =\frac{1}{4} (\alpha +*(\alpha \wedge \Phi )). \end{aligned}$$

(2.24)

In contrast we do not have such a nice description for the projection map $\pi ^4_{7}$ since the definition of the space $\Lambda ^4_7$ depends on $\Lambda ^2_7$ and its action on $\Phi$ by the operator $\diamond$. Below we shall give a more practical method of computing the map $\pi ^4_7$ in our framework.

Connections on $\textrm{Spin}(7)$ manifolds. Let E be a vector bundle over a $\textrm{Spin}(7)$ manifold $M^8$ with a connection A. Then we say that A is a $\textrm{Spin}(7)$-instanton if

$$\begin{aligned} \pi ^2_7(F_A) =0 \Longleftrightarrow *(F_A \wedge \Phi )=-F_A \end{aligned}$$

(2.25)

i.e. $F_A \in \Lambda ^2_{21}$. It is worth pointing out that as $\textrm{Spin}(7)$ modules we have that

$$\begin{aligned} S^2(\Lambda ^2_{21}) \cong \langle \Phi \rangle \oplus \Lambda ^4_{27} \oplus \Lambda ^4_{35}\oplus V_{0,2,0}, \end{aligned}$$

where $V_{0,2,0}$ denotes an irreducible 168-dimensional space cf. [4]. Since latter decomposition contains no 7-dimensional module, it follows that $\pi ^4_{7}(F_A \wedge F_A)=0$ for any $\textrm{Spin}(7)$-instanton A; observe that the latter equation is precisely (1.3). As in the dHYM case (see Remark 2.1), it is now easy to see from (1.4) that if $A_k$ denotes a sequence of deformed $\textrm{Spin}(7)$-instanton such that $\Vert F_{A_k}\Vert \rightarrow 0$ then we have that $A_k$ converges to a $\textrm{Spin}(7)$-instanton.

Remark 2.2

Recall that the deformed $\textrm{Spin}(7)$-instanton equations are given by the pair (1.3)-(1.4) and that these two equations are in general independent of each other. However, Kawai-Yamamoto showed in [22, Proposition 3.3] that if A satisfies (1.4) and additionally we have that $*F_A^4\ne 24$ i.e. $*F_A^4- 24$ is a nowhere vanishing function, then (1.3) automatically holds. The latter result can be interpreted as the mirror version of Theorem 2.20 in [15, Chapter IV] for Cayley submanifolds. More precisely, Cayley graphs in $\mathbb {R}^8$ are defined by mirror equations to (1.3), (1.4), these correspond to (1)–(2) in [15, Theorem 2.20], and Harvey–Lawson showed therein that (1) implies (2) provided the mirror equation to ‘$*F_A^4\ne 24$’ holds. Harvey-Lawson also showed in [15, Pg. 132] that there are in fact explicit examples when (the mirror condition to) ‘$*F_A^4=24$’ occurs and hence this shows that indeed (1) does not imply (2) in general. In Sect. 5 we shall show that there exist explicit examples of the deformed $\textrm{Spin}(7)$-instantons satisfying $*F_A^4=24$ and as such we cannot appeal to [22, Proposition 3.3]. In particular, our examples illustrate that the mirror picture is again consistent.

Inclusion of $\textrm{Sp}(2)$ in $\textrm{Spin}(7)$. Given an $\textrm{Sp}(2)$-structure it is easy to see by comparing the pointwise models given in (2.1), (2.3) and (2.18) that we can define a $\textrm{Spin}(7)$ 4-form by

$$\begin{aligned} \Phi \ {}&=\frac{1}{2}(\omega _1^2+\omega _2^2-\omega _3^2). \end{aligned}$$

(2.26)

This exhibits $\textrm{Sp}(2)$ as a subgroup of $\textrm{Spin}(7)$. Note however that $\textrm{Sp}(2)$ can be embedded in $\textrm{Spin}(7)$ in many ways; these embeddings are parametrised by sections of a bundle whose fibre is isomorphic to the homogeneous space $\textrm{Spin}(7)/\textrm{Sp}(2)$. Given a hyperKähler triple $\langle \omega _1, \omega _2, \omega _3\rangle$, in this paper we shall be interested in the ‘canonical’ triple of $\textrm{Spin}(7)$ 4-forms defined by

$$\begin{aligned} \Phi _i := \frac{1}{2}(-\omega _i^2+\omega _j^2+\omega _k^2), \end{aligned}$$

(2.27)

where (i, j, k) denotes as usual a cyclic permutation of (1, 2, 3). Observe that $\Phi _3$ corresponds precisely to (2.26). Note also that while each $\Phi _i$ induces the same metric and orientation on $M^8$ (which of course coincide with hyperKähler one), they are all nonetheless distinct as $\textrm{Spin}(7)$-structures and hence, in particular, will have different $\textrm{Spin}(7)$- and deformed $\textrm{Spin}(7)$-instantons.

If the $\textrm{Spin}(7)$ 4-form is given by (2.27) then we can express the irreducible $\textrm{Spin}(7)$ summands of $\Lambda ^2$ and $\Lambda ^4$ in terms of the irreducible $\textrm{Sp}(2)$-modules. A straightforward computation and application of Schur’s lemma gives the next result.

Proposition 2.3

The irreducible $\textrm{Spin}(7)$ summands of $\Lambda ^2$ and $\Lambda ^4$ with respect to $\Phi _3$, split into $\textrm{Sp}(2)$-modules as follows:

$$\begin{aligned} \Lambda ^2_7= & {} \langle \omega _1, \omega _2 \rangle \oplus E_3\\ \Lambda ^2_{21}= & {} \langle \omega _3 \rangle \oplus E_1 \oplus E_2 \oplus \Lambda ^2_{10} \\ \Lambda ^4_7= & {} \langle \omega _1 \wedge \omega _3, \omega _2 \wedge \omega _3 \rangle \oplus F^+_3 \\ \Lambda ^4_{27}= & {} \langle \omega _1^2+3\ \!\omega _3^2, \omega _2^2+3\ \!\omega _3^2, \omega _1 \wedge \omega _2 \rangle \oplus F_1^+ \oplus F_2^+ \oplus \Lambda ^{4+}_{14}\\ \Lambda ^4_{35}= & {} \Lambda ^{4-}_5 \oplus K_1^- \oplus K_2^- \oplus K_3^- \end{aligned}$$

If the $\textrm{Spin}(7)$ 4-form is instead given by $\Phi _1$ or $\Phi _2$, then it suffices to cyclically permute the indices 1, 2, 3 of the $\textrm{Sp}(2)$-modules in the above decompositions.

From the definitions of $E_i$ and the irreducible $\textrm{Sp}(2)$ summands of $\Lambda ^4$, it follows for instance that $F_3^+$ can also be characterised as

$$\begin{aligned} F_3^+= \{\alpha \in \Lambda ^4 \ |\ *\alpha =\alpha , \text { } \ J_3(\alpha )=\alpha ,\ J_1(\alpha )=J_2(\alpha )=-\alpha \text { and } g(\alpha ,\omega _1\wedge \omega _2)=0\}. \end{aligned}$$

(2.28)

This provides a rather practical algorithm to compute the $F_3^+$-component of an arbitrary 4-form $\alpha$: first we self-dualise it by $\alpha \mapsto \pi ^{4+}(\alpha ):=(\alpha + * \alpha )/2$, then we map it to $\pi ^{4+}(\alpha ) - J_1(\pi ^{4+}(\alpha ))-J_2(\pi ^{4+}(\alpha ))+J_3(\pi ^{4+}(\alpha ))$ to obtain a self-dual 4-form invariant by $J_3$ but on which $J_1$ and $J_2$ act as $-\textrm{Id}$, and finally we project the latter in the orthogonal complement of $\omega _1 \wedge \omega _2$. Together with the components of $\alpha$ in $\omega _1 \wedge \omega _3$ and $\omega _2 \wedge \omega _3$, this allows us to compute the $\textrm{Spin}(7)$ projection map $\pi ^4_7:\Lambda ^4 \rightarrow \Lambda ^4_7$.

Inclusion of connections. We now consider the special connections on $(M^8,\omega _1,\omega _2,\omega _3)$ as connections on a manifold with a $\textrm{Spin}(7)$-structure given by (2.27). From Proposition 2.3 it is easy to see that A is a traceless $\omega _i$-HYM connection if and only if it is a $\textrm{Spin}(7)$-instanton both with respect to $\Phi _j$ and $\Phi _k$. Moreover, A is a hyper-holomorphic connection if and only if it is a $\textrm{Spin}(7)$-instanton with respect to the triple $\Phi _1,\Phi _2$ and $\Phi _3$. We state the analogous results demonstrated by Kawai–Yamamoto in [24, Lemma 7.2] and [23, Theorem 7.5] which relate dHYM connections and deformed $\textrm{Spin}(7)$-instantons in our set up:

Theorem 2.4

Let $(M^8,\omega _1,\omega _2,\omega _3)$ denote a hyperKähler manifold, A a connection on a line bundle L, and $\Phi _i$ the $\textrm{Spin}(7)$ 4-form defined by (2.27). Then the following hold:

• (1) If A is a Hermitian connection such that the (0, 2)-part of $F_A$ with respect to $J_i$ vanishes, then A is an $\omega _i$-dHYM connection with phase 1 if and only if A is a deformed $\textrm{Spin}(7)$-instanton with respect to $\Phi _k$ (or equivalently $\Phi _j$).

• (2) If $M^8$ is a compact connected manifold, and there exists an $\omega _i$-dHYM connection with phase 1, then any deformed $\textrm{Spin}(7)$-instanton with respect to $\Phi _k$ (or equivalently $\Phi _j$) is a $\omega _i$-dHYM connection with phase 1.

Note that compactness assumption in Theorem 2.4 (2) is indeed a necessary condition as our examples in Sect. 5 shall demonstrate. Next, in the same spirit as [29, Example 3.4], we describe the simplest examples of deformed instantons in our context.

Elementary examples Let M be a hyperKähler manifold as above and suppose that $[c \omega _i]\in H^2(M,\mathbb {Z})$ for some constant $c\in \mathbb {R}$. Then this defines a line bundle L on M and there exists a unitary connection A such that $F_A=c \omega _i$. It is easy to see from (2.17) that such a connection A is an $\omega _i$-dHYM connection with phase 1 if and only if $c=0,\pm 1$. From Proposition 2.3 we also see that A is a deformed $\textrm{Spin}(7)$-instanton with respect to $\Phi _i$ for any c. In particular, together with Proposition 2.3 we deduce that A is both a $\textrm{Spin}(7)$-instanton and deformed $\textrm{Spin}(7)$-instanton with respect to $\Phi _i$ for any c. This shows that there is a non-trivial overlap of $\textrm{Spin}(7)$-instantons and deformed $\textrm{Spin}(7)$-instantons on Calabi Yau 4-folds. On the other hand, A is a deformed $\textrm{Spin}(7)$-instanton with respect to $\Phi _j$ or $\Phi _k$ if and only if $c=0,\pm 1$ but is a $\textrm{Spin}(7)$-instanton if and only if $c=0$.

Note that it is not generally true that if A is a (traceless) HYM connection then it is dHYM connection, likewise a $\textrm{Spin}(7)$-instanton is not generally a deformed $\textrm{Spin}(7)$-instanton. However, we do have that:

Proposition 2.5

If A is a hyper-holomorphic connection then A is a $\omega _i$-dHYM connection with phase 1 for $i=1,2,3$.

Proof

By assumption we have that $F_A \in \Lambda ^2_{10}\cong \mathfrak {sp}(2)$ and hence it follows that $F_A \wedge \omega ^3_i=0$. On the other hand we have that the $\textrm{Sp}(2)$ module $S^3(\mathfrak {sp}(2))$ i.e. the 3-symmetric product of $\mathfrak {sp}(2)$, consists of 5 irreducible modules, 2 of which correspond to $\mathfrak {sp}(2)$ while the remaining 3 modules have dimensions 35, 81 and 84. Hence we deduce that $*F_A^3\in \Lambda ^2_{10}$ and in particular, $F_A^3 \wedge \omega _i=0$. Thus, we have shown that (2.17) holds with $\theta =0$ and this concludes the proof. $\square$

In particular, Proposition 2.5 also implies that hyper-holomorphic connections are both $\textrm{Spin}(7)$- and deformed $\textrm{Spin}(7)$-instantons. In other words, hyper-holomorphic connections lie in the intersection of all these special classes of connections. With these preliminaries in mind we next proceed to consider the dHYM equations on $T^*\mathbb {C}\mathbb {P}^1$.

3 Deformed Hermitian Yang–Mills Connections on $T^*\mathbb {C}\mathbb {P}^1$

In this section we construct dHYM connections on the cotangent bundle of $\mathbb {C}\mathbb {P}^1$ endowed with the Eguchi-Hanson hyperKähler structure [11]. This set up provides a lower dimensional model of the situation we shall investigate in Sect. 5 and hence it will serve to illustrate certain key similarities and differences. It is well-known that the Eguchi-Hanson metric is invariant under a cohomogeneity one action of $\textrm{SU}(2)$. Hence the problem of constructing $\textrm{SU}(2)$-invariant dHYM connections reduces to solving a non-linear system of ODEs. We shall show that the solutions to these ODEs are generally implicitly given by certain polynomials of degree equal to n, where n is as in (1.2), and the coefficients are functions of the geodesic distance. This feature will also occur in the Sect. 5 but the equations will be more involved since in that case $n=4$. It is worth pointing out that such a phenomenon was also observed in [18] whereby the authors construct dHYM connections on the blow up of $\mathbb {C}\mathbb {P}^n$ (which in contrast to our set up is a compact manifold but similar to our set up also admits a cohomogeneity one action).

Let $\eta _i$ denote the standard left invariant coframing of $S^3=\textrm{SU}(2)$ satisfying $d\eta _i=\sum _{j,k}\varepsilon _{ijk}\eta _{jk}/2$, where $\varepsilon _{ijk}$ denotes the Levi-Civita symbol and $\eta _{jk}$ is shorthand for $\eta _j \wedge \eta _k$. The Eguchi–Hanson metric on $T^*\mathbb {C}\mathbb {P}^2\cong \mathcal {O}_{\mathbb {C}\mathbb {P}^1}(-2)$ can then be expressed as

$$\begin{aligned} g_{EH}=\Big (1-\frac{c}{r^4}\Big )^{-1} dr^2 + \frac{r^2}{4}\Big (1-\frac{c}{r^4}\Big ) \eta _1^2+ \frac{r^2}{4} (\eta _{2}^2+\eta _{3}^2), \end{aligned}$$

(3.1)

where c is a positive constant determining the size of the zero section $\mathbb {C}\mathbb {P}^1$ and $r\in [c^{1/4},\infty )$. Note that when $c=0$ the above metric corresponds to the flat metric on $\mathbb {C}^2/\mathbb {Z}_2$ and the Eguchi-Hanson space can be viewed as the resolution of this orbifold singularity obtained by blowing up the origin.

The associated hyperKähler triple to the Eguchi-Hanson metric are given by

$$\begin{aligned} \omega _1= & {} \frac{1}{2} r dr \wedge \eta _1+\frac{r^2}{4}\eta _{23}, \end{aligned}$$

(3.2)

$$\begin{aligned} \omega _2= & {} \frac{r}{2}\Big (1-\frac{c}{r^4}\Big )^{-1/2}dr \wedge \eta _2+\frac{r^2}{4}\Big (1-\frac{c}{r^4}\Big )^{1/2} \eta _{31}, \end{aligned}$$

(3.3)

$$\begin{aligned} \omega _3= & {} \frac{r}{2}\Big (1-\frac{c}{r^4}\Big )^{-1/2}dr \wedge \eta _3+\frac{r^2}{4}\Big (1-\frac{c}{r^4}\Big )^{1/2}\eta _{12}. \end{aligned}$$

(3.4)

One can indeed easily verify that $\nabla \omega _i=0$, or equivalently, that $d\omega _i=0$ for $i=1,2,3$. Note that the Kähler form $\omega _1$ generates the cohomology group $H^2(T^*\mathbb {C}\mathbb {P}^1,\mathbb {Z})\cong \mathbb {Z}$; in particular, $\omega _1$ is not exact. By contrast, we have that $\omega _2=d({\sqrt{r^4-c}}{}\ \eta _2/4)$ and $\omega _3=d({\sqrt{r^4-c}}{}\ \eta _3/4)$. The principal orbits of the $\textrm{SU}(2)$ action on $T^*\mathbb {C}\mathbb {P}^1$ are all diffeomorphic to $\textrm{SO}(3)\cong \textrm{SU}(2)/\mathbb {Z}_2\cong \mathbb {R}\mathbb {P}^3$ while the singular orbit is $S^2\cong \textrm{SU}(2)/\textrm{U}(1)$ (this corresponds to the zero section of $T^*\mathbb {C}\mathbb {P}^1$ at $r=c^{1/4}$). Strictly speaking, since $\textrm{SO}(3)$ is the $\mathbb {Z}_2$ quotient of $\textrm{SU}(2)$, we shall also consider $\eta _i$ as a global coframing on $\textrm{SO}(3)$ by suitably truncating the range of one of the Euler angles defining $\eta _i$ on $S^3$ cf. [11, Section IIC]; this is indeed implicit in the expressions (3.1)–(3.4).

A general $\textrm{SU}(2)$-invariant abelian connection 1-form over the open set $\mathcal {O}_{\mathbb {C}\mathbb {P}^1}(-2)\backslash \mathbb {C}\mathbb {P}^1 \cong \mathbb {R}_{r>c^{1/4}} \times \textrm{SO}(3)$ can be expressed as

$$\begin{aligned} A= \sum _{i=1}^3f_i(r)\eta _i, \end{aligned}$$

(3.5)

where $f_i$ are arbitrary functions of $r\in (c^{1/4},\infty ).$ Note that here we are expressing A without a ‘$f_0(r)dr$’ term since we can always set $f_0=0$ by a suitable gauge transformation; in other words we are expressing the connection A in temporal gauge. A simple computation now shows that:

Proposition 3.1

The curvature form of A, as defined by (3.5), is given by

$$\begin{aligned} F_A = dr \wedge (f_1'\eta _1+f_2'\eta _2+f_3'\eta _3)+f_1\eta _{23}+f_2\eta _{31}+f_3\eta _{12}. \end{aligned}$$

(3.6)

Since $H^2(\textrm{SO}(3),\mathbb {Z})\cong \mathbb {Z}_2$ there are only 2 complex line bundles over $\textrm{SO}(3)$: the trivial one and a non-trivial one (which squares to the trivial one). The connection form A is defined over the former bundle when $\eta _i$ are viewed as a coframing of $\textrm{SO}(3)$ and over the latter bundle when $\eta _i$ are viewed as a coframing of the double cover $S^3$ instead. In any case, the space of line bundles over $T^*\mathbb {C}\mathbb {P}^1$ are parametrised by $H^2(T^*\mathbb {C}\mathbb {P}^1,\mathbb {Z})\cong H^2(\mathbb {C}\mathbb {P}^1,\mathbb {Z})\cong \mathbb {Z}.$ As can easily be seen from (3.2) the latter cohomology class is generated by the Fubini-Study form of the zero section which, without loss of generality, we can identify with $k[\eta _{23}]=k[d\eta _1]$, where $k\in \mathbb {Z}$. Thus, in order for A(r) to extend to a connection across the zero section $\mathbb {C}\mathbb {P}^1$, we need that $f_1(c^{1/4})=k$ and $f_2(c^{1/4})=f_3(c^{1/4})=0$, in which case the bundle over this $\mathbb {C}\mathbb {P}^1$ will be diffeomorphic to $\mathcal {O}_{\mathbb {C}\mathbb {P}^1}(k).$ With this in mind we can now proceed to the problem of constructing instantons on $T^*\mathbb {C}\mathbb {P}^1$.

The dHYM equations (1.1), (1.2) with respect to the Kähler form $\omega _i$ are given by the pair

$$\begin{aligned} F_A \wedge \omega _j= & {} F_A \wedge \omega _k=0, \end{aligned}$$

(3.7)

$$\begin{aligned} 2 F_A \wedge \omega _i= & {} \tan (\theta ) (\omega _i^2-F_A^2), \end{aligned}$$

(3.8)

where (i, j, k) denotes as usual cyclic permutations of (1, 2, 3). Note that on a compact Kähler surface, Jacob-Yau showed that the existence of dHYM connections are equivalent a certain notion of stability on L cf. [19, Theorem 1.2], see also [7]; however, we cannot appeal to their result since $T^*\mathbb {C}\mathbb {P}^1$ is a non-compact manifold. From (3.7) and (3.8) we see that when $\tan (\theta )=0$, the dHYM equation is equivalent to asking that $F_A$ is the curvature form of a hyper-holomorphic line bundle i.e. $F_A \wedge \omega _i =0$ for $i=1,2,3$, or equivalently, $F_A \in \Lambda ^2_-\cong \mathfrak {su}(2)$ i.e. A is an anti-self-dual instanton. For our purpose, it is worth pointing out that there are many such explicit examples on non-compact hyperKähler manifolds owing to the results of Hitchin in [17] (see also Remark 3.3 below). Note that the fact that phase 1 dHYM connections coincide with traceless HYM connections only occurs in dimension 4.

For the sake of comparison with the usual HYM condition we first begin by describing all such examples in our set up.

Proposition 3.2

Let $C_1,C_2, C_3$ be real constants, then on the open set $\mathcal {O}_{\mathbb {C}\mathbb {P}^1}(-2)\backslash \mathbb {C}\mathbb {P}^1 \cong \mathbb {R}_{r>c^{1/4}}\times \textrm{SO}(3)$ we have that A as given by (3.5) is a HYM connection with respect to

• (1) $\omega _1$ such that $F_A \wedge \omega _1 = \lambda \omega _1 \wedge \omega _1$ if and only if

  $$\begin{aligned} A = \frac{\lambda r^4+C_1}{4 r^2}\eta _1+\frac{C_2}{\sqrt{r^4-c}}\eta _2+\frac{C_3}{\sqrt{r^4-c}}\eta _3. \end{aligned}$$

  In particular, if $C_1=C_2=C_3=0$ then $F_A = \lambda \omega _1$.

• (2) $\omega _2$ such that $F_A \wedge \omega _2 = \lambda \omega _1 \wedge \omega _1$ if and only if

  $$\begin{aligned} A = \frac{C_1}{r^2}\eta _1+\frac{\lambda (r^4-c)+C_2}{4\sqrt{r^4-c}}\eta _2+\frac{C_3}{\sqrt{r^4-c}}\eta _3. \end{aligned}$$

  In particular, if $C_1=C_2=C_3=0$ then $F_A = \lambda \omega _2$.

• (3) $\omega _3$ such that $F_A \wedge \omega _3 = \lambda \omega _1 \wedge \omega _1$ if and only if

  $$\begin{aligned} A = \frac{C_1}{r^2}\eta _1+\frac{C_2}{\sqrt{r^4-c}}\eta _2+\frac{\lambda ( r^4-c)+C_3}{4\sqrt{r^4-c}}\eta _3. \end{aligned}$$

  In particular, if $C_1=C_2=C_3=0$ then $F_A = \lambda \omega _3$.

When $\lambda =0$, A is a hyper-holomorphic connection. If $C_2$ or $C_3$ is non-zero, the above connections do not extend smoothly across the zero section $\mathbb {C}\mathbb {P}^1$ but instead these correspond to meromorphic HYM connections which blow up along the zero section of $T^*\mathbb {C}\mathbb {P}^1$. Thus, there is (up to scaling) a unique globally well-defined hyper-holomorphic connection on $T^*\mathbb {C}\mathbb {P}^1$, namely $A=\eta _1/r^2$.

Proof

The first part is a straightforward computation using (3.6) and (3.7), (3.8) with $\theta =0$; we refer the reader to the proof of Proposition 3.4 below for the details. As discussed above the locally defined connection A on $\mathcal {O}_{\mathbb {C}\mathbb {P}^1}(-2)\backslash \mathbb {C}\mathbb {P}^1$ extends to $T^*\mathbb {C}\mathbb {P}^2$ when the coefficient functions of $\eta _{2}$ and $\eta _{3}$ vanish at $r=c^{1/4}$ and this proves the last statement. $\square$

Remark 3.3

In [17, Sect. 2.2] Hitchin showed that the aforementioned globally well-defined connection is in the fact the unique $S^1$-invariant hyper-holomorphic connection on $T^*\mathbb {C}\mathbb {P}^1$ which coincides with the Fubini-Study form on the zero section. Here the $S^1$ action is generated by the left-invariant vector field $X_1$ which is dual to $\eta _1$ i.e. $\eta _i(X_1)=\delta _{i1}$. This Killing circle action is permuting i.e. $\mathcal {L}_{X_1}\omega _1=0$ but $\mathcal {L}_{X_1}\omega _2=-\omega _3$ and $\mathcal {L}_{X_1}\omega _3=+\omega _2$. Such examples of hyper-holomorphic connections arising from permuting Killing vector fields were first constructed by Haydys in [16]. In particular, this applies to any hyperKähler manifold of the form $T^*(G/H)$, where G/H is a Hermitian symmetric space and the permuting $S^1$ action corresponds to a rotation in the fibres. We shall again encounter this hyper-holomorphic connection in the next section where we shall have that $G/H=\textrm{SU}(3)/\textrm{U}(2)\cong \mathbb {C}\mathbb {P}^2$.

Next we consider the dHYM equations.

Proposition 3.4

Let $C_1,C_2, C_3$ be real constants, then on the open set $\mathcal {O}_{\mathbb {C}\mathbb {P}^1}(-2)\backslash \mathbb {C}\mathbb {P}^1\cong \mathbb {R}_{r>c^{1/4}}\times \textrm{SO}(3)$ the connection form A as given by (3.5) is a dHYM connection with respect to

• (1) $\omega _1$ if and only if $f_2(r)=\frac{C_2}{\sqrt{r^4-c}}$, $f_3(r)=\frac{C_3}{\sqrt{r^4-c}}$ and $f_1(r)$ is implicitly defined by

  $$\begin{aligned} 2\tan (\theta )f^2_1+r^2f_1-\frac{1}{8}\tan (\theta )\Big (r^4-\frac{16(C_2^2+C_3^2)}{r^4-c}\Big )=C_1. \end{aligned}$$

  (3.9)

• (2) $\omega _2$ if and only if $f_1(r)=\frac{C_1}{r^2}$, $f_3(r)=\frac{C_3}{\sqrt{r^4-c}}$ and $f_2(r)$ is implicitly defined by

  $$\begin{aligned} 2\tan (\theta )f^2_2+\sqrt{r^4-c}f_2+2\tan (\theta )\Bigg (\frac{C_1^2}{r^4}+\frac{C_3^2}{r^4-c}-\frac{r^4}{16}\Bigg )=C_2. \end{aligned}$$

  (3.10)

• (3) $\omega _3$ if and only if $f_1(r)=\frac{C_1}{r^2}$, $f_2(r)=\frac{C_2}{\sqrt{r^4-c}}$ and $f_3(r)$ is implicitly defined by

  $$\begin{aligned} 2\tan (\theta )f^2_3+\sqrt{r^4-c}f_3+2\tan (\theta )\Bigg (\frac{C_1^2}{r^4}+\frac{C_2^2}{r^4-C}-\frac{r^4}{16}\Bigg )=C_3. \end{aligned}$$

  (3.11)

Proof

We begin with case $\mathrm {(1)}$. A direct computation using (3.2)–(3.4) and (3.6) shows that $F_A\wedge \omega _2=F_A \wedge \omega _3=0$ is equivalent to

$$\begin{aligned} (r^4-c)f_i'=-2r^3f_i, \end{aligned}$$

for $i=2,3$ and solving the latter we get $f_2(r)=\frac{C_2}{\sqrt{r^4-c}}$, $f_3(r)=\frac{C_3}{\sqrt{r^4-c}}$. Using (3.2) and (3.6), we find that

$$\begin{aligned} F_A \wedge \omega _1 = \frac{2}{r^2}(rf_1'+2f_1)\ \omega _1 \wedge \omega _1 \end{aligned}$$

and

$$\begin{aligned} \omega _1^2-F_A^2=(-\frac{8f_1f_1'}{r^3}+(1+16\frac{C_2^2+C_3^2}{(r^4-c)^2}))\ \omega _1 \wedge \omega _1. \end{aligned}$$

Thus, it follows that (3.8) is given by the non-linear ODE:

$$\begin{aligned} 2(r^2+4f_1 \tan (\theta ))f_1'+4rf_1-r^3\Big (1+16\frac{C_2^2+C_3^2}{(r^4-c)^2}\Big )\tan (\theta )=0. \end{aligned}$$

A close inspection of the latter shows that it is in fact an exact ODE. Indeed the reader will find no difficulty in verifying that differentiating (3.9) yields the latter ODE. Thus, this proves the first part.

For case (2), by an analogous calculation as above we find that $F_A \wedge \omega _1=F_A \wedge \omega _3 =0$ if and only if $f_1=\frac{C_1}{r^2}$ and $f_3=\frac{C_3}{\sqrt{r^4}-c}$. On the other hand from (3.8) we find that $f_2(r)$ has to solve the non-linear ODE:

$$\begin{aligned} (4\tan (\theta )f_2+\sqrt{r^4-c})f_2'+\frac{2r^3}{\sqrt{r^4-c}}f_2-2\Bigg (\frac{4 C_1^2}{r^5}+\frac{4C_3^2 r^3}{(r^4-c)^2}+\frac{r^3}{4}\Bigg )\tan (\theta )=0. \end{aligned}$$

Again it is not hard to see that the latter is an exact ODE and that the general solution is given by (3.10). Case (3) is identical by swapping $f_2$ and $f_3$, and this concludes the proof. $\square$

Since the $\tan (\theta )=0$ case is already covered by Proposition 3.2 we shall now restrict to the case when $\tan (\theta )\ne 0$. Observe that in this case the solutions $f_i$ are implicitly defined as the solutions to certain quadratic polynomials (and that these reduce to linear equations precisely when $\tan (\theta )=0$). In order to determine which of the above locally defined dHYM connections extend to $T^*\mathbb {C}\mathbb {P}^1$, we need to impose the boundary conditions to the functions $f_i(r)$ at $r=c^{1/4}$. Moreover, we also need to verify that these are indeed well-defined for all values of $r\in [c^{1/4},\infty )$ since this is not a priori obvious from Proposition 3.4.

Theorem 3.5

Let $k\in \mathbb {Z}$ and suppose that $\tan (\theta ) \ne 0$, then on $T^*\mathbb {C}\mathbb {P}^1$ the connection form A as given by (3.5) is a dHYM connection with respect to

• (1) $\omega _1$ if and only if $f_2(r)=f_3(r)=0$ and

  $$\begin{aligned} f_1(r)=\frac{-r^2\pm \sqrt{\sec ^2(\theta )r^4+\tan ^2(\theta )(16k^2-c)+8kc^{1/2}\tan (\theta )}}{4\tan (\theta )}, \end{aligned}$$

  (3.12)

  where we only take $+(-)$ if $4k\tan (\theta )+c^{1/2}>(<)0$.

• (2) $\omega _2$ if and only if $f_1(r)=\frac{c^{1/2}k}{r^2}$, $f_3(r)=0$ and

  $$\begin{aligned} f_2(r)=\sqrt{r^4-c}\ \Bigg (\frac{-r^2\pm \sqrt{\sec ^2(\theta )r^4+16k^2\tan ^2(\theta )}}{4r^2\tan (\theta )}\Bigg ). \end{aligned}$$

  (3.13)

• (3) $\omega _3$ if and only if $f_1(r)=\frac{c^{1/2}k}{r^2}$, $f_2(r)=0$ and

  $$\begin{aligned} f_3(r)=\sqrt{r^4-c}\ \Bigg (\frac{-r^2\pm \sqrt{\sec ^2(\theta )r^4+16k^2\tan ^2(\theta )}}{4r^2\tan (\theta )}\Bigg ). \end{aligned}$$

  (3.14)

Proof

Imposing the conditions that $f_2(c^{1/4})=f_3(c^{1/4})=0$ and $f_1(c^{1/4})=k$ in solution (1) of Proposition 3.4 gives that $C_2=C_3=0$ and

$$\begin{aligned} C_1=2\tan (\theta ) k^2+c^{1/2}k-\frac{c}{8}\tan (\theta ). \end{aligned}$$

With $C_1$ as above and assuming $\tan (\theta )\ne 0$, we solve for $f_1$ in (3.9) to find (3.12). It is not hard to see from (3.12) that $f_1(r)$ is indeed well-defined for all values of $r\in [c^{1/4},\infty )$. Next we consider case (2).

Imposing the same boundary conditions as before in solution (2) of Proposition 3.4 we now get that $C_1=c^{1/2}k$, $C_2=(16k^2-c)\tan (\theta )/8$ and $C_3=0$. Assuming that $\tan (\theta )\ne 0$, we solve for $f_2$ in (3.10) to find (3.13). Again it is clear that $f_2(r)$ is well-defined for all values of $r\in [c^{1/4},\infty )$. The same argument applies for case (3) and this concludes the proof. $\square$

Remark 3.6

• (1) In case (1) of Theorem 3.5, for generic values of k and $\theta$ there is exactly one dHYM connection. Indeed when $r=c^{1/4}$, the term under the square root is given by $(4k\tan (\theta )+c^{1/2})^2$ and thus, we see that there is exactly one solution (among the two defined by (3.12)) which satisfies $f_1(c^{1/4})=k$, except if $4k\tan (\theta )+c^{1/2}=0$ in which case we get 2 distinct solutions. It is also worth pointing out that by taking $\tan (\theta )=\frac{8 k c^{1/2}}{c-16 k^2}$ we recover the HYM solution $f_1=\lambda r^2$ of Proposition 3.2$\mathrm {(1)}$ with $\lambda =k/c^{1/2}, -c^{1/2}/16k$.

• (2) By contrast to case (1), in cases (2) and (3) for generic values of k and $\theta$ there are two dHYM connections. It is also worth pointing out that by taking $k=0$ we recover the HYM solution $f_2=\lambda \sqrt{r^4-c}$ as in Proposition 3.2$\mathrm {(2)}$ i.e. $F_A$ is just a multiple of $\omega _2$, with $\lambda =(-1\pm |\sec (\theta )|)/4\tan (\theta )$. The analogous statement holds for case (3).

Our results in this section show that although each Kähler form $\omega _i$ is compatible with the same metric $g_{EH}$, the associated dHYM connections are rather different. In particular, for fixed value of phase angle $\theta$ and k i.e. fixing the line bundle, the number of dHYM connections can be used to distinguish $\omega _1$ from $\omega _2$ and $\omega _3$. We shall show in Sect. 5 that a similar result also holds on $T^*\mathbb {C}\mathbb {P}^2$. On a compact manifold the phase angle $\theta$ is determined by the line bundle L itself, so this freedom of varying both $\theta$ and k is only occurs on non-compact manifolds; we illustrate this difference in an explicit example in next section.

4 Deformed Hermitian Yang–Mills Connections on $\textrm{SU}(3)/\mathbb {T}^2$

In this section we construct $\textrm{SU}(3)$-invariant dHYM connections on the flag manifold $F_{1,2}:=\textrm{SU}(3)/\mathbb {T}^2$ with its standard homogeneous Kähler Einstein structure. This section serves two main purposes. Firstly it illustrates certain key differences between dHYM connections in the compact and non-compact setting, and secondly it will provide the basic set up for the cohomogeneity one $\textrm{SU}(3)$-invariant connections that we shall investigate in the next section on $T^*\mathbb {C}\mathbb {P}^2$.

We begin by expressing the Maurer-Cartan form of $\textrm{SU}(3)$ as

$$\begin{aligned} \Theta :=\begin{pmatrix} i(\theta _1+\theta _2) &{} i\theta _3-\theta _4 &{} \theta _5+i\theta _6 \\ i\theta _3+\theta _4 &{} i(\theta _1-\theta _2) &{} i\theta _7+\theta _8 \\ -\theta _5+i\theta _6 &{} i\theta _7-\theta _8 &{} -2i\theta _1 \end{pmatrix}. \end{aligned}$$

From the latter, the structure equations can be easily computed using $d\Theta +\Theta \wedge \Theta =0$. Since we shall use these frequently in the calculations throughout this article, we state them explicitly for the reader’s convenience:

$$\begin{aligned} d\theta _{1}= & {} -\theta _{56}+\theta _{78},\\ d\theta _{2}= & {} -2\theta _{34}-\theta _{56}-\theta _{78},\\ d\theta _{3}= & {} 2\theta _{24}-\theta _{57}+\theta _{68},\\ d\theta _{4}= & {} -2\theta _{23}-\theta _{58}-\theta _{67},\\ d\theta _{5}= & {} 3\theta _{16}+\theta _{26}+\theta _{37}+\theta _{48},\\ d\theta _{6}= & {} -3\theta _{15}-\theta _{25}-\theta _{38}+\theta _{47},\\ d\theta _{7}= & {} -3\theta _{18}+\theta _{28}-\theta _{35}-\theta _{46},\\ d\theta _{8}= & {} 3\theta _{17}-\theta _{27}+\theta _{36}-\theta _{45}, \end{aligned}$$

where as before $\theta _{i...j}$ is shorthand for $\theta _i \wedge \cdots \wedge \theta _j$. In what follows, we denote by $e_i$ the dual left invariant vector field to $\theta _i$ i.e. $\theta _i(e_j)=\delta _{ij}$.

Recall that the Hopf fibration is given by

$$\begin{aligned} \textrm{U}(1) \hookrightarrow S^5 = \frac{\textrm{SU}(3)}{\textrm{SU}(2)} \rightarrow \mathbb {C}\mathbb {P}^2 = \frac{\textrm{SU}(3)}{\textrm{U}(1)\textrm{SU}(2)}, \end{aligned}$$

where the $\textrm{U}(1)$ action is generated by $e_1$ and the $\textrm{SU}(2)$ action is generated by $\langle e_2,e_3,e_4\rangle$. The Fubini–Study Kähler form of the base $\mathbb {C}\mathbb {P}^2$ can be identified with $\theta _{56}-\theta _{78}=:\omega _{\mathbb {C}\mathbb {P}^2}$ (this corresponds to the curvature of the connection 1-form $-\theta _1$). In view of twistor theory we also have the $S^2$ bundle

$$\begin{aligned} \pi : F_{1,2}:=\textrm{SU}(3)/\mathbb {T}^2 \rightarrow \mathbb {C}\mathbb {P}^2, \end{aligned}$$

where the $\mathbb {T}^2$ action here is generated by $e_1$ and $e_3$; we refer the reader to [2, Chap. 13] and [32, Chap. 4] for basic facts on twistor theory. We can express the Kähler Einstein structure of the flag manifold $F_{1,2}$ by

$$\begin{aligned} g_{KE}= & {} 2\theta _{2}^2+2\theta _{4}^2+\theta _{5}^2+\theta _{6}^2+\theta _{7}^2+\theta _{8}^2, \end{aligned}$$

(4.1)

$$\begin{aligned} \omega _{KE}= & {} d\theta _{3}=2\theta _{24}-\theta _{57}+\theta _{68}. \end{aligned}$$

(4.2)

We should point out that the twistor map $\pi$ is not holomorphic with respect to the natural complex structure of $\mathbb {C}\mathbb {P}^2$. Instead the twistor fibration can be viewed as the 2-sphere bundle associated to the quaternionic Kähler structure of $\overline{\mathbb {C}\mathbb {P}^2}$ (which has the opposite orientation to the Fubini–Study one); indeed the quaternionic Kähler structure of $\overline{\mathbb {C}\mathbb {P}^2}$ is generated by the self-dual 2-forms $\theta _{56}+\theta _{78}$, $\theta _{57}-\theta _{68}$ and $\theta _{58}+\theta _{67}$, and $F_{1,2}$ can be identified with its unit sphere bundle. Since $\omega _{\mathbb {C}\mathbb {P}^2}$ is anti-self-dual with respect to the quaternionic Kähler structure it follows from twistor theory that $\pi ^*\omega _{\mathbb {C}\mathbb {P}^2}$ is of type (1, 1) with respect to the Kähler Einstein structure of $F_{1,2}$.

Remark 4.1

If we had considered the $\mathbb {T}^2$ action generated by $e_1$ and $e_2$ to define the flag manifold then we would have obtain an equivalent Kähler Einstein structure which is instead given by

$$\begin{aligned} g= & {} 2\theta _{3}^2+2\theta _{4}^2+\theta _{5}^2+\theta _{6}^2+\theta _{7}^2+\theta _{8}^2,\\ \omega= & {} d\theta _{2}=-2\theta _{34}-\theta _{56}-\theta _{78}. \end{aligned}$$

Similarly if one considers the $\mathbb {T}^2$ action generated by $e_1$ and $e_4$, we get another equivalent Kähler Einstein structure. This gives rise to the so-called triality. Since these are all related by a $\mathbb {Z}_3$-automorphism it suffices to restrict to only one case.

Before describing the $\textrm{SU}(3)$-invariant abelian connections on $F_{1,2}$, we first note that $H^2(F_{1,2},\mathbb {Z})\cong \mathbb {Z}^2$ and hence the space of complex line bundles on $F_{1,2}$ is determined by a pair of integers. It is not hard to see that the general $\textrm{SU}(3)$-invariant abelian connection 1-form on the flag $F_{1,2}$ is given by

$$\begin{aligned} A=a_1\theta _{1}+a_3 \theta _{3}, \end{aligned}$$

where $(a_1,a_3)\in \mathbb {Z}^2\cong H^2(F_{1,2},\mathbb {Z})$ i.e. A corresponds to an $\textrm{SU}(3)$-invariant connection on each of the distinct line bundles. Moreover, using the structure equations we have that

$$\begin{aligned} F_A= a_1(-\theta _{56}+\theta _{78})+a_3(2\theta _{24}-\theta _{57}+\theta _{68})=-a_1 \pi ^*\omega _{\mathbb {C}\mathbb {P}^2}+a_3\omega _{KE} \end{aligned}$$

(4.3)

and it follows from the above discussion that A is a holomorphic connection with respect to the Kähler Einstein structure of $F_{1,2}$.

Remark 4.2

On a general compact Kähler manifold $M^{2n}$, the phase angle $\theta$ is determined by the Chern class of the line bundle itself, i.e. by $[F_A]$, via

$$\begin{aligned} \theta =\arg \left( \int _M(\omega +iF_A)^n\right) . \end{aligned}$$

(4.4)

This is not generally applicable in the non-compact set up since the latter integral might be infinite; this was indeed the case in our cohomogeneity one framework in the last section. This explains why we had the freedom to vary both k and $\theta$ independently in the Theorem 3.5. In the $F_{1,2}$ situation, however, it follows that $\theta$ is completely determined by the pair $(a_1,a_2)$. We shall now describe the dHYM connections in this case explicitly.

Theorem 4.3

The connection form $A=a_1\theta _{1}+a_3 \theta _{3}$ is a dHYM connection with respect to $\omega _{KE}$ if and only if

$$\begin{aligned} \tan (\theta )=\frac{a_3(a_3^2-a_1^2-3)}{3a_3^2-a_1^2-1}, \end{aligned}$$

(4.5)

or equivalently,

$$\begin{aligned} \theta \equiv \arctan ({a_3}) + \arctan (a_3+a_1)+\arctan (a_3-a_1) \mod 2\pi . \end{aligned}$$

(4.6)

Proof

First setting $n=3$ in (1.2) gives

$$ 3 F_A \wedge \omega _{KE}^2- F_A^3 = \tan (\theta ) (\omega _{KE}^3- 3 F_A^2 \wedge \omega _{KE}).$$

Since we already know that A is a holomorphic connection, it suffices to solve the latter equation. A straightforward computation using (4.2) and (4.3) shows that

$$\begin{aligned} F_A \wedge \omega _{KE}^2= & {} 12 a_3 \theta _{245678},\\ F_A^3= & {} -12(a_1^2 a_3-a_3^3)\theta _{245678},\\ \omega _{KE}^3= & {} 12\theta _{245678},\\ F_A^2 \wedge \omega _{KE}= & {} -4(a_1^2-3a_3^2)\theta _{245678}, \end{aligned}$$

and from this we easily deduce (4.5). To obtain the second expression we follow the approach in [9, Section 2]. First using $\omega _{KE}$ we can view the (1, 1)-form $F_A$ as an endomorphism of holomorphic tangent bundle $T^{1,0}$; concretely it is can expressed as

$$\begin{aligned} \begin{pmatrix} {a_3} &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad a_3 &{} \quad -a_1 \\ 0 &{} \quad -a_1 &{} \quad a_3 \end{pmatrix} \end{aligned}$$

and has eigenvalues $a_3, a_3+a_1$ and $a_3-a_1$. Thus, we have that

$$\begin{aligned} re^{i\theta }= \frac{(\omega _{KE}+iF_A)^3}{\omega _{KE}^3}=(1+i{a_3})(1+i(a_3+a_1))(1+i(a_3-a_1)), \end{aligned}$$

for some positive number r, and using (4.4) we get (4.6). $\square$

In [19] Jacob-Yau showed that solutions to the dHYM equations on a compact Kähler manifold are unique in a given cohomology class. Hence it follows that the solutions in Theorem 4.3 in fact consist of all the dHYM connections on $F_{1,2}$ i.e. not just the invariant ones. Figure 1 illustrates how the phase angle depends on the choice on the line bundle over $F_{1,2}$. Having given a concrete description of $\textrm{SU}(3)$, next we proceed to our main goal, namely, the construction of the deformed instantons equations on $T^*\mathbb {C}\mathbb {P}^2$.

Fig. 1

The shaded region corresponds to $\tan (\theta )\ge 0$, bold line to $\tan (\theta )=0$ and dotted line to $\cot (\theta )=0$ in (4.5)

5 Deformed Hermitian Yang–Mills and Deformed $\textrm{Spin}(7)$ Instantons on $T^*{\mathbb {C}\mathbb {P}^2}$

In this section we construct dHYM connections on the cotangent bundle of $\mathbb {C}\mathbb {P}^2$ endowed with the Calabi hyperKähler structure [6]. It is well-known that the latter is invariant under a cohomogeneity one action of $\textrm{SU}(3)$ and hence the problem of constructing $\textrm{SU}(3)$-invariant dHYM connections reduces to solving a non-linear system of ODEs. This is a higher dimensional analogue of the problem considered in Sect. 3 albeit more involved. As we already saw in Sect. 2 one can also define ‘natural’ $\textrm{Spin}(7)$-structures on a hyperKähler 8-manifold by (2.27). Hence we shall also consider deformed $\textrm{Spin}(7)$ instantons in this set up and compare the two cases.

We begin by describing the cohomogeneity one hyperKähler structure of $T^*\mathbb {C}\mathbb {P}^2$ following the same notation as introduced in the previous section. First we recall the well-known fact that for each pair of integers k, l there are different embeddings of $\textrm{U}(1)$ in the maximal torus of $\textrm{SU}(3)$. Concretely, we denote these by $\textrm{U}(1)_{k,l}:=\textrm{diag}(e^{ik\varphi },e^{il\varphi }, e^{-i(k+l)\varphi })$, where $\varphi \in [0,2\pi )$. Observe that the infinitesimal vector field associated to this $\textrm{U}(1)_{k,l}$ action is given by $\frac{k+l}{2}e_1+\frac{k-l}{2}e_2$. The homogeneous spaces $N_{k,l}:=\textrm{SU}(3)/\textrm{U}(1)_{k,l}$ are called the Aloff-Wallach spaces. Here we shall only consider the case when $k=l=1$ since $N_{1,1}$ is precisely the principal orbit of the cohomogeneity one action of $\textrm{SU}(3)$ on $T^*\mathbb {C}\mathbb {P}^2$ and for convenience we shall denote $\textrm{U}(1)_{1,1}$ simply by $\textrm{U}(1)$. On the other hand, the singular orbit of the $\textrm{SU}(3)$ action is isomorphic to $\mathbb {C}\mathbb {P}^2\cong \textrm{SU}(3)/\textrm{U}(2)$, where the $\textrm{U}(2)$ is generated by $\langle e_1,e_2,e_7,e_8\rangle$ (see Proposition 5.1 below). It is worth pointing out that in fact the Aloff Wallach spaces $N_{1,1}, N_{1,-2}$ and $N_{-2,1}$ are all diffeomorphic under the action of the Weyl group of $\textrm{SU}(3)$; however, when viewed as bundles over the latter $\mathbb {C}\mathbb {P}^2$ they correspond to either an $\mathbb {R}\mathbb {P}^3$ or $S^3$ bundle and this subtlety turns out to be rather important as we shall see below. In [10] the authors show that the Calabi hyperKähler structure in fact belongs to a 1-parameter family of complete $\textrm{Spin}(7)$ metrics and moreover that there also exist $\textrm{SU}(3)$-invariant $\textrm{Spin}(7)$ metrics with $N_{k,l}$ as the principal orbits for each k, l. In this paper we shall not consider these more general families since these are not explicit and hence more involved to study.

For the homogeneous space $N_{1,1}$, we have the reductive splitting

$$\begin{aligned} \mathfrak {su}(3)\cong \mathfrak {u}(1)\oplus \mathfrak {m}, \end{aligned}$$

where $\mathfrak {u}(1)\cong \langle e_1 \rangle$ and $\mathfrak {m}$ denotes its $\textrm{Ad}\big |_{\textrm{U}(1)}$ complement. As a $\textrm{U}(1)$-module we have that

$$\begin{aligned} \mathfrak {m} \cong 3\mathbb {R}\oplus 2\mathbb {C}, \end{aligned}$$

where the three copies of $\mathbb {R}$ are generated by $\langle e_2,e_3,e_4\rangle$ and the two copies of $\mathbb {C}$ are spanned by $\langle e_5, e_6 \rangle$ and $\langle e_8, e_7 \rangle$. Note that the $\textrm{U}(1)$ acts with weight 3 on each of these copies of $\mathbb {C}$ i.e. we have that $[e_1,e_5]=3e_6$, $[e_1,e_6]=-3e_5$ and $[e_1,e_8]=3e_7$, $[e_1,e_7]=-3e_8$. The tangent space of $N_{1,1}$ at a point can be identified with the isotropy representation $\mathfrak {m}$. It follows that the space of $\textrm{SU}(3)$-invariant 2-forms on $N_{1,1}$ can then be identified with $\textrm{U}(1)$-invariant 2-forms on $\mathfrak {m}$. A simple computation shows that as $\textrm{U}(1)$-module

$$\begin{aligned} \Lambda ^2(\mathfrak {m})\cong 7\mathbb {R}\oplus 6 \mathbb {C}\end{aligned}$$

and hence there are 7 such invariant 2-forms: 3 of them are obtained as combinations of $\langle \theta _2,\theta _3,\theta _4\rangle$ and the other 4 are given by

$$\begin{aligned} \langle \theta _{56}, \theta _{78}, \theta _{57}+\theta _{86},\theta _{58}+\theta _{67}\rangle . \end{aligned}$$

Using these invariant forms on $N_{1,1}$ we can now express the $\textrm{Sp}(2)$-structure of $T^*\mathbb {C}\mathbb {P}^2$ as follows:

Proposition 5.1

The Calabi hyperKähler structure of $T^*{\mathbb {C}\mathbb {P}^2}$ is explicitly given by

$$\begin{aligned} g_{Ca}= & {} f_0^2dr^2+f_2^2\theta ^2_2+f_3^2(\theta _3^2+\theta _4^2)+ f_5^2 (\theta _5^2+\theta _6^2)+f_7^2(\theta _7^2+\theta _8^2), \end{aligned}$$

(5.1)

$$\begin{aligned} \omega _1= & {} f_0f_2dr \wedge \theta _2 +f_3^2 \theta _{34}+f_5^2\theta _{56}+f_7^2\theta _{78}, \end{aligned}$$

(5.2)

$$\begin{aligned} \omega _2= & {} f_0f_3dr \wedge \theta _3 +f_2f_3 \theta _{42}+f_5f_7\sigma _{2}, \end{aligned}$$

(5.3)

$$\begin{aligned} \omega _3= & {} f_0f_3dr \wedge \theta _4 +f_2f_3 \theta _{23}+f_5f_7\sigma _{3}, \end{aligned}$$

(5.4)

where $\sigma _{2}:=\theta _{57}+\theta _{86}$, $\sigma _{3}:=\theta _{58}+\theta _{67}$,

$$\begin{aligned} f_0= & {} -(1-4c^2r^{-4})^{-1/2},\\ f_2= & {} r(1-4c^2r^{-4})^{1/2}=\frac{2}{r}(r^2/2+c)^{1/2}(r^2/2-c)^{1/2},\\ f_3= & {} r,\\ f_5= & {} (r^2/2+c)^{1/2},\\ f_7= & {} (r^2/2-c)^{1/2}, \end{aligned}$$

and c is a constant determining the size of the zero section $\mathbb {C}\mathbb {P}^2$. Without loss of generality, we shall assume that $c>0$ and take $r\in [\sqrt{2c},+\infty )$. The bolt (or zero section) in this case is a $\mathbb {C}\mathbb {P}^2$ spanned by $\langle e_3,e_4,e_5,e_6\rangle$ and where the Fubini-Study form is, up to a constant, given by $\theta _{34}+\theta _{56}$.

Proof

First we need to verify that $d\omega _1=d\omega _2=d\omega _3=0.$ Consider the $\textrm{Sp}(2)$-structure defined by (5.1)-(5.4) whereby $f_i(r)$ are arbitrary functions. Then a straightforward calculation using the structure equations of $\textrm{SU}(3)$ shows that $d\omega _1=0$ is equivalent to

$$\begin{aligned} f_3^2= & {} f_5^2+f_7^2,\\ f_0f_2= & {} -\frac{1}{2}(f_3^2)'=-(f_5^2)'=-(f_7^2)'. \end{aligned}$$

On the other hand, the condition that $d\omega _2=0$ (which coincides with $d\omega _3=0$) is equivalent to

$$\begin{aligned} f_2f_3= & {} 2f_5f_7,\\ f_0f_3= & {} -\frac{1}{2}(f_2f_3)'=-(f_5f_7)'. \end{aligned}$$

The functions $f_i(r)$ as given in the proposition follow immediately by solving the above system of ODEs. To conclude we still need to show that the metric $g_{Ca}$ extends smoothly across the zero section and hence is indeed complete. Since this was already demonstrated in [6], see also [10], this concludes the proof. $\square$

As already mentioned in Remark 3.3, there exists again a permuting Killing vector field, given by $e_2/2$, such that $\mathcal {L}_{e_2}\omega _1=0$, $\mathcal {L}_{e_2}\omega _2=+2\omega _3$ and $\mathcal {L}_{e_2}\omega _3=-2\omega _2$. As in Sect. 3 this will again give rise to a hyper-holomorphic connection (see Lemma 5.5 below). Note that we can always set the constant $c=1$ by a suitably reparametrising the radial coordinate r, however we shall find it more useful to keep it so as to facilitate comparison with the conical situation whereby $c=0$, see Sect. 6. To compute the deformed $\textrm{Spin}(7)$ instanton equations later on we shall need the next result.

Proposition 5.2

With respect to the Calabi hyperKähler structure, as given in Proposition 5.1, the space of $\textrm{SU}(3)$-invariant 2-forms in each of the spaces $E_1, E_2,E_3$, as defined by (2.8), is one dimensional and these are generated by

$$\begin{aligned} \alpha _1= & {} f_0f_2dr \wedge \theta _2 +f_3^2 \theta _{34}-f_5^2\theta _{56}-f_7^2\theta _{78}, \end{aligned}$$

(5.5)

$$\begin{aligned} \alpha _2= & {} f_0f_3dr \wedge \theta _3 +f_2f_3 \theta _{42}-f_5f_7\sigma _{2}, \end{aligned}$$

(5.6)

$$\begin{aligned} \alpha _3= & {} f_0f_3dr \wedge \theta _4 +f_2f_3 \theta _{23}-f_5f_7\sigma _{3}, \end{aligned}$$

(5.7)

respectively. In particular, from (2.11) we have that $F_i^+=\langle \alpha _j \wedge \omega _k \rangle$.

Proof

We already showed that a general $\textrm{SU}(3)$-invariant 2-form on $N_{1,1}$ is given by a linear combination of $\langle \theta _{23},\theta _{24},\theta _{34},\theta _{56},\theta _{78},\sigma _{2},\sigma _{3}\rangle$. Thus, it follows that a general $\textrm{SU}(3)$-invariant 2-form on $T^*\mathbb {C}\mathbb {P}^2$ consists of a combination of the latter seven 2-forms together with the triple of 2-forms $\langle dr \wedge \theta _2, dr \wedge \theta _3, dr \wedge \theta _4\rangle$. Using the definition of the spaces $E_i$ as given by (2.8), we find that in this ten dimensional family there is exactly one such 2-form that lies in each of $E_i$. In fact in this ten dimensional family of $\textrm{SU}(3)$-invariant 2-forms, three correspond to $\omega _1,\omega _2,\omega _3$, one lies in each of $E_i$, and the remaining four lie in $\Lambda ^2_{10}$. The last statement follows from (2.11). $\square$

Observe that when $c=0$, the hyperKähler metric (5.1) corresponds to the cone metric over $N_{1,1}$ with its well-known 3-Sasakian structure cf. [13]. Galicki–Salamon showed in [13, Proposition 2.4] that squashing a 3-Sasakian metric along its 3-dimensional foliation (given in our case by $\langle e_2,e_3,e_4\rangle$) by a factor of $1/\sqrt{5}$ yields a strictly nearly parallel $\textrm{G}_2$ metric: the latter is characterised by the fact the cone metric has holonomy equal to $\textrm{Spin}(7)$, in contrast to the 3-Sasakian case whereby the cone metric has holonomy equal to $\textrm{Sp}(2)$. For the squashed structure on $S^7$ the associated conical $\textrm{Spin}(7)$ metric can in fact be smoothed to a complete metric on the spinor bundle of $S^4$: this is the so-called Bryant-Salamon metric cf. [5, Sect. 4. Theorem 2]. In our set up, the analogous $\textrm{SU}(3)$-invariant Bryant-Salamon $\textrm{Spin}(7)$ metric is not smooth but instead has orbifold singularities since $\mathbb {C}\mathbb {P}^2$ is not a spin manifold. Explicitly, it is given as follows:

Proposition 5.3

The orbi-spinor bundle of $\mathbb {C}\mathbb {P}^2$ admits the Bryant-Salamon $\textrm{Spin}(7)$-structure and is explicitly given by

$$\begin{aligned} g_{BS}=&h_0^2dr^2+h_2^2(\theta _2^2+\theta _3^2+\theta _4^2)+h_5^2(\theta _5^2+\theta _6^2+\theta _7^2+\theta _8^2), \end{aligned}$$

(5.8)

$$\begin{aligned} \Phi _{BS}=&h_0dr \wedge (h_2^3\theta _{234}-h_2 h_5^2(\theta _{256}+\theta _{278}+\theta _{357}-\theta _{368}+\theta _{458}+\theta _{467})-\nonumber \\&h_2^2h_5^2(\theta _{2358}+\theta _{2367}-\theta _{2457}+\theta _{2468}+\theta _{3456}+\theta _{3478})+h_5^4\theta _{5678} \end{aligned}$$

(5.9)

where

$$\begin{aligned} h_0= & {} \frac{2}{(r^2+c)^{1/5}},\\ h_2= & {} \frac{2 r}{(r^2+c)^{1/5}},\\ h_5= & {} \sqrt{10} (r^2+c)^{3/10}, \end{aligned}$$

$r\in [0,+\infty )$ and c is a non-negative constant determining the size of the zero section $\mathbb {C}\mathbb {P}^2$ spanned by $\theta _5,\theta _6,\theta _7,\theta _8$. When $c=0$ we get a cone metric over $N^{1,1}$ with its strictly nearly parallel $G_2$-structure, and when $c>0$ we get a metric which has a $(\mathbb {R}^4/\mathbb {Z}_2)$-orbifold singularity along the zero section.

Proof

It suffices to verify that $d\Phi _{BS}=0$ which is a straightforward computation using the structure equations. $\square$

Observe that the singular orbits in Propositions 5.1 and 5.3 correspond to two different $\mathbb {C}\mathbb {P}^2$s. In the former case the principal orbit $N_{1,1}$ corresponds to an $S^3$ bundle over this $\mathbb {C}\mathbb {P}^2$ whereas in the latter case it corresponds to an $\mathbb {R}\mathbb {P}^3$ bundle which explains why the metric $g_{BS}$ is not smooth but instead has orbifold singularities, see also [10, Sect. 3.1.3] for a general description of $N_{k,l}$ as a lens space bundle over $\mathbb {C}\mathbb {P}^2$. For the sake of comparison we shall also describe the deformed $\textrm{Spin}(7)$ instantons for the hyperKähler and $\textrm{Spin}(7)$ cones (see Sect. 6).

5.1 Invariant connections

To classify abelian $\textrm{SU}(3)$-invariant connections on the principal orbit $N^{1,1}:=\textrm{SU}(3)/\textrm{U}(1)$ we appeal to Wang’s theorem cf. [25, Chap. 10 Th. 2.1]. In our context, Wang’s theorem asserts that given a group homomorphism $\lambda _k:\textrm{U}(1)\rightarrow \textrm{U}(1)$ such that $e^{i\theta }\mapsto e^{ik\theta }$, all $\textrm{SU}(3)$-invariant connections on the associated principal $\textrm{U}(1)$-bundle

$$\begin{aligned} \textrm{SU}(3)\times _{\lambda _k} \textrm{U}(1) \rightarrow N^{1,1} \end{aligned}$$

are classified by $\textrm{U}(1)$-equivariant maps

$$\begin{aligned} \Lambda : (3\mathbb {R}\oplus 2\mathbb {C}, \textrm{Ad}\big |_{\textrm{U}(1)})\rightarrow (\mathfrak {u}(1), \textrm{Ad}\circ \lambda _k). \end{aligned}$$

An easy application of Schur’s lemma shows that there is a 3-parameter family of such connections for each given $k\in \mathbb {Z}$. Concretely, we have that abelian $\textrm{SU}(3)$-invariant connections on the principal orbits $\{r\}\times N^{1,1}$ are of the form

$$\begin{aligned} A(r)=k\theta _1+a_2(r) \theta _2+a_3(r) \theta _3+a_4(r) \theta _4. \end{aligned}$$

(5.10)

where k is a constant. We should point out that these connections were also described in [1] whereby the authors used them to construct $\textrm{G}_2$-instantons on the Aloff-Wallach spaces $N_{k,l}$. Next we compute the curvature form as follows:

Proposition 5.4

The curvature form of A, as defined by (5.10), is given by

$$\begin{aligned} F_A =&dr\wedge ( a_2'\theta _2+a_3' \theta _3+a_4' \theta _4)-2a_2 \theta _{34}-(a_2+k) \theta _{56}-(a_2-k) \theta _{78}\nonumber \\&+2a_3 \theta _{24}-a_3 \sigma _{2}-2a_4 \theta _{23}-a_4 \sigma _{3}, \end{aligned}$$

(5.11)

where we recall that $\sigma _{2}:=\theta _{57}+\theta _{86}$ and $\sigma _{3}:=\theta _{58}+\theta _{67}$.

Observe that $F_A$ is indeed a well-defined 2-form on the open set $\mathbb {R}_{r>\sqrt{2c}}\times N^{1,1}$. However, in order for A(r) to extend to a connection on $T^*\mathbb {C}\mathbb {P}^2$ we need it to extend to a smooth connection on the zero section $\mathbb {C}\mathbb {P}^2$ as well. From Proposition 5.1 it is easy to see that the tangent space of the latter $\mathbb {C}\mathbb {P}^2$ is spanned by $\langle e_3,e_4, e_5,e_6\rangle$ and that $\omega _1\big |_{\mathbb {C}\mathbb {P}^2}=\sqrt{2c}(\theta _{34}+\theta _{56})$ corresponds precisely to its Fubini-Study form. Since $H^2(\mathbb {C}\mathbb {P}^2,\mathbb {Z})\cong \mathbb {Z}$ is generated by the cohomology class of the latter form, it follows from Chern-Weil theory that we need A to extend to a connection 1-form whose curvature form is an integer multiple of $\theta _{34}+\theta _{56}$ after suitably fixing the value of c. Inspecting the structure equations of $\textrm{SU}(3)$, we see that

$$\begin{aligned} d(\theta _1+\theta _2)=-2(\theta _{34}+\theta _{56}). \end{aligned}$$

Thus, we deduce that the connection A(r) of the form (5.10) on $\mathbb {R}_{r>\sqrt{2c}}\times N^{1,1}$ will extend to a connection on $T^*\mathbb {C}\mathbb {P}^2$ provided that $a_3(\sqrt{2c})=a_4(\sqrt{2c})=0$ and $a_2(\sqrt{2c})=k\in \mathbb {Z}$. In other words, we need that $A(\sqrt{2c})=k(\theta _{1}+\theta _{2})$. With this in mind, we can now proceed to the construction of instantons.

5.2 HYM connections and $\textrm{Spin}(7)$-instantons

We begin by describing the $\textrm{SU}(3)$-invariant holomorphic connections on the open set $N_{1,1}\times \mathbb {R}_{r>\sqrt{2 c}}$.

Lemma 5.5

Let $k \in \mathbb {Z}$, then the connection $A=k \theta _1+a_2\theta _2+a_3\theta _3+a_4\theta _4$ is holomorphic with respect to

• (1) $\omega _1$ if and only if $a_3=a_4=0$.

• (2) $\omega _2$ if and only if $a_2= 2ck/r^2$ and $a_4=0$.

• (3) $\omega _3$ if and only if $a_2= 2ck/r^2$ and $a_3=0$.

In particular, A is hyper-holomorphic i.e. $F_A \in \Lambda ^2_{10}$ if and only if $a_2=2ck/r^2$ and $a_3=a_4=0$.

Proof

Recall that A is a holomorphic connection with respect to $\omega _i$ if and only if $F_A \wedge (\omega _j+i\omega _k)^2=0$. The result follows easily by computing the latter using (5.2)–(5.4) and (5.11). $\square$

Observe when $r=\sqrt{2 c}$ we indeed have that the above hyper-holomorphic connection satisfies $A(\sqrt{2 c})=k(\theta _1+\theta _2)$ and hence it extends smoothly to a connection on $T^*\mathbb {C}\mathbb {P}^2$. As we already mentioned in Remark 3.3, this hyper-holomorphic connection is precisely the one described by Hitchin in [17, Section 2.2]. Moreover, Hitchin showed that this connection is characterised as the unique hyper-holomorphic connection on $T^*\mathbb {C}\mathbb {P}^2$ which is invariant under the $S^1$ action generated by the permuting Killing vector field $e_2/2$ and whose curvature form restricts to Fubini-Study form on the zero section $\mathbb {C}\mathbb {P}^2$. Furthermore, it is easy to verify that its curvature form is given by

$$\begin{aligned} F_A = \omega _1 + dd^c\mu , \end{aligned}$$

where $\mu$ is the moment map defined by $d\mu =e_2 \lrcorner \omega _1/2$, which is consistent with [17, Theorem 1].

Having described the holomorphic connections, next we investigate which among them correspond to HYM connections.

Proposition 5.6

Let $k \in \mathbb {Z}$, then the connection $A=k\theta _1 + a_2\theta _2 + a_3\theta _3 + a_4\theta _4$ is an

• (1) $\omega _1$-HYM connection on $N_{1,1}\times \mathbb {R}_{r>\sqrt{2 c}}$ such that $F_A \wedge \omega _1^3 =\lambda \omega _1^4$ if and only if $a_3=a_4=0$ and

  $$\begin{aligned} a_2=\frac{C_0-(r^4-4c^2)(\lambda (r^4-4c^2)-4ck)}{2 r^2(r^4-4c^2)}, \end{aligned}$$

  (5.12)

  where $C_0$ is a constant. A extends smoothly to $T^*\mathbb {C}\mathbb {P}^2$ only when $C_0=0$, in which case $F_A$ is the sum of $\lambda \omega _1$ and the curvature of the hyper-holomorphic connection.

• (2) $\omega _2$-HYM on $N_{1,1}\times \mathbb {R}_{r>\sqrt{2 c}}$ such that $F_A \wedge \omega _2^3 =\lambda \omega _2^4$ if and only if $a_2=2ck/r^2$, $a_4=0$ and

  $$\begin{aligned} a_3=\frac{C_0-\lambda (r^4-4c^2)^2}{2(r^4-4c^2)^{3/2}}, \end{aligned}$$

  (5.13)

  where $C_0$ is a constant. A extends smoothly to $T^*\mathbb {C}\mathbb {P}^2$ only when $C_0=0$, in which case $F_A$ is the sum of $\lambda \omega _2$ and the curvature of the hyper-holomorphic connection.

• (3) $\omega _3$-HYM on $N_{1,1}\times \mathbb {R}_{r>\sqrt{2 c}}$ such that $F_A \wedge \omega _2^3 =\lambda \omega _3^4$ if and only if $a_2=2ck/r^2$, $a_3=0$ and

  $$\begin{aligned} a_4=\frac{C_0-\lambda (r^4-4c^2)^2}{2(r^4-4c^2)^{3/2}}, \end{aligned}$$

  (5.14)

  where $C_0$ is a constant. A extends smoothly to $T^*\mathbb {C}\mathbb {P}^2$ only when $C_0=0$, in which case $F_A$ is the sum of $\lambda \omega _3$ and the curvature of the hyper-holomorphic connection.

Proof

In case (1), from Lemma 5.5, we already know that we need $a_3=a_4=0$. Using Propositions 5.1 and 5.4 we find that $F_A\wedge \omega _1^3=\lambda \omega _1^4$ is equivalent to the

$$\begin{aligned} r^2a_2'(r^4-4c^2)+2ra_2(3r^4-4c^2)+4\lambda r^7-8cr^3(k+2c\lambda )=0. \end{aligned}$$

The latter is a linear ODE and one easily solves it to get (5.12). For case (2), with $a_2=2ck/r^2$ and $a_3=0$, we have that $F_A\wedge \omega _2^3=\lambda \omega _1^4$ is equivalent to

$$\begin{aligned} a_3'(r^4-4c^2)+6r^3 a_3+4\lambda r^3\sqrt{r^4-4c^2}=0 \end{aligned}$$

and solving the latter yields (5.13). For case (3) it suffices to swap $a_3$ and $a_4$ in case (2), and this concludes the proof. $\square$

Next we consider the $\textrm{Spin}(7)$-instantons with respect $\Phi _i$ as defined by (2.27).

Theorem 5.7

Let $k\in \mathbb {Z}$ and $C_i \in \mathbb {R}$, then

• (1) the general $\textrm{SU}(3)$-invariant abelian $\textrm{Spin}(7)$-instanton on $N_{1,1}\times \mathbb {R}_{r>\sqrt{2 c}}$ with respect to $\Phi _1$ is given by

  $$\begin{aligned} A=k\theta _{1}+\frac{C_1 (r^4-4c^2)+2ck}{r^2}\theta _2+\frac{C_2}{(r^4-4c^2)^{3/2}}\theta _3+\frac{C_3}{(r^4-4c^2)^{3/2}}\theta _4. \end{aligned}$$

  (5.15)

  The latter includes a traceless $\omega _2$-HYM connection (when $C_1=C_3=0$) and a traceless $\omega _3$-HYM connection (when $C_1=C_2=0$). Only the solutions with $C_2=C_3=0$ extend globally to $T^*\mathbb {C}\mathbb {P}^2$.

• (2) the general $\textrm{SU}(3)$-invariant abelian $\textrm{Spin}(7)$-instanton on $N_{1,1}\times \mathbb {R}_{r>\sqrt{2 c}}$ with respect to $\Phi _2$ is given by

  $$\begin{aligned} A=k\theta _{1}+\Bigg (\frac{2ck}{r^2}+\frac{C_1}{r^2 (r^4-4c^2)}\Bigg )\theta _2+{C_2}{(r^4-4c^2)^{1/2}}\theta _3+\frac{C_3}{(r^4-4c^2)^{3/2}}\theta _4. \end{aligned}$$

  (5.16)

  The latter includes a traceless $\omega _1$-HYM connection (when $C_2=C_3=0$) and a traceless $\omega _3$-HYM connection (when $C_1=C_2=0$). Only the solutions with $C_1=C_3=0$ extend globally to $T^*\mathbb {C}\mathbb {P}^2$.

• (3) the general $\textrm{SU}(3)$-invariant abelian $\textrm{Spin}(7)$-instanton on $N_{1,1}\times \mathbb {R}_{r>\sqrt{2 c}}$ with respect to $\Phi _3$ is given by

  $$\begin{aligned} A=k\theta _{1}+\Bigg (\frac{2ck}{r^2}+\frac{C_1}{r^2 (r^4-4c^2)}\Bigg )\theta _2+\frac{C_2}{(r^4-4c^2)^{3/2}}\theta _3+{C_3}{(r^4-4c^2)^{1/2}}\theta _4. \end{aligned}$$

  (5.17)

  The latter includes a traceless $\omega _1$-HYM connection (when $C_2=C_3=0$) and a traceless $\omega _2$-HYM connection (when $C_1=C_3=0$). Only the solution with $C_1=C_2=0$ extends globally to $T^*\mathbb {C}\mathbb {P}^2$.

Proof

For case (1) using Propositions 5.1 and 5.4 one easily shows that $*(F_A\wedge \Phi _1)=-F_A$ is equivalent to the system

$$\begin{aligned} ra_2'(r^4-4c^2)-2a_2(r^4+4c^2)+8ckr^2= & {} 0,\\ a_3'(r^4-4c^2)+6a_3 r^3= & {} 0,\\ a_4'(r^4-4c^2)+6a_4 r^3= & {} 0. \end{aligned}$$

Solving the latter yields (5.15). Similarly for case (2) we have that $*(F_A\wedge \Phi _2)=-F_A$ is equivalent to

$$\begin{aligned} ra_2'(r^4-4c^2)+2a_2(3r^4-4c^2)-8ckr^2= & {} 0,\\ a_3'(r^4-4c^2)-2a_3 r^3= & {} 0,\\ a_4'(r^4-4c^2)+6a_4 r^3= & {} 0, \end{aligned}$$

and the solution is given by (5.16). For case (3) it suffices to swap $a_3$ and $a_4$ in case (2). $\square$

Observe that the $\textrm{Spin}(7)$-instantons which are globally well-defined on $T^*\mathbb {C}\mathbb {P}^2$ are precisely the HYM connections from Proposition 5.6. Note also that the system of ODEs arising in the above theorem are all linear and moreover since the gauge group is $\textrm{U}(1)$ the system is not coupled; as such solving the ODEs was rather simple. Next we shall consider the deformed instanton equations which in contrast to the above will yield systems of ODEs that are both non-linear and coupled.

5.3 The deformed HYM connections and deformed $\textrm{Spin}(7)$-instantons

Theorem 5.8

The connection $A(r)=k \theta _1+a_2\theta _2+a_3\theta _3+a_4\theta _4$ is a dHYM connection with phase $e^{i\theta }$ with respect to $\omega _1$ on $T^*\mathbb {C}\mathbb {P}^2$ only if $a_3=a_4=0$ and $a_2(r)$ is implicitly defined by

$$\begin{aligned} \tan (\theta )=\frac{2(a_2 r^2-2ck)(a_2^2-r^4/4+c^2-k^2)}{(a_2^2-r^4/4+c^2-k^2)^2-(a_2 r^2-2ck)^2}. \end{aligned}$$

(5.18)

For generic values of $k,\theta$, there are 2 distinct $\omega _1$-dHYM connections on $T^*\mathbb {C}\mathbb {P}^2$ i.e. these are the solutions $a_2(r)$ satisfying $a_2(\sqrt{2c})=k \in \mathbb {Z}$.

Proof

First from Lemma 5.5 we already know that we need $a_3=a_4=0$. A long but straightforward calculation using Propositions 5.1 and 5.4 then shows that

$$\begin{aligned} F_A \wedge \omega _1^3= & {} -\frac{r(r^4-4c^2)a_2'+2(3r^4-4c^2)a_2-8ckr^2}{4r^2(r^4-4c^2)} \omega _1^4,\\ F_A^4= & {} \frac{8a_2(a_2^2-k^2)a_2'}{r^3(r^4-4c^2)}\omega _1^4,\\ F_A^2 \wedge \omega _1^2= & {} \frac{((3r^4-4c^2)a_2-4ckr^2)a_2'-2k^2r^3+6r^3a_2^2-8ckra_2}{r^3(3r^4-4c^2)}\omega _1^4. \end{aligned}$$

Using the above one finds that (2.17) is given by

$$\begin{aligned}&\Big (8\tan (\theta )a_2^3-12a_2^2r^2-\big ((6r^4-8c^2+8k^2)\tan (\theta )-16ck\big )a_2\nonumber \\&\quad +r^2(r^4-4c^2+4k^2+8ck\tan (\theta ))\Big )a_2' -8r a_2^3-12 r^3 a_2^2 \tan (\theta )\nonumber \\&\quad +(6r^5-8c^2r+8k^2r+16ckr \tan (\theta ))a_2+(\tan (\theta )(r^4-4c^2+4k^2)-8ck)r^3 =0. \end{aligned}$$

(5.19)

It turns out that the latter is an exact ODE: indeed, it is not hard to verify that differentiating the expression

$$\begin{aligned}&{2\tan (\theta )}a_2^4-4r^2a_2^3-{\big ((3r^4-4c^2+4k^2)\tan (\theta )-8ck\big )}a_2^2\nonumber \\&\quad +r^2(r^4-4c^2+4k^2+8ck\tan (\theta ))a_2\ \nonumber \\&\quad +\tan (\theta )\left( \frac{r^4}{8}-c^2+k^2\right) r^4-2ckr^4-C=0, \end{aligned}$$

(5.20)

with respect to r yields (5.19). Thus, this shows that (5.20) is the general solution to (5.19) and hence we have constructed all $\textrm{SU}(3)$-invariant $\omega _1$-dHYM connections on the open set $\mathbb {R}_{r>\sqrt{2 c}}\times N^{1,1}$. Observe that $a_2(r)$ is implicitly given as the solution to a quartic polynomial whose coefficients are functions of r.

Since C is an arbitrary constant of integration we have a 1-parameter family of solutions on the open set $\mathbb {R}_{r>\sqrt{2 c}}\times N^{1,1}$. However, we still need to impose that $a_2(\sqrt{2 c})=k$ in order to get a smooth extension of the connection A at the zero section $\mathbb {C}\mathbb {P}^2$. Substituting $r=\sqrt{2c}$ and $a_2(\sqrt{2 c})=k$ in (5.20) shows that we need

$$C=-2\tan (\theta )((c^2-k^2)^2-4c^2 k^2)-8ck(c^2-k^2).$$

With C as above, one can rewrite (5.20) very neatly as (5.18). Observe that if $\tan (\theta )=0$ then (5.18) is a cubic polynomial in $a_2(r)$, otherwise (5.18) is a quartic polynomial and hence generically there will be 4 distinct solutions $a_4(r)$. Among these solutions only those satisfying $a_2(\sqrt{2c})=k$ will extend to global solutions on $T^*\mathbb {C}\mathbb {P}^2$. We shall determine those next.

Setting $r=\sqrt{2c}$ in (5.18) and solving for $a_2(\sqrt{2c})$ shows that $a_2(\sqrt{2c})=k$ occurs as a double root while the other two roots satisfy

$$\begin{aligned} (a_2(\sqrt{2c})+k)^2\tan (\theta )-4c(a_2(\sqrt{2c})+k)-4c^2\tan (\theta )=0. \end{aligned}$$

From the latter expression we deduce easily that $a_2(\sqrt{2c})=k$ occurs as a triple root in the special case that $\tan (\theta )= \frac{2ck}{k^2-c^2}$ (this situation is illustrated in Fig. 4); the fourth root in this case is $a_2(\sqrt{2c})=-(2c^2+k^2)/k$. Hence for generic values of $k,\theta$ we get two distinct solutions satisfying $a_2(\sqrt{2c})=k$ (this is illustrated in Fig. 2 and 3).

To conclude the proof we still need to show that $a_2(r)$ is well-defined for all values of $r\in [\sqrt{2c},+\infty )$ since this is not a priori obvious from expression (5.18).

Let us first consider the situation when $\tan (\theta )=0$. In this case from (5.18) we have that

$$\begin{aligned} (a_2 r^2 -2ck)(4a_2^2-r^4+4c^2-4k^2)=0 \end{aligned}$$

i.e.

$$\begin{aligned} a_2 = \frac{2ck}{r^2},\ a_2 = \pm \frac{1}{2}\sqrt{r^4-4c^2+4k^2}. \end{aligned}$$

(5.21)

Hence, clearly there are 2 distinct solutions with $a_2(\sqrt{2c})=k$ and they are well-defined for $r\in [\sqrt{2c},+\infty )$. On the other hand, when $\cot (\theta )=0$, from (5.18) we have that

$$\begin{aligned} a_2&=\frac{r^2}{2} \pm \frac{1}{2}\sqrt{2r^4-4c^2-8ck+4k^2},\\ a_2&=-\frac{r^2}{2}\pm \frac{1}{2}\sqrt{2r^4-4c^2+8ck+4k^2}. \end{aligned}$$

Again it is not hard to see that we get two distinct solutions satisfying $a_2(\sqrt{2c})=k$ which are well-defined for $r\in [\sqrt{2c},\infty )$, except in case $c=-3k$ when $a_2(\sqrt{2c})=k$ occurs as a triple root as we already noted above.

For the more general case when $\cot (\theta ),\tan (\theta )\ne 0$, we can view (5.18) as a quartic polynomial in $a_2$. Setting $c=1$ for convenience, we find that the discriminant of this polynomial (up to a positive constant) is given by

$$\begin{aligned} \Big ((k^4-6k^2+1)\tan ^2(\theta )-4k(k^2-1)\tan (\theta )+\frac{r^4/4+k^2-1}{\tan (\theta )}\Big )(r^4-4)^2(r^4+4k^2). \end{aligned}$$

When $r=\sqrt{2c}=\sqrt{2}$, we see that the above expression is zero which is consistent with the fact that there are two solutions $a_2(r)$ emanating from $(\sqrt{2c},k)$. Since the discriminant is positive for $r> 2$, we deduce that indeed $a_2(r)$ is well-defined for all $r\in [\sqrt{2c},\infty )$ and this concludes the proof. $\square$

Remark 5.9

There are 3 distinct $\omega _1$-dHYM connections corresponding to the case when $\theta =0$. One is the hyper-holomorphic connection of Lemma 5.5 (as expected from Proposition 2.5). If furthermore, we set $k=\pm c$ then the other two solutions correspond to $F_A= \mp \omega _1$ (these are the elementary examples described in Sect. 2).

Fig. 2

Graphical solutions to (5.20) with $c=1$, $k=3$ and $\theta =0$ illustrating a generic case of Theorem 5.8. The asymptotically zero solution corresponds to the hyper-holomorphic connection of Lemma 5.5, the vertical dotted line to $r=c$ and the horizontal dotted line to $a_2=k$. There are 2 distinct solutions emanating from (c, k) which are well-defined for $r\in [1,+\infty ).$

Fig. 3

Graphical solutions to (5.20) with $c=1$, $k=3$ and $\theta =2$ illustrating a generic case of Theorem 5.8 with non-zero phase angle $\theta$

Fig. 4

Graphical solutions to (5.20) with $c=1$, $k=3$ and $\theta =\arctan (3/4)$ illustrating a non-generic case of Theorem 5.8 whereby there are 3 distinct solutions

Fig. 5

Graphical solutions to (5.22) with $c=k=1$ and $\theta =\arctan (2)$ illustrating a generic case of Theorem 5.10

Theorem 5.10

The connection $A(r)=k \theta _1+a_2\theta _2+a_3\theta _3+a_4\theta _4$ is a dHYM connection with phase $e^{i\theta }$ with respect to $\omega _2$ on $T^*\mathbb {C}\mathbb {P}^2$ if and only if $a_2= 2ck/r^2$, $a_4=0$ and $a_3(r)$ is implicitly defined by

$$\begin{aligned} \tan (\theta ) =\frac{2(4a_3r^4\sqrt{r^4-4c^2})(r^8-4 r^4(a_3^2+c^2-k^2)-16c^2k^2)}{(4a_3r^4\sqrt{r^4-4c^2})^2-(r^8-4 r^4(a_3^2+c^2-k^2)-16c^2k^2)^2}, \end{aligned}$$

(5.22)

where $k \in \mathbb {Z}$. For generic values of $k,\theta$, there are 4 distinct $\omega _2$-dHYM connections on $T^*\mathbb {C}\mathbb {P}^2$ i.e. these are the solutions $a_3(r)$ satisfying $a_3(\sqrt{2c})=0$.

Proof

Since the proof is similar as in the previous theorem, we shall only highlight the main computations. From Lemma 5.5 we already know that we need $a_2=2ck/r^2$ and $a_4=0$. A straightforward calculation then shows that

$$\begin{aligned} F_A \wedge \omega _2^3= & {} -\frac{a_3'(r^4-4c^2)+6r^3 a_3}{4r^3(r^4-4c^2)^{1/2}} \omega _2^4,\\ F_A^4= & {} \frac{8(r^4a_3^2-k^2r^4+4c^2k^2)(r^5a_3a_3'-8c^2k^2)}{r^{12}(r^4-4c^2)}\omega _2^4,\\ F_A^2 \wedge \omega _2^2= & {} \frac{3r^5(r^4-4c^2)a_3 a_3'+6r^8 a_3^2-2k^2(r^8-16c^4)}{3r^8(r^4-4c^2)}\omega _2^4. \end{aligned}$$

As before computing (2.17) we find that the resulting ODE is exact and that the general solution on the open set $\mathbb {R}_{r>\sqrt{2 c}}\times N^{1,1}$ is given by

$$\begin{aligned}&16\tan (\theta )a_3^4-32(r^4-4c^2)^{1/2} a_3^3-24\tan (\theta )(r^4-4c^2)(1+\frac{4k^2}{3r^4}) a_3^2\\&\quad + \Bigg (\frac{32k^2}{r^4}+8\Bigg )(r^4-4c^2)^{3/2}a_3\\&\quad +\tan (\theta )((r^4-4c^2)^2+\frac{8k^2}{r^8}(r^4(r^8+16c^4)-16c^2k^2(r^4-2c^2)))-C=0. \end{aligned}$$

Imposing that $a_3(\sqrt{2c})=0$ we find that we need

$$\begin{aligned} C=16k^2\tan (\theta )(4c^2-k^2). \end{aligned}$$

Rewriting the resulting expression with C as above gives (5.22). A similar argument as in the previous theorem shows that the solutions $a_3(r)$ are well-defined for all $r\in [\sqrt{2c},\infty )$ (see also Remark 5.15 (3) below). $\square$

Remark 5.11

There are 3 distinct $\omega _2$-dHYM connections corresponding to the case when $\theta =0$. One is the hyper-holomorphic connection of Lemma 5.5 (as expected from Proposition 2.5). If furthermore, we set $k=0$ then the other two solutions correspond to $F_A= \pm \omega _2$ (these are again the elementary examples described in Sect. 2).

Note that for dHYM connections with respect to $\omega _3$ it suffices to swap $a_3$ and $a_4$ in Theorem 5.10. Next we investigate the deformed $\textrm{Spin}(7)$ instanton condition.

Theorem 5.12

The connection form $A(r)=k \theta _1+a_2\theta _2+a_3\theta _3+a_4\theta _4$ is a deformed $\textrm{Spin}(7)$-instanton on $T^*\mathbb {C}\mathbb {P}^2$ with respect to $\Phi _1$ if and only if either of the following holds:

• (1) $a_3(r)=a_4(r)=0$ and

  $$\begin{aligned} a_2(r)=\frac{2ck}{r^2}\theta _2, \end{aligned}$$

  (5.23)

• (2) $a_2(r)=2ck/r^2$, $a_3(r)=C_3 p(r)$ and $a_4(r)=C_4 p(r)$, where

  $$\begin{aligned} p(r)=\frac{\sqrt{(r^4+4k^2)(r^4-4c^2)}}{2r^2\sqrt{C_3^2+C_4^2 }}, \end{aligned}$$

  (5.24)

• (3) $a_3(r)=a_4(r)=0$ and

  $$\begin{aligned} a_2(r)= \Bigg (\frac{1}{2}C r^2 \pm \frac{1}{2}\sqrt{C^2 r^4-8 C c k+{r^4}-4c^2+4k^2}\Bigg ), \end{aligned}$$

  (5.25)

  where if $cC\ge (\le ) k$ we take the − ($+$) sign so that $a_2(\sqrt{2c})=k$.

Proof

First, from Proposition 5.4 we have that

$$\begin{aligned}&F_A \wedge F_A = \\&\quad -dr\wedge \big (4(a_2' a_2+a_3' a_3+a_4' a_4)\theta _{234}+2a_2'(a_2+k)\theta _{256}+2a_2'a_3\theta _{2}\wedge \sigma _{2}\\&\quad +2a_2' a_4 \theta _{2}\wedge \sigma _3+2a_2'(a_2-k)\theta _{278}+2a_3'(a_2+k)\theta _{356}+2a_3'a_3\theta _3\wedge \sigma _{2}\\&\quad +2 a_3'a_4\theta _3\wedge \sigma _3+2a_3'(a_2-k)\theta _{378}+2a_4'(a_2+k)\theta _{456}+2a_4'a_3\theta _{4}\wedge \sigma _{2}\\&\quad +2a_4'a_4 \theta _4 \wedge \sigma _{3}+2a_4'(a_2-k)\theta _{478}\big ) +4a_4(a_2+k)\theta _{2356}+4a_3 a_4\theta _{23}\wedge \sigma _2\\&\quad +4a_4^2\theta _{23}\wedge \sigma _3+4a_4(a_2-k)\theta _{2378}-4a_3(a_2+k)\theta _{2456}\\&\quad -4a_3^2\theta _{2457}-4a_3 a_4 \theta _{24}\wedge \sigma _{3}+4a_3^2\theta _{2468}-4a_3(a_2-k)\theta _{2478}+4a_2(a_2+k)\theta _{3456}\\&\quad +4a_2a_3\theta _{34}\wedge \sigma _2+4a_2a_4\theta _{34}\wedge \sigma _3+4a_2(a_2-k)\theta _{3478}+2(a_2^2+a_3^2+a_4^2-k^2)\theta _{5678}. \end{aligned}$$

On the other hand from Propositions 2.3 and 5.2 we know that the $\textrm{SU}(3)$-invariant forms in $\Lambda ^4_{7}$ (defined with respect to $\Phi _1$) are spanned by $\alpha _2\wedge \omega _3= - \alpha _3\wedge \omega _2\in F_1^+$, $\omega _1\wedge \omega _2$ and $\omega _1\wedge \omega _3$. Hence $\pi ^4_7(F_A\wedge F_A)=0$ is equivalent to the condition that $F_A \wedge F_A$ is orthogonal to the latter triple. Using the above expression for $F_A \wedge F_A$ and using Proposition 5.1 we have that $g_{Ca}(F_A\wedge F_A,\omega _1\wedge \omega _2)=0$ is given by

$$\begin{aligned}{} & {} 4f_0f_2f_5f_7 a_2 a_3+ 2f_0f_3f_5^2 a_3(a_2-k)+2f_0f_3f_7^2 a_3(a_2+k)-f_2f_3f_5^2a_3'(a_2-k)\\{} & {} \quad -f_2f_3f_7^2a_3'(a_2+k)-2f_3^2f_5f_7 a_2'a_3=0, \end{aligned}$$

where $f_i$ are as defined in Proposition 5.1. Simplifying, it turns out that the latter is equivalent to

$$\begin{aligned} \frac{d}{dr}((r^2a_2-2ck)(r^4-4c^2)^{1/2}a_3)=0. \end{aligned}$$

(5.26)

Similarly, we compute $g_{Ca}(F_A\wedge F_A,\alpha _2\wedge \omega _3)=0$, $g_{Ca}(F_A\wedge F_A,\omega _1\wedge \omega _3)=0$ and find that these are given by

$$\begin{aligned}{} & {} a_3'a_4=a_4' a_3, \end{aligned}$$

(5.27)

$$\begin{aligned}{} & {} \frac{d}{dr}((r^2a_2-2ck)(r^4-4c^2)^{1/2}a_4 )=0, \end{aligned}$$

(5.28)

respectively. Considering all the possible distinct solutions to the above system of ODEs, we deduce that A(r) is either of the form

$$\begin{aligned} A=k\theta _1 +\frac{2ck}{r^2} \theta _2 + p(r)(C_3\theta _3+C_4\theta _4), \end{aligned}$$

(5.29)

where p(r) is an arbitrary function and $C_3,C_4$ are constants, or

$$\begin{aligned} A=k\theta _1 +a_2 \theta _2 + \frac{1}{(r^2 a_2-2ck)\sqrt{r^4-4c^2}}(C_3\theta _3+C_4\theta _4). \end{aligned}$$

(5.30)

With A as given by (5.29), a rather long but straightforward computation using Proposition 5.1, (5.11) and (2.24) shows that

$$\begin{aligned}&\pi ^2_7(F_A-\frac{1}{6}* F_A^3) \\&\quad = \frac{1}{4r^8(r^4-4c^2)^{3/2}} \Big (r(r^4-4c^2)(12r^4(C_3^2+C_4^2)p^2-(r^4-4k^2)(r^4-4c^2))p'(r)\\&\qquad +8r^8(C_3^2+C_4^2)p^3-6(r^4-4c^2)(r^8+4k^2r^4/3+32c^2k^2/3)p \Big )(C_3 f_0f_3dr \wedge \theta _3\\&\quad +C_4 f_0f_3dr \wedge \theta _4+C_4 f_2f_3\theta _{23}-C_3 f_2f_3\theta _{24}+C_3 f_5f_7\sigma _2+C_4f_5f_7 \sigma _3). \end{aligned}$$

Simplifying the above, we find that (1.4) is equivalent to

$$\begin{aligned} \frac{d}{dr}\Big ((r^4-4c^2)^{1/2}(C_3^2+C_4^2)p(r)^3-\frac{(r^4+4k^2)(r^4-4c^2)^{3/2}}{4r^4}p(r)\Big )=0 \end{aligned}$$

and hence p(r) is implicitly defined as the solution to a cubic polynomial whereby the coefficients are functions of r.

On the other hand, when A as given by (5.30), similarly as above we find that (1.4) is equivalent to

$$\begin{aligned} \frac{d}{dr}\Bigg (\frac{a_2^2-r^4/4+c^2-k^2}{a_2r^2-2ck}+\frac{C_3^2+C_4^2}{(a_2r^2-2ck)^3(r^4-4c^2)}\Bigg )=0 \end{aligned}$$

and hence $a_2(r)$ is implicitly defined by

$$\begin{aligned}&\Bigg (\frac{4a_2}{r^2}+\frac{8ck}{r^4}-C\Bigg ){(r^4-4c^2)(a_2r^2-2ck)^3}-\nonumber \\&\quad \frac{(r^4-4c^2)^2(r^4+4k^2)(a_2r^2-2ck)^2}{r^4} +4(C_3^2+C_4^2)=0. \end{aligned}$$

(5.31)

Thus, we have found all the $\textrm{SU}(3)$-invariant deformed $\textrm{Spin}(7)$-instantons with respect to $\Phi _1$ on $\mathbb {R}_{r>\sqrt{2c}}\times N^{1,1}$. To conclude it remains to apply the initial conditions to the above local solutions.

Imposing $a_2(\sqrt{2c})=k$ in (5.31) shows that we need $C_3=C_4=0$. Solving for $a_2(r)$ we get the deformed $\textrm{Spin}(7)$-instanton given by (5.25). Similarly, imposing $p(\sqrt{2c})=0$ we get the hyper-holomorphic connection (5.23) and the deformed $\textrm{Spin}(7)$-instanton given by (5.24). $\square$

Remark 5.13

• (1) Observe that the solution given by (5.23) corresponds to the hyper-holomorphic connection of Lemma 5.5. This was indeed to be expected as we showed in Proposition 2.5.

• (2) Comparing with Theorem 5.10, it is easy to see that the solutions given by (5.24) correspond to the $\omega _2$-dHYM connections with phase 1 when $C_4=0$ and to the $\omega _3$-dHYM connections with phase 1 when $C_3=0$. More generally, it is not hard to verify that these deformed $\textrm{Spin}(7)$-instantons are in fact also dHYM connections with phase 1 with respect to the 1-parameter family of Kähler forms given by $\frac{C_3}{\sqrt{C_3^2+C_4^2}}\omega _2+\frac{C_4}{\sqrt{C_3^2+C_4^2}}\omega _3$; this is again a consequence of Theorem 2.4 (1). Furthermore, these deformed $\textrm{Spin}(7)$-instantons all satisfy $*F_A^4=24$, cf. Remark 2.2.

• (3) Setting $\tan (\theta )=\frac{2 C}{C^2-1}$ in (5.18) one can verify directly that the solutions given by (5.18) coincide with those given by (5.25). Thus, the deformed $\textrm{Spin}(7)$-instantons given by (5.25) in fact correspond to the $\omega _1$-dHYM connections for all phase angle $\theta$. In particular, we see that asymptotically $a_2(r) \sim (C \pm \sqrt{C^2+1})r^2/2$ i.e. $a_2$ always has quadratic growth but the coefficient depends on the phase angle. Unlike the deformed $\textrm{Spin}(7)$-instantons given by (5.24), these examples, as well as the hyper-holomorphic connection, do not satisfy $*F_A^4=24$ except when $C=0$.

Theorem 5.14

The connection form $A(r)=k \theta _1+a_2\theta _2+a_3\theta _3+a_4\theta _4$ is a deformed $\textrm{Spin}(7)$-instanton on $T^*\mathbb {C}\mathbb {P}^2$ with respect to $\Phi _2$ if and only if either of the following holds:

• (1) $a_3(r)=a_4(r)=0$ and

  $$\begin{aligned} a_2(r)=\frac{2ck}{r^2} \end{aligned}$$

  (5.32)

• (2) $a_3(r)=a_4(r)=0$ and

  $$\begin{aligned} a_2(r)=\pm \frac{1}{2}\sqrt{r^4-4c^2+4k^2} \end{aligned}$$

  (5.33)

• (3) $a_3(r)=0$,

  $$\begin{aligned} a_2(r)=\frac{2ck}{r^2}+Ca_4(r)\cdot \frac{(r^4-4c^2)^{1/2}}{r^2} \end{aligned}$$

  (5.34)

  and

  $$\begin{aligned} a_4(r)=\frac{4 Cck\pm r^2\sqrt{C^2(r^4-4c^2+4k^2)+r^4+4k^2}}{-2C^2(r^4-4c^2)-2r^4}(r^4-4c^2)^{1/2}, \end{aligned}$$

  (5.35)

• (4) $a_2(r)=2ck/r^2$, $a_4(r)=0$ and

  $$\begin{aligned} a_3(r)=\frac{C r^2 \pm \sqrt{(C^2+1)r^4+4k^2}}{2r^2}(r^4-4c^2)^{1/2}. \end{aligned}$$

  (5.36)

Proof

From Propositions 2.3 and 5.2 we have that the $\textrm{SU}(3)$-invariant forms in $\Lambda ^4_{7}$ (defined with respect to $\Phi _2$) are spanned by $\alpha _3\wedge \omega _1 = - \alpha _1\wedge \omega _3\in F_2^+$, $\omega _2\wedge \omega _3$ and $\omega _1\wedge \omega _2$. Hence $\pi ^4_7(F_A\wedge F_A)=0$ is equivalent to the condition that $F_A \wedge F_A$ is orthogonal to the latter triple. A direct but lengthy calculation as before shows this is given by the system

$$\begin{aligned}{} & {} \frac{d}{dr}(a_3 a_4 (r^4-4c^2) )=0, \end{aligned}$$

(5.37)

$$\begin{aligned}{} & {} (a_2 r^2 -2ck)^2 \cdot \frac{d}{dr}\Bigg (\frac{a_4(r^4-4c^2)^{1/2}}{a_2 r^2 -2ck}\Bigg )=0, \end{aligned}$$

(5.38)

$$\begin{aligned}{} & {} \frac{d}{dr}((a_2 r^2-2ck)(r^4-4c^2)^{1/2}a_3)=0. \end{aligned}$$

(5.39)

Imposing the initial conditions $a_3(\sqrt{2c})=a_4(\sqrt{2c})=0$ in (5.37) implies that either $a_3(r)=0$ or $a_4(r)=0$.

Assuming $a_3(r)=0$ automatically solves (5.39) and from (5.38) we find that we need $a_2(r)$ of the form (5.34). Substituting the latter in (1.4), a long calculation yields the exact ODE:

$$\begin{aligned}&\frac{d}{dr}\Bigg (\frac{(C^2(r^4-4c^2)+r^4)(r^4-4c^2)^{1/2}}{r^4}a_4^3+\frac{4Cck(r^4-4c^2)}{r^4}a_4^2\\&\quad -\frac{(r^4+4k^2)(r^4-4c^2)^{3/2}}{4r^4}a_4\Bigg )=0 \end{aligned}$$

Solving for $a_4(r)$ with the initial condition that $a_4(\sqrt{2c})=0$ shows that either $a_4(r)=0$, in which case we get (5.32), or $a_4(r)$ is given by (5.35).

On the other hand, assuming instead that $a_4(r)=0$ automatically solves (5.38) and we have that (5.39) is equivalent to

$$\begin{aligned} (a_2 r^2-2ck)(r^4-4c^2)^{1/2}a_3 = C, \end{aligned}$$

(5.40)

where C is a constant. Imposing the initial conditions $a_2(\sqrt{2c})=k$ and $a_3(\sqrt{2c})=0$ in the latter shows that $C=0$ and hence either $a_2=2ck/r^2$ or $a_3=0$. In the latter case from Lemma 5.5 we know that A is holomorphic with respect to $\omega _1$. Thus, by Theorem 2.4 (1), the deformed $\textrm{Spin}(7)$-instantons with respect to $\Phi _2$ coincide with $\omega _1$-dHYM connections with phase 1 and this gives (5.33). In the former case again from Lemma 5.5 we now see that A is holomorphic with respect to $\omega _2$. However, we cannot appeal to Theorem 2.4 as before since we are now considering deformed $\textrm{Spin}(7)$-instantons with respect to $\Phi _2$. With the hypothesis that $a_2=2ck/r^2$ and $a_4=0$, we find after a long computation that (1.4) is equivalent to

$$\begin{aligned}&\big (4a_3^2 r^4-(r^4-4c^2)(r^4+4k^2)\big )^3\frac{d}{dr}\\&\quad \Bigg (\frac{\big (4a_3^2 r^4+(r^4-4c^2)(r^4-4k^2)\big )^2+16k^2r^4(r^4-4c^2)^2}{\big (4a_3^2 r^4-(r^4-4c^2)(r^4+4k^2)\big )^2}\Bigg )=0. \end{aligned}$$

Solving for $a_3(r)$ with $a_3(\sqrt{2c})=0$ gives (5.36). $\square$

Note that the deformed $\textrm{Spin}(7)$-instantons with respect to $\Phi _3$ can be deduced from Theorem 5.14 by swapping $a_3$ and $a_4$.

Remark 5.15

• (1) Again as expected from Proposition 2.5 the solution given by (5.32) corresponds to the hyper-holomorphic connection of Lemma 5.5.

• (2) The solutions given by (5.33) correspond to the $\omega _1$-dHYM connections with phase 1. Recall also that the $\omega _3$-dHYM connections can be deduced from Theorem 5.10 by swapping $a_3$ and $a_4$. The solutions given by (5.34)–(5.35) when $C=0$ correspond precisely to the $\omega _3$-dHYM connections with phase 1; this is again a consequence of Theorem 2.4 (1). More generally, one can show that these deformed $\textrm{Spin}(7)$-instantons are in fact also dHYM connections with phase 1 with respect to the 1-parameter family of the Kähler forms given by $\frac{C}{\sqrt{C^2+1}}\omega _1+\frac{1}{\sqrt{C^2+1}}\omega _3$. The solution (5.33) is obtained from (5.34)–(5.35) by setting $C=1/\tilde{C}$ and taking the limit as $\tilde{C}\rightarrow 0$. It is not hard to verify that all these deformed $\textrm{Spin}(7)$-instantons also satisfy $*F_A^4=24$, cf. Remark 2.2.

• (3) Setting $\tan (\theta )=\frac{2 C}{C^2-1}$ in (5.22) one can verify directly that the solutions given by (5.22) coincide with those given by (5.36). Note that for each fixed C and k (5.36) consists of two solutions and on the other hand given $\theta$ there are in general two values of C solving $\tan (\theta )=\frac{2 C}{C^2-1}$. Hence we see that this indeed yields a total of four solutions $a_3(r)$ for given $\theta$ and k which is consistent with Theorem 5.10. Thus, we have that the deformed $\textrm{Spin}(7)$-instantons given by (5.36) correspond to the $\omega _2$-dHYM connections for all phase angle $\theta$. In particular, we see that asymptotically $a_3(r) \sim (C \pm \sqrt{C^2+1})r^2/2$ i.e. $a_3$ always has quadratic growth but the coefficient depends on the phase angle. Unlike the previous deformed $\textrm{Spin}(7)$-instantons, these examples do not satisfy $*F_A^4=24$ except when $C=0$.

Note that when $\tan (\theta )\ne 0$ equation (2.17) is quartic in $F_A$ while equation (1.4) is only cubic, so in general one would not expect the dHYM connections and deformed $\textrm{Spin}(7)$-instantons to coincide. However, as we saw in Remark 5.13 (3) and 5.15 (3), these instantons also satisfy $*F_A^4=24$ which might explain the above observations. It is also worth pointing out that the condition $*F_A^4 \ne 24$ is necessary in order to obtain a good description of the moduli space of deformed $\textrm{Spin}(7)$-instantons, see [22, Theorem 1.2]. So it would be interesting to understand the geometrical significance of this condition. We also saw in this section that all the ODEs arising from the deformed instanton equations were always exact and hence we were able to find explicit solutions. However, this is not the case in general as we shall show in the next section.

6 Deformed $\textrm{Spin}(7)$-Instantons on 3-Sasakian Cones

Recall that the cone metric over a strictly nearly parallel $\textrm{G}_2$ manifold has holonomy group equal to $\textrm{Spin}(7)$ while the cone metric over a 3-Sasakian 7-manifold has holonomy group equal to $\textrm{Sp}(2)$. In [29] Lotay-Oliveira constructed deformed $\textrm{G}_2$-instantons on the Aloff-Wallach space $N_{1,1}$ endowed with its 3-Sasakian and strictly nearly parallel $\textrm{G}_2$-structures. They showed that the moduli space of deformed $\textrm{G}_2$-instantons in each case is different: in particular, in the 3-Sasakian case the moduli space is one dimensional while in the strictly nearly parallel case it is two dimensional cf. [29, Corollary 4.12]. Hence their result shows that the deformed $\textrm{G}_2$-instantons can be used to distinguish between these two $\textrm{G}_2$-structures. In this section we shall show that a similar result also holds for deformed $\textrm{Spin}(7)$ instantons on the associated $\textrm{Spin}(7)$ cones. As before we shall again use the notation introduced in Sect. 4.

Proposition 6.1

The connection form $A(r)=k \theta _1+a_2\theta _2+a_3\theta _3+a_4\theta _4$ is a deformed $\textrm{Spin}(7)$-instanton on $N_{1,1}\times \mathbb {R}^+$ for the Bryant-Salamon $\textrm{Spin}(7)$-structure as given in Proposition 5.3 if and only if

$$\begin{aligned} A = k \theta _1+ p(r)\big (C_2\theta _2+C_3\theta _3+C_4\theta _4\big ), \end{aligned}$$

(6.1)

where $C_i$ are constants and p(r) solves

$$\begin{aligned}{} & {} \big ((9r^2+10c)C^2p^2+k^2r^2+100r^2(r^2+c)^{6/5}\big )p'-40r(3r^2+5c)(r^2+c)^{1/5}p\nonumber \\{} & {} \quad +2r(C^2p^2-k^2)p=0, \end{aligned}$$

(6.2)

where $C^2:=C_2^2+C_3^2+C_4^2$.

Proof

Recall that in Proposition 5.2 we showed that the space of $\textrm{SU}(3)$-invariant 2-forms are spanned by $\langle \theta _{23},\theta _{24},\theta _{34},\theta _{56},\theta _{78},\sigma _{2},\sigma _{3}, dr \wedge \theta _2, dr \wedge \theta _3, dr \wedge \theta _4\rangle$. In this 10 dimensional family using (2.24) we find that $\Lambda ^2_7$ (defined with respect to $\Phi _{BS}$) is 3-dimensional and in fact is spanned by

$$\begin{aligned}{} & {} h_0h_2 dr \wedge \theta _2+h_2^2 \theta _{34}-h_5^2 \theta _{56}-h_5^2 \theta _{78},\\{} & {} h_0h_2 dr \wedge \theta _3-h_2^2 \theta _{24}-h_5^2 \theta _{57}+h_5^2 \theta _{68},\\{} & {} h_0h_2 dr \wedge \theta _4+h_2^2 \theta _{23}-h_5^2 \theta _{58}-h_5^2 \theta _{67}. \end{aligned}$$

where $h_i$ are as defined in Proposition 5.3. It follows that the space of $\textrm{SU}(3)$-invariant forms in $\Lambda ^4_7$ is also 3 dimensional and a rather laborious calculation shows that it is spanned by

$$\begin{aligned}{} & {} h_0h_2h_5^2dr \wedge (\theta _{358}+\theta _{367}-\theta _{457}+\theta _{468} )-h_2^2h_5^2(\theta _{2357}-\theta _{2368}+\theta _{2458}+\theta _{2467}),\\{} & {} h_0h_2h_5^2dr \wedge (\theta _{258}+\theta _{267}-\theta _{456}-\theta _{478} )-h_2^2h_5^2(\theta _{2356}+\theta _{2378}-\theta _{3458}-\theta _{3467}),\\{} & {} h_0h_2h_5^2dr \wedge (\theta _{257}-\theta _{268}-\theta _{356}-\theta _{378}) +h_2^2h_5^2(\theta _{2456}+\theta _{2478}+\theta _{3457}-\theta _{3468}). \end{aligned}$$

Hence $\pi ^4_7(F_A \wedge F_A)=0$ if and only if $F_A \wedge F_A$ is orthogonal to latter triple. A direct computation shows that 1.3 is equivalent to

$$\begin{aligned} a_2' a_3= & {} a_2 a_3', \end{aligned}$$

(6.3)

$$\begin{aligned} a_2' a_4= & {} a_2 a_4', \end{aligned}$$

(6.4)

$$\begin{aligned} a_3' a_4= & {} a_3 a_4'. \end{aligned}$$

(6.5)

Thus, it follows that we need A of the form (6.1). Using Proposition 5.4, a long calculation shows that

$$\begin{aligned} F_A - \frac{1}{6}*F_A^3 =&\Bigg (\frac{50r(r^2+c)^{6/5} p'+C^2 p^3-k^2p}{200 (r^2+c)^{4/5}r^2}\Bigg ) h_0 h_2 (\theta _1 \wedge (C_2\theta _2+C_3\theta _3+C_4\theta _4)) -\\&\Bigg (\frac{200(r^2+c)^{6/5} p^2+r(C^2p^2-k^2)p'}{400 (r^2+c)^{4/5}r^2}\Bigg ) h_2^2 (C_2\theta _{34}+C_3\theta _{42}+C_4\theta _{23})+\\&\Bigg (\frac{C^2(r^2+c)^{4/5}(k-C_2p)pp'-8r^3(C_2p+k)}{80 (r^2+c)^{3/5}r^3}\Bigg ) h_5^2 \theta _{56}+\\&\Bigg (\frac{C^2(r^2+c)^{4/5}(-k-C_2p)pp'-8r^3(C_2p-k)}{80 (r^2+c)^{3/5}r^3}\Bigg ) h_5^2 \theta _{78}-\\&\Bigg (\frac{(C^2(r^2+c)^{4/5}pp'+8r^3)p}{80 (r^2+c)^{3/5}r^3}\Bigg ) h_5^2 (C_3\sigma _2+C_4\sigma _3). \end{aligned}$$

Using (2.24) we can then compute (1.4) and another lengthy computation yields (6.2). $\square$

In contrast to the ODEs we obtained in Sect. 5, equation (6.2) is not exact and hence we have been unable to find the general solution to (6.2). However in the special case when $k=c=0$, i.e. when $g_{BS}$ corresponds to the conical metric and L is the trivial line bundle, we have the following result:

Theorem 6.2

Consider $N_{1,1}\times \mathbb {R}^+$ endowed with the Bryant-Salamon conical $\textrm{Spin}(7)$-structure of Proposition 5.3. Then the $\textrm{SU}(3)$-invariant connection $A(r)=a_2\theta _2+a_3\theta _3+a_4\theta _4$ is a deformed $\textrm{Spin}(7)$-instanton if and only if

$$\begin{aligned} A = \frac{10 r^{6/5}}{3\sqrt{W(C_0 r^{128/45})}}\Bigg (\frac{C_2\theta _2+C_3\theta _3+C_4\theta _4}{\sqrt{C_2^2+C_3^2+C_4^2}}\Bigg ), \end{aligned}$$

(6.6)

where W denotes the Lambert W-function, implicitly defined by $x=W(x)\exp (W(x))$, and $C_0$ is a positive constant. On the other hand we have that A is a $\textrm{Spin}(7)$-instanton if and only if

$$\begin{aligned} A = r^{6/5}\big (C_2\theta _2+C_3\theta _3+C_4\theta _4\big ). \end{aligned}$$

(6.7)

Proof

Setting $c=k=0$ in (6.2) we get

$$\begin{aligned} \big (9C^2rp^2+100r^{17/5}\big )p'-120r^{12/5}p+2C^2p^3=0. \end{aligned}$$

Solving the latter yields (6.6). On the other hand, a straightforward computation shows that $*(F_A\wedge \Phi _{BS})=-F_A$ is equivalent to the system

$$\begin{aligned} r(r^2+c)a_i'-2\left( \frac{3r^2}{5}+c\right) a_i=0. \end{aligned}$$

The solution with $c=0$ is given by (6.7). $\square$

Note that since asymptotically $W(x)\sim \log (x) +o(1)$, it follows that the deformed $\textrm{Spin}(7)$-instantons given by (6.6) have slower growth rate than the $\textrm{Spin}(7)$-instantons given by (6.7). We should point out that Theorem 6.2 does not only apply to the Aloff-Wallach space $N_{1,1}$ but to any 3-Sasakian 7-manifold with its squashed strictly nearly parallel $\textrm{G}_2$-structure as given by [13, Proposition 2.4]; in this case the 1-forms $\theta _{2},\theta _{3},\theta _{4}$ in Theorem 6.2 are then replaced by the dual 1-forms to the Reeb Killing vector fields. Moreover, it is worth noting that Proposition 6.1 with $k=0$ also corresponds to the deformed $\textrm{Spin}(7)$ condition on the trivial line bundle over the spinor bundle of $S^4$ endowed with the complete Bryant-Salamon $\textrm{Spin}(7)$ metric [5]: this is simply because the structure equations are the same. In this case the initial condition we need for A to extend smoothly to the zero section $S^4$ is $p(0)=0$. Thus, equation (6.2) corresponds to a singular initial value problem and as such one cannot appeal standard theory to obtain existence and uniqueness of a solution. With $k=0$ and $C=c=1$, seeking for a power series solution to (6.2) around $r=0$ gives

$$\begin{aligned} p(r)=a-\frac{a^2-100}{10a} r^2+\frac{a^4-2000}{20a^3}r^4+\cdots \end{aligned}$$

where $a\in \mathbb {R}$. In particular, we see that such a solution cannot vanish at $r=0$. We saw a similar phenomenon when investigating the deformed $\textrm{G}_2$-instanton equations on the spinor bundle of $S^3$ in [12, Section 4.2], but nonetheless there can still exist smooth solutions as demonstrated therein. We hope to analyse equation (6.2) in more detail in future work to find global solutions. Note that we also obtained singular initial value problems in the last section but this was not a difficulty since we were already able to solve the equations explicitly.

We now want to compare the above instantons on the strictly nearly parallel $\textrm{G}_2$ cone with those on the 3-Sasakian cone. From the results of Sect. 5 we can easily deduce the following:

Theorem 6.3

Consider $N_{1,1}\times \mathbb {R}^+$ endowed with the Calabi conical hyperKähler structure of Proposition 5.1. Then the $\textrm{SU}(3)$-invariant connection $A(r)=a_2\theta _2+a_3\theta _3+a_4\theta _4$ is a deformed $\textrm{Spin}(7)$-instanton with respect to $\Phi _1$ if and only if either $A=C_2r^2\theta _2$ or

$$\begin{aligned} A=p(r)\Bigg (\frac{C_3\theta _3+C_4\theta _4}{2\sqrt{C_3^2+C_4^2}}\Bigg ), \end{aligned}$$

(6.8)

where p(r) is implicitly defined by

$$\begin{aligned} p^3-{r^4}p-\frac{C_0}{r^2}=0. \end{aligned}$$

(6.9)

On the other hand we have that A is a $\textrm{Spin}(7)$-instanton with respect to $\Phi _1$ if and only if

$$\begin{aligned} A = C_2r^2\theta _2+\frac{1}{r^6}\Big (C_3\theta _3+C_4\theta _4\big ). \end{aligned}$$

(6.10)

The instantons with respect to $\Phi _2$ and $\Phi _3$ are obtained by permuting $\theta _2,\theta _3$ and $\theta _4$ in the above expressions for A.

Observe that when $A=C_2r^2\theta$ in Theorem 6.3, we have that $F_A$ is a constant multiple of $\omega _1$ and this corresponds to both a $\textrm{Spin}(7)$-instanton and deformed $\textrm{Spin}(7)$-instanton with respect to $\Phi _1$. On the other hand when $C_0=0$ and $p(r)=\pm r^2$, from (6.8) we have that $F_A=\mp \frac{C_3\omega _2+C_4\omega _3}{\sqrt{C_3^2+C_4^2}}$ and this corresponds to a deformed $\textrm{Spin}(7)$-instanton with respect to $\Phi _1$, but not a $\textrm{Spin}(7)$-instanton. These examples coincide with the elementary examples described in Sect. 2. Hence we only get non-elementary examples when $C_0\ne 0$, in which case from (6.9) we see that asymptotically either $p(r) \sim \pm r^2$ or $p(r) \sim 0$. Furthermore, we should also point out that for each $C_0 \ne 0$ exactly one of these three solutions is well-defined on $\mathbb {R}^+$, while the other two are only defined on a subset of $\mathbb {R}^+$.

Comparing the deformed $\textrm{Spin}(7)$-instantons of Theorem 6.2 and 6.3, we see that the moduli spaces in each case have different dimensions and furthermore, they have different number of connected components.

Remark 6.4

In [29, Corollary 4.5] Lotay-Oliveira showed that there exist a circle of non-trivial deformed $\textrm{G}_2$-instantons on $N_{1,1}$ with respect to (one of) its 3-Sasakian $\textrm{G}_2$-structure and a 2-sphere of non-trivial deformed $\textrm{G}_2$-instantons with respect to its strictly nearly parallel $\textrm{G}_2$-structure. In particular, this shows that deformed $\textrm{G}_2$-instantons can discriminate between these two $\textrm{G}_2$-structures cf. [29, Remark 4.6]. If we fix the constant $C_0$ in Theorem 6.2 and 6.3 above and we restrict the deformed $\textrm{Spin}(7)$-instantons A(r) to a hypersurface $r=r_0$, then we obtain a circle and 2-sphere family of connections respectively; these coincide precisely with the aforementioned deformed $\textrm{G}_2$-instantons of Lotay–Oliveira up to an overall constant factor.