SU(2)\(^2\)invariant \(G_2\)instantons
Abstract
We initiate the systematic study of \(G_2\)instantons with SU(2)\(^2\)symmetry. As well as developing foundational theory, we give existence, nonexistence and classification results for these instantons. We particularly focus on \(\mathbb {R}^4\times S^3\) with its two explicitly known distinct holonomy \(G_2\) metrics, which have different volume growths at infinity, exhibiting the different behaviour of instantons in these settings. We also give an explicit example of sequences of \(G_2\)instantons where “bubbling” and “removable singularity” phenomena occur in the limit.
1 Introduction
In this article we study \(G_2\)instantons: these are examples of Yang–Mills connections on Riemannian manifolds whose holonomy group is contained in the exceptional Lie group \(G_2\) (socalled \(G_2\)manifolds). These connections are, in a sense, analogues of antiselfdual connections in dimension 4, and are likewise hoped to be used to understand the geometry and topology of \(G_2\)manifolds, via the construction of enumerative invariants. Our focus is on \(G_2\)instantons on \(G_2\)manifolds where both the connections and ambient \(G_2\) geometry enjoy SU(2)\(^2\)symmetry. In particular, as a \(G_2\)manifold is Ricci flat, for it to admit continuous symmetries it must be noncompact. By restricting to this case, we are able to shed light on the still rather poorly understood theory of \(G_2\)instantons, in an explicit setting. In particular, we give new existence and nonexistence results for \(G_2\)instantons. Furthermore, we can see how general theory works in practice, examine how the ambient geometry affects the \(G_2\)instantons and give local models for the behaviour of \(G_2\)instantons on compact \(G_2\)manifolds.
1.1 \(G_2\)instantons
More recently, the study of \(G_2\)instantons has gained a special interest, primarily due to Donaldson–Thomas’ suggestion [10] that it may be possible to use \(G_2\)instantons to define invariants for \(G_2\)manifolds, inspired by Donaldson’s pioneering work on antiselfdual connections on 4manifolds. Later Donaldson–Segal [9], Haydys [14], and Haydys–Walpuski [15] gave further insights regarding that possibility.
On a compact holonomy \(G_2\)manifold \((X^7, \varphi )\) any harmonic 2form is “antiselfdual” as in (1.2), hence any complex line bundle L on X admits a \(G_2\)instanton, namely that whose curvature is the harmonic representative of \(c_1(L)\). However, the construction of nonabelian \(G_2\)instantons on compact \(G_2\)manifolds is much more involved. In the compact case, the first such examples were constructed by Walpuski [25], over Joyce’s \(G_2\)manifolds (see [16]). Sá Earp and Walpuski’s work [22, 26] gives an abstract construction of \(G_2\)instantons, and currently one example, on the other known class of compact \(G_2\)manifolds, namely “twisted connected sums” (see [7, 17]).
The goal of this paper is to perform a general analysis of \(G_2\)instantons on some noncompact \(G_2\)manifolds. In the noncompact setting, the first examples of \(G_2\)instantons where found by Clarke [8], and further examples were given by the second author in [20]. In this article we primarily study \(G_2\)instantons on \(\mathbb {R}^4 \times S^3\), which has two known complete and explicit \(G_2\)holonomy metrics, namely: the Bryant–Salamon (BS) metric [5] and the Brandhuber et al. (BGGG) metric [2]. Both these metrics have \(\{0\} \times S^3\) as an associative submanifold: such areaminimizing submanifolds in \(G_2\)manifolds have both known and expected relationships with \(G_2\)instantons, so studying these metrics allows us to verify known theory and test expectations. Of particular note is that the BS and BGGG metrics have different volume growths at infinity, and are in a sense analagous to the flat and TaubNUT hyperkähler metrics on \(\mathbb {R}^4\). Our results exhibit the similarities and differences in the existence theory for \(G_2\)instantons for these metrics.
1.2 Summary
The aim of the article is to start the systematic study of SU(2)\(^2\)invariant \(G_2\)instantons. We now summarize the organization of our paper and the main results.
Both the BS and BGGG metric have SU(2)\(^2\) as a subgroup of their isometry group: in fact, SU(2)\(^2\) acts with cohomogeneity1. All known complete SU(2)\(^2\)invariant \(G_2\)manifolds of cohomogeneity1 actually have SU(2)\(^2\times U(1)\)symmetry. These facts are summarized in Sect. 2, where we also deduce the ODEs for SU(2)\(^2\)invariant \(G_2\)instantons. In Sect. 2.5, we give some explicit elementary solutions to the equations, namely flat connections and abelian ones. Already in this simple abelian setting we see a marked difference between the \(G_2\)instantons for the BS and BGGG metric.
In Sect. 3 we focus on the BS metric, which has isometry group \(\mathrm{SU}(2)^3\). This group also acts with cohomogeneity1 and has a unique singular orbit which is the associative \(S^3\). We describe \(\mathrm{SU}(2)^3\)invariant \(G_2\)instantons with gauge group \(\mathrm{SU}(2)\). A dichotomy arises from the two possible homogeneous bundles over the associative \(S^3\) on which the instantons can extend: let \(P_1\) and \(P_{{{\mathrm{id}}}}\) denote these two bundles.
In the \(P_1\) case, by combining our study in Sect. 3 with our work in Sect. 4 we obtain our first main result.
Theorem 1
Let A be an irreducible \(\mathrm{SU}(2)^2 \times U(1)\)invariant \(G_2\)instanton with gauge group \(\mathrm{SU}(2)\) on the BS metric. If A smoothly extends over \(P_1\), then it is one of Clarke’s \(G_2\)instantons in [8].
See Theorems 4 and 7 for more precise statements and an explicit formula for the instantons, and see Corollary 1 for a classification of the reducible instantons. Here we mention that Clarke’s \(G_2\)instantons form a family \(\lbrace A^{x_1} \rbrace \), parametrized by \(x_1\ge 0\), and the curvature of these connections decays at infinity.
In the \(P_{{{\mathrm{id}}}}\) case, we find (in Theorem 5) a new explicit \(G_2\)instanton \(A^{\lim }\). We show in Theorem 6 and Corollary 2 that \(A^{\lim }\) is, in a certain (precise) sense, the limit of Clarke’s ones as \(x_1 \rightarrow + \infty \). We state our second main result informally, which confirms expectations from [23, 24].
Theorem 2
 (a)
After a suitable rescaling, the family \(\lbrace A^{x_1} \rbrace \) bubbles off an antiselfdual connection transversely to the associative \( S^3 = \lbrace 0 \rbrace \times S^3\).
 (b)
The connections \(A^{x_1}\) converge uniformly with all derivatives to \(A^{\lim }\) on every compact subset of \((\mathbb {R}^4 {\setminus } \{0\})\times S^3\).
 (c)
The functions \(\vert F_{A^{x_1}} \vert ^2  \vert F_{A_{\lim }} \vert ^2\) are integrable and converge to \(8 \pi ^2 \delta _{\lbrace 0 \rbrace \times S^3}\), where \(\delta _{\lbrace 0 \rbrace \times S^3}\) denotes the delta current associated with the associative \(S^3\).
Whilst (a) gives the familiar “bubbling” behaviour of sequences of instantons, with curvature concentrating on an associative \(S^3\) by (c), we can interpret (b) as a “removable singularity” phenomenon since \(A^{\lim }\) is a smooth connection on \(\mathbb {R}^4\times S^3\). In proving Theorem 2, we show that as \(\lbrace A^{x_1} \rbrace \) bubbles along the associative \(S^3\) one obtains a Fueter section, as in [9, 14, 27]. Here this is just a constant map from \(S^3\) to the moduli space of antiself dual connections on \(\mathbb {R}^4\) (thought of as a fibre of the normal bundle), taking value at the basic instanton on \(\mathbb {R}^4\). Since \(8\pi ^2\) is the Yang–Mills energy of the basic instanton, we can also view (c) as the expected “conservation of energy”.
We also give a local existence result for \(G_2\)instantons in a neighbourhood of the associative \(S^3\) that extend over \(P_{{{\mathrm{id}}}}\) in Proposition 4. The outcome is that there is a local oneparameter family of such instantons. Of these only one, i.e. \(A^{\lim }\), is shown to extend over the whole of \(\mathbb {R}^4 \times S^3\). The other ones may blow up at a finite distance to \(\lbrace 0 \rbrace \times S^3\), as suggested by numeric simulations. Some of the necessary analysis leading to our local existence results is given in Appendix A.
In order to use similar techniques for \(G_2\)intantons on the BGGG metric, we must reduce the symmetry group to \(\mathrm{SU}(2)^2 \times U(1)\). This acts with cohomogeneity1 both on BGGG and BS and, as before, its only singular orbit is the associative \(\lbrace 0 \rbrace \times S^3\). Hence, in Sect. 4 we describe \(\mathrm{SU}(2)^2 \times U(1)\)invariant \(G_2\)instantons on cohomogeneity1 metrics with that symmetry on \(\mathbb {R}^4 \times S^3\). As a result, the same dichotomy appears in that the \(G_2\)instantons can extend over the associative \(S^3\) either on the homogeneous bundle \(P_1\) or \(P_{{{\mathrm{id}}}}\). We can thus compare the existence of \(G_2\)instantons for the BS and BGGG metrics. While there is a 1parameter family of \(G_2\)instantons (Clarke’s ones) that smoothly extend over \(P_1\) on the BS metric, for the BGGG metric we instead have the following.
Theorem 3
 (a)
The instantons in U have quadratically decaying curvature.
 (b)
The map \(Hol_{\infty } : U \rightarrow U(1) \subset \mathrm{SU}(2)\), which evaluates the holonomy of the \(G_2\)instanton along the finite size circle at \(+\infty \), is surjective.
The more precise version of this result appears as Theorem 9 and Corollary 3. It is typical in gauge theory to assume a bound on the curvature of the connection. One might be tempted to impose an \(L^2\)bound, but this is too restrictive in the \(G_2\) setting: in particular, Clarke’s examples do not satisfy this. Therefore, we impose a weak natural curvature bound in deriving Theorem 3, namely that the curvature stays bounded. We also prove that there is a 2parameter family of locally defined instantons on \(P_1\) for the BGGG metric which do not extend globally with bounded curvature: this is Theorem 8.
Finally, we give local existence results for \(G_2\)instantons with \(\mathrm{SU}(2)^2 \times U(1)\)symmetry in a neighbourhood of an associative \(S^3\), on any \(\mathrm{SU}(2)^2 \times U(1)\)invariant \(G_2\)metric. In Proposition 7, we show the existence of a 2parameter family of locally defined \(G_2\)instantons smoothly extending over \(P_1\), whereas in Proposition 8 we show the existence of a 1parameter family of \(G_2\)instantons smoothly extending over \(P_{{{\mathrm{id}}}}\). This yields the possibility for further study of \(G_2\)instantons even on the wellknown Bryant–Salamon metric on \(\mathbb {R}^4\times S^3\).
2 The SU(2)\(^2\)invariant equations
In this section we derive the ordinary differential equations (ODEs) which describe invariant \(G_2\)instantons on SU(2)\(^2\)invariant \(G_2\)manifolds of cohomogeneity1. We begin by giving the general framework of the evolution equations approach to \(G_2\)manifolds and \(G_2\)instantons in Sect. 2.1. We then apply this theory in Sect. 2.2 to the case of the invariant \(G_2\)manifolds we wish to study, leading to systems of ODEs describing the \(G_2\)manifolds, and summarise the known complete examples which arise from this approach. We then give a short presentation of the theory of invariant fields on homogeneous bundles in Sect. 2.3 so that we can obtain the general expression for an invariant connection on a principal orbit and its curvature. Combining these considerations yields our desired ODEs in Sect. 2.4, which we then solve in elementary cases in Sect. 2.5.
2.1 Evolution equations
Lemma 1
Proof
Proposition 1
In the setting of Lemma 1, suppose that the family of \(\mathrm{SU}(3)\)structures \((\omega (t), \Omega _2(t))\) depends real analytically on t, and let a(0) be a real analytic connection on P such that \(\Lambda _0 F_a(0)=0\). Then there is \(\epsilon >0\) and a \(G_2\)instanton A on \((\epsilon , \epsilon ) \times M^6\) with \(A\vert _{\lbrace 0 \rbrace \times M^6}=a(0)\).
Proof
This is immediate from applying the Cauchy–Kovalevskaya theorem to (2.7). \(\square \)
2.2 SU(2)\(^2\)invariant \(G_2\)manifolds of cohomogeneity1
2.2.1 The Bryant–Salamon (BS) metric
The Bryant–Salamon metric on \(\mathbb {R}^4\times S^3\) [5] is one of the first examples of a complete metric with \(G_2\)holonomy. It is not only SU(2)\(^2\)invariant, but actually \(\mathrm{SU}(2)^3\)invariant, having group diagram \(I(\mathrm{SU}(2)^3, \mathrm{SU}(2), \mathrm{SU}(2)^2)\); i.e. the principal orbits are \(\mathrm{SU}(2)^3/\mathrm{SU}(2)\cong S^3\times S^3\) and the (unique) singular orbit is \(\mathrm{SU}(2)^3/\mathrm{SU}(2)^2\cong S^3\). (Here, the \(\mathrm{SU}(2)\) in \(\mathrm{SU}(2)^3\) is the subgroup \(\mathrm{SU}(2)_3 = 1 \times 1 \times \mathrm{SU}(2)\), and \(\mathrm{SU}(2)^2 \subset \mathrm{SU}(2)^3\) is the subgroup \( \mathrm{SU}(2)_3 \times \Delta \mathrm{SU}(2)\), where \(\Delta \mathrm{SU}(2) \subset \mathrm{SU}(2)^2 \times 1\) is the diagonal.) In terms of the SU(2)\(^2\)invariant point of view above, the metric can be explicitly written as follows.
There is a oneparameter family of \(\mathrm{SU}(2)^3\)invariant \(G_2\)instantons for this Bryant–Salamon torsionfree \(G_2\)structure constructed by Clarke [8], where the parameter can be interpreted as how concentrated the instanton is around the associative \(S^3\). We shall prove, in Theorem 4 and Proposition 7, a uniqueness result for these \(G_2\)instantons in the class of \(\mathrm{SU}(2)^2 \times U(1)\)invariant ones.
Remark 1
In [5] Bryant–Salamon constructed \(G_2\)holonomy metrics on the total spaces of the bundles of antiselfdual 2forms over \(\mathbb {CP}^2\) and \(\mathbb {S}^4\), i.e. \(\Lambda ^2_ \mathbb {CP}^2\) and \(\Lambda ^2_\mathbb {S}^4\). Such metrics are also of cohomogeneity1 with respect to SO(5) and SU(3) respectively and asymptotically conical. Instantons on these \(G_2\)manifolds are also known to exist and some explicit examples can be found in [20].
It follows from Proposition 3 in [20] [or easily from (2.5), (2.6)] that on an asymptotically conical \(G_2\)manifold, a \(G_2\)instanton whose curvature is decaying pointwise at infinity will have as a limit (if it exists) a pseudoHermitian–Yang–Mills connection \(a_{\infty }\) (or nearly Kähler instanton): i.e. if \(\varphi _{\infty }=t^2dt\wedge \omega _{\infty }+t^3\Omega _{1,\infty }\) and \(\psi _{\infty }=t^4\omega _{\infty }^2/2t^3dt\wedge \Omega _{2,\infty }\) is the conical \(G_2\)structure on the asymptotic cone then \(F_{a_{\infty }}\wedge \omega _{\infty }^2=0\) and \(F_{a_{\infty }}\wedge \Omega _{2,\infty }=0\).
2.2.2 The Brandhuber et al. (BGGG) metric
In this setting, the geometry at infinity presents a new feature (that also exists in the BB manifolds below): there is a circle that remains of finite length at infinity. More precisely, the metric is asymptotic to a metric on a circle bundle over a 6dimensional cone with the fibres of the fibration having constant finite length. The length of this circle is the limit of \(A_1\) at infinity: for the family depending on the parameters \(\lambda , c\) this is \(2c\lambda /3\). One also sees that the volume of the associative \(S^3\) is \(B_1^{\lambda }(c\lambda ^2)B_2^{\lambda }(c\lambda ^2)^2 \sim (c \lambda )^3\), and so, using this family, it is impossible to vary the size of the circle while keeping the volume of the singular orbit fixed.
In [3], Bogoyavlenskaya constructed a 1parameter family (up to scaling) of \(\mathrm{SU}(2)^2\times U(1)\)invariant, cohomogeneity1, \(G_2\)holonomy metrics on \(\mathbb {R}^4\times S^3\), obtained by continuously deforming the BGGG metric. With these metrics, one can independently vary the size of the circle at infinity and the associative \(S^3\), and thus, in particular, obtain the BS metric as a limit of the family.
2.2.3 The Bazaikin–Bogoyavlenskaya (BB) metrics
The Bazaikin–Bogoyavlenskaya \(G_2\)manifolds X [1] (BB manifolds for short) have group diagram \(I(\mathrm{SU}(2)^2; \mathbb {Z}_4 ; U(1))\), i.e. the principal orbits are of the form \(S^3 \times S^3/\mathbb {Z}_4\) and the (unique) singular orbit is \( \mathrm{SU}(2)^2/U(1) \cong S^2 \times S^3\). In fact, X is diffeomorphic to \(L^4 \times S^3\), where \(L \rightarrow S^2\) is the complex line bundle canonically associated with the Hopf bundle and \(L^4\) denotes its fourth tensor power.
In [1] some complete torsionfree \(G_2\)structures with an extra U(1)symmetry, i.e. with \(A_2=A_3\) and \(B_2=B_3\), are constructed. These structures give rise to a 1parameter family of holonomy \(G_2\)metrics on \(X=L^4 \times S^3\), which have \((A_1(0),A_2(0),B_1(0), B_2(0))= ( \mu , \lambda ,0 , \lambda )\) for some values of \(\lambda , \mu \in \mathbb {R}\) with \(\lambda ^2 + \mu ^2 =1\). In particular, the volume of the singular orbit \(S^2 \times S^3\) is proportional to \(\lambda ^4 \mu =(1\mu ^2)^2 \mu \) and that of any 3sphere \(*\times S^3\) is proportional to \(\lambda ^2 \mu = (1\mu ^2)\mu \): these 3spheres are not associative. At least some of these metrics are asymptotic to an \(S^1\)bundle over the conifold.
We have some preliminary results on \(G_2\)instantons on these \(G_2\)manifolds and intend to investigate them further in future work.
Remark 2
The examples of \(G_2\)manifolds in Sects. 2.2.2–2.2.3 are asymptotic at infinity to a circle bundle over a cone: such manifolds are called asymptotically locally conical (ALC), and it is wellknown that the asymptotic cone is Calabi–Yau. One can see, under suitable assumptions, that \(G_2\)instantons on ALC \(G_2\)manifolds are asymptotic to Calabi–Yau monopoles on the Calabi–Yau cone. See [21] for some examples and results on Calabi–Yau monopoles in the asymptotically conical and conical settings.
2.3 Homogeneous bundles and invariant fields
We will now classify invariant connections on bundles over the SU(2)\(^2\)principal orbits in the \(G_2\)manifolds X of Sect. 2.2 so \(X\cong \mathbb {R}^4\times S^3\) or \(L^4\times S^3\).
We start with a review of the general setup on a homogeneous manifold K / H. First, Khomogeneous Gbundles over K / H (which will be our principal orbits) are determined by their isotropy homomorphism. These are group homomorphisms \(\lambda : H \rightarrow G\), associated with which we construct the bundle \(P_{\lambda }=K \times _{(H,\lambda )} G\). The reductive splitting \(\mathfrak {k} = \mathfrak {h} \oplus \mathfrak {m}\) equips \(K \rightarrow K/H\) with a connection whose horizontal space is \(\mathfrak {m}\). This is the socalled canonical invariant connection and its connection form \(A^c_{\lambda } \in \Omega ^1(K, \mathfrak {g})\) is the leftinvariant translation of \(d \lambda \oplus 0 : \mathfrak {h} \oplus \mathfrak {m} \rightarrow \mathfrak {g}\). Other invariant connections are classified by Wang’s theorem [28] and are in correspondence with morphisms of Hrepresentations \(\Lambda : (\mathfrak {m} , {{\mathrm{Ad}}}) \rightarrow (\mathfrak {g} , {{\mathrm{Ad}}}\circ \lambda )\).
Remark 3
We can always use an SU(2)\(^2\)invariant gauge transformation \(g : \mathbb {R}^+ \rightarrow G\) to put any invariant connection A in temporal gauge. This amounts to solving the ODE \(\dot{g}g^{1} + gA(\partial _t)g^{1} =0\), which has a unique solution g converging to 1 as \(t \rightarrow + \infty \).
Lemma 2
2.4 The SU(2)\(^2\)invariant ODEs
We may now write down the ODEs arising from Eqs. (2.5) and (2.6) which describe our invariant \(G_2\)instantons.
Lemma 3
Proof
The proof amounts to inserting the formula for the curvature \(F_a\) from Lemma 2 into (2.5), (2.6). For this we need to use the \(\mathrm{SU}(3)\)structure on the principal orbits given in (2.10)–(2.12). For convenience we shall write \(\eta ^{\pm }_{a\cdots b}=\eta ^{\pm }_a\wedge \cdots \wedge \eta ^{\pm }_{b}\).
2.5 Elementary solutions
In this subsection we consider elementary cases of SU(2)\(^2\)invariant \(G_2\)instanton equations on any of the SU(2)\(^2\)invariant \(G_2\)manifolds of cohomogeneity1 described in Sect. 2.2. We will let X denote such a \(G_2\)manifold.
We verify that flat connections satisfy our \(G_2\)instanton equations in Sect. 2.5.1 and we classify and describe all abelian \(G_2\)instantons explicitly in Sect. 2.5.2.
2.5.1 Flat connections
Any flat connection on X is obviously a \(G_2\)instanton and so must be a solution to our equations (for any gauge group G). As the fundamental group \(\pi _1(X)\) is trivial, any flat connection in this setting is gauge equivalent to the trivial connection. However, on a homogeneous bundle there may be invariant flat connections that are not gauge equivalent to the trivial connection through invariant gauge transformations.
2.5.2 Abelian instantons
On circle bundles, equivalently complex line bundles, the Lie algebra structure of the gauge group is trivial and the \(G_2\)instanton equations in Lemma 3 become linear. Consequently, it is then easy to integrate them, which we shall now proceed to do.
Proposition 2
Proof
The principal orbits on \(\mathbb {R}^4\times S^3\) are \(S^3 \times S^3\) and the singular one is \(S^3=\mathrm{SU}(2)^2/ \Delta \mathrm{SU}(2)\). The extensions of a circle bundle P on \(\mathbb {R}^+\times S^3\times S^3\) to \(S^3\) are parametrized by isotropy homomorphisms \(\lambda :\Delta \mathrm{SU}(2)\rightarrow U(1)\). The only such homomorphism \(\lambda \) is the trivial one, so the unique extension of P to the singular orbit is as the trivial bundle.
The canonical invariant connection on the trivial homogeneous bundle vanishes as an element of \(\Omega ^1(\mathrm{SU}(2)^2 , \mathbb {R})\). Any other invariant connection on this bundle is then given as an element of \(\Omega ^1(\mathrm{SU}(2)^2 , \mathbb {R})\) by the pullback of a biinvariant 1form on \(S^3=\mathrm{SU}(2)^2/ \Delta \mathrm{SU}(2)\). However, the only such 1form is the zero form, so the connection A extends over the singular orbit if and only if Lemma 9 in Appendix A applies to the 1form \(a=\sum _{i=1}^3 a_i^+ \eta _i^+ + a_i^ \eta _i^\).
In the BS or BGGG case, we can evaluate the integrals in Proposition 2 to give the following.
Corollary 1
 (a)In the BS case, A can be written asfor some \(x_1,x_2,x_3\in \mathbb {R}\), where \(r\in [1,+\infty )\) is determined by (2.22).$$\begin{aligned} A =\frac{r^31}{r}\sum _{i=1}^3x_i\eta _i^+ \end{aligned}$$
 (b)In the BGGG case, A can be written asfor some \(x_1,x_2,x_3\in \mathbb {R}\), where \(r\in [9/4,+\infty )\) is given by (2.26). When \(x_2=x_3=0\), A is a multiple of the harmonic 1form dual to the Killing field generating the U(1)action.$$\begin{aligned} A=\frac{(r9/4)(r+9/4)}{(r3/4)(r+3/4)}x_1\eta _1^++ \frac{(r9/4)e^r}{\sqrt{r}(r+9/4)^2}\left( x_2\eta _2^++x_3\eta _3^+\right) \end{aligned}$$
We already observe a marked difference in the behaviour of \(G_2\)instantons on the BS and BGGG \(\mathbb {R}^4\times S^3\) in this simple abelian setting. In particular, the instantons in the BS case all have bounded curvature, whereas those in the BGGG case have bounded curvature only when \(x_2=x_3=0\), in which case the curvature also decays to 0 as \(r\rightarrow \infty \).
Remark 4
Of course, for any abelian gauge group all Lie brackets vanish and the ODE system decouples into several independent linear ODEs for instantons on circle bundles. Hence, the construction of abelian \(G_2\)instantons here reduces to the U(1) case given in Proposition 2.
3 \(\mathrm{SU}(2)^3\)invariant \(G_2\)instantons
The only possible homogeneous SU(2)bundle P on the principal orbits \(S^3\times S^3\) is \(P=\mathrm{SU}(2)^2 \times \mathrm{SU}(2)\), i.e. the trivial SU(2)bundle. We consider connection 1forms with the extra SU(2)symmetry existent in the underlying geometry.
We begin in Sect. 3.1 by simplifying the ODEs and constraint system in Lemma 3 to this more symmetric situation, and then derive the conditions necessary to extend the solution to this system across the singular orbit in Sect. 3.2. We give classification results for the solutions to these equations in Sect. 3.3. We also examine the asymptotic behaviour of the solutions in terms of a connection on \(S^3\times S^3\), and give a compactness result for the space of solutions. The latter result is related to the familiar “bubbling” and “removable singularities” phenomena.
3.1 The SU(2)\(^3\)invariant ODEs
We simplify the invariant \(G_2\)instanton equations from Lemma 3 in this setting.
Proposition 3
Proof
We start by realizing SU(2)\(^2\) as \(\mathrm{SU}(2)^3/\Delta \mathrm{SU}(2)\). Isomorphism classes of \(\mathrm{SU}(2)^3\)equivariant bundles over SU(2)\(^2\) are then in correspondence with conjugation classes of homomorphisms \(\mu : \mathrm{SU}(2) \rightarrow \mathrm{SU}(2)\). There are only two such conjugation classes, namely those represented by the identity and the trivial homomorphism.
We turn now to the case when \(\mu : \mathrm{SU}(2) \rightarrow \mathrm{SU}(2)\) is the trivial homomorphism. Here, the canonical invariant connection \(d \mu \) vanishes as a 1form on \(\mathrm{SU}(2)^3\) with values in \(\mathfrak {su}(2)\). By Wang’s theorem, any other invariant connection is then given by a morphism of \(\Delta \mathrm{SU}(2)\)representations \(\Lambda : ( \mathfrak {m}, {{\mathrm{Ad}}}) \rightarrow (\mathfrak {su}(2) , {{\mathrm{Ad}}}\circ \mu )\). The lefthand side splits into two copies of the adjoint representation of \(\mathrm{SU}(2)\) while the righthand side decomposes into three trivial representations. Schur’s lemma then implies that \(\Lambda \) must vanish and so the trivial connection is the unique invariant one on this homogeneous bundle. This corresponds to taking \(x=y=0\) in the statement. \(\square \)
3.2 Initial conditions
Lemma 4

either \(x(t)=x_1t+ x_3 t^3+\cdots , \ y(t)= y_2 t^2 + \cdots \), in which case A extends smoothly as a connection on \(P_{1}\);

or \(x(t)=\frac{2}{t}+x_1t+\cdots , \ y(t)=y_0 + y_2 t^2 +\cdots \), in which case A extends smoothly as a connection on \(P_{{{\mathrm{id}}}}\).
Proof
We only analyze the case \(\lambda ={{\mathrm{id}}}\) in detail, as both situations are similar.

\(A_1(t)x(t)\), \(B_1(t)y(t)\) are both even,

\(\lim _{t \rightarrow 0} A_1(t)x(t)=1\) and \(\lim _{t\rightarrow 0}B_1(t)y(t)\) is finite.
To carry over the analysis in the case where \(\lambda =1\) we apply Lemma 10 directly to the 1form A, giving \(A_1x\), \(B_1y\) are even with \(\lim _{t\rightarrow 0}A_1x=\lim _{t\rightarrow 0}B_1y=0\). \(\square \)
3.3 Solutions and their properties
We now describe solutions of the \(\mathrm{SU}(2)^3\)invariant \(G_2\)instanton equations, which splits into two cases: when the bundle \(P=P_1\) and when \(P=P_{{{\mathrm{id}}}}\), in the notation of the previous subsection. In the first case we recover the \(G_2\)instantons constructed in [8], and in the second case we find a new example of a \(G_2\)instanton. We then analyse the asymptotic behaviour of the instantons, and finally show that the \(\mathbb {R}_{\ge 0}\)family of solutions on \(P_1\) admits a natural compactification.
3.3.1 Solutions smoothly extending on \(P_1\)
Clarke [8] constructed a 1parameter family of \(G_2\)instantons on the Bryant–Salamon \(\mathbb {R}^4\times S^3\). These instantons live on the bundle \(P_1\) given by (3.5), i.e. when the homomorphism \(\lambda \) is trivial. Moreover, they have \(y=0\) in the notation of Proposition 3, and so the ODEs there reduce to a single ODE for x which can be explicitly integrated. We shall reconstruct these \(G_2\)instantons in the proof of the next result, which classifies and explicitly describes the \(G_2\)instantons that smoothly extend over the singular orbit on the bundle \(P_1\).
Theorem 4
Proof
It will be enough to show that any instanton as in the statement defined on a neighbourhood of the singular orbit must coincide with one of Clarke’s examples there. For that, let (x(t), y(t)) be a solution to the ODEs (3.3), (3.4). We shall show that if the resulting instanton A extends over the singular orbit then \(y(t)=0\) for all t.
3.3.2 Solutions smoothly extending on \(P_{{{\mathrm{id}}}}\)
We now turn to solutions defined on the bundle \(P_{{{\mathrm{id}}}}\) given by (3.5) with the homomorphism \(\lambda ={{\mathrm{id}}}\). We first give a local existence result for instantons on \(P_{{{\mathrm{id}}}}\).
Proposition 4
Proof
Theorem 5
Proof
Remark 5
The reader may wonder about potential \(G_2\)instantons A arising from the local solutions with \(y_0\ne 0\) in Proposition 4. Numerical investigation appears to indicate that such local solutions do not extend globally, if we impose the condition that the curvature of A decays at infinity. We hope to study this situation further.
3.3.3 Asymptotics of the solutions
We now consider the asymptotic behaviour of the \(G_2\)instantons \(A^{x_1}\) and \(A^{\lim }\) constructed in Theorems 4 and 5.
Proposition 5
 if \(A=A^{x_1}\) for some \(x_1 \in \mathbb {R}^+\), then for \(t \gg 1\)where \(c>0\) is some constant independent of \(x_1\);$$\begin{aligned} \vert A^{x_1}  a_{\infty } \vert \le \frac{c}{ x_1 t^3}, \end{aligned}$$

if \(A=A^{\lim }\), then for \(t \gg 1\), \(\vert A^{\lim }  a_{\infty } \vert = O(t^{4})\).
Remark 6
As previously mentioned, any \(G_2\)instanton on an asymptotically conical \(G_2\)manifold which has a welldefined limit at infinity and has pointwise decaying curvature will be asymptotic to a pseudoHYM connection on the link of the asymptotic cone [20]. Proposition 5 refines this result in this setting.
3.3.4 Compactness properties of the moduli of solutions
Next we show that as \(x_1 \rightarrow + \infty \) Clarke’s \(G_2\)instantons \(A^{x_1}\) “bubble off” an antiselfdual (ASD) connection along the normal bundle to the associative \(S^3= \{0\}\times S^3 \subset \mathbb {R}^4\times S^3\). We shall also show that in the same limit Clarke’s \(G_2\)instantons converge outside the associative \(S^3\) to \(A^{\lim }\). The fact that \(A^{\lim }\) smoothly extends over \(S^3\) can then be interpreted as a removable singularity phenomenon.
Theorem 6
 (a)
Given any \(\lambda >0\), there is a sequence of positive real numbers \(\delta =\delta (x_1,\lambda ) \rightarrow 0\) as \(x_{1} \rightarrow + \infty \) such that: for all \(p \in S^3\), \((s^p_{\delta })^* A^{x_1}\) converges uniformly with all derivatives to the basic ASD instanton \(A^{\text {ASD }}_{\lambda }\) on \(B_1\subseteq \mathbb {R}^4\) as in (3.16).
 (b)
The connections \(A^{x_1}\) converge uniformly with all derivatives to \(A^{\lim }\) given in Theorem 5 on every compact subset of \((\mathbb {R}^4 {\setminus } \{0\})\times S^3 \) as \(x_1\rightarrow +\infty \).
Proof
 (a)We view the basic instanton \(A^{\text {ASD}}_{\lambda }\) in (3.16) as defined on \(\mathbb {R}^4\times \{p\}\). Using the formula for \(A^{x_1}\) in Theorem 4 and the expansions of \(A_1\) and \(B_1\) near 0 in Example 1 from Appendix A, we compute, for \(t< 1\),Hence, setting \(\delta = \delta (x_1,\lambda )= \sqrt{2 \lambda / x_1}\) we have that for every \(k \in \mathbb {N}_0\), there is \(c_k>0\), independent of \(\lambda \) and \(x_1\), such that$$\begin{aligned} \left( s^p_{\delta }\right) ^* A^{x_1}= & {} A_1(\delta t ) x(\delta t ) T_i\otimes \eta _i^+ = \frac{2x_1 A_1^2(\delta t)}{1+x_1 \left( B_1^2(\delta t)  \frac{1}{3} \right) } T_i\otimes \eta _i^+ \\= & {} \frac{x_1\delta ^2 t^2/2 + O(x_1 \delta ^4 t^4)}{1+x_1\delta ^2 t^2/2 + O(x_1 \delta ^4 t^4)} T_i \otimes \eta _i^+ . \end{aligned}$$Therefore, given \(\epsilon >0\), we have for any \(x_1\ge c_k \lambda ^2/ \epsilon \) that$$\begin{aligned} \left\ \left( s^p_{\delta }\right) ^* A^{x_1}  A^{\text {ASD}}_{\lambda } \right\ _{C^k(B_1)} \le c_k \frac{\lambda ^2}{x_1}. \end{aligned}$$demonstrating the claimed convergence.$$\begin{aligned} \left\ \left( s^p_{\delta }\right) ^* A^{x_1}  A^{\text {ASD}}_{\lambda } \right\ _{C^k(B_1)} \le \epsilon , \end{aligned}$$
 (b)We take the explicit formulas for \(A^{x_1}\) and \(A^{\lim }\) in Theorems 4 and 5 and computefor some constant \(c>0\). Recall that in the coordinate \(r\in [1,+\infty )\) from (2.22) we have \(B_1(r)=r/\sqrt{3}\) by (2.23). Hence, \(B_1^2\frac{1}{3}\) is bounded and bounded away from zero on every compact \(K\subseteq (\mathbb {R}^4 {\setminus } \{0\})\times S^3 \). Thus, for every such K there is some (possibly other) constant \(c>0\) such that$$\begin{aligned} \vert A^{x_1}  A^{\lim } \vert= & {} \frac{A_1^2(t)}{\frac{1}{2}\big (B_1^2(t)\frac{1}{3}\big ) } \ \left \frac{x_1 \big (B_1^2(t)\frac{1}{3}\big )}{1+x_1 \big (B_1^2(t)\frac{1}{3}\big ) } 1 \right \left \sum _{i=1}^3T_i\otimes \eta _i^+ \right \\\le & {} \frac{c A_1(t)}{\frac{1}{2}\big (B_1^2(t)\frac{1}{3}\big ) } \frac{1}{1+x_1 \big (B_1^2(t)\frac{1}{3}\big ) }, \end{aligned}$$and we have similar estimates for the derivatives of \(A^{x_1}A^{\lim }\). By letting \(x_1 \rightarrow + \infty \) the righthand side of (3.17) tends to zero as required.$$\begin{aligned} \vert A^{x_1}  A^{\lim } \vert \le \frac{c}{1+x_1}, \end{aligned}$$(3.17)
Remark 7
As already mentioned, the fact that \(A^{\lim }\) smoothly extends over \(S^3\) is an example of a removable singularity phenomenon. It follows from Tian and Tao’s work [23, 24] that such phenomena occur more generally provided that the \(G_2\)instanton is invariant under a group action all of whose orbits have dimension greater than or equal to 3 (codimension less than or equal to 4).
Even though the \(G_2\)instantons \(A^{x_1}\) and \(A_{\lim }\) do not have finite energy, and so the results of [23] do not immediately apply, we now show that we do have the expected energy concentration along the associative \(S^3\). Below, we let \(\delta _{\lbrace 0 \rbrace \times S^3}\) denote the delta current associated with \(\lbrace 0 \rbrace \times S^3\).
Corollary 2
Proof
Remark 8
The sequence of instantons with curvature concentrating along the associative \(S^3\) determines a Fueter section, as in [9, 14, 27], from \(S^3\) to the bundle of moduli spaces of antiselfdual connections associated to the normal bundle. The section thus determined is constant, taking value at the basic instanton on \(\mathbb {R}^4\). The Yang–Mills energy of the basic instanton is \(8\pi ^2\), so Corollary 2 confirms the expected “conservation of energy” formula (c.f. [23]).
4 \(\mathrm{SU}(2)^2\times U(1)\)invariant \(G_2\)instantons
The main goal of this section is to investigate SU(2)\(^2\)invariant \(G_2\)instantons on the Brandhuber et al. (BGGG) \(G_2\)manifold \(\mathbb {R}^4\times S^3\) from Sect. 2.2.2. We will restrict ourselves to instantons that enjoy an extra U(1)symmetry present in the underlying geometry. As already mentioned, all of the known complete SU(2)\(^2\)invariant \(G_2\)manifolds of cohomogeneity1 enjoy an extra U(1)symmetry and so the analysis of \(\mathrm{SU}(2)^2\times U(1)\)invariant \(G_2\)instantons provides a natural stepping stone to a complete understanding of SU(2)\(^2\)invariant \(G_2\)instantons.
We begin in Sect. 4.1 by deriving the ODEs determining \(G_2\)instantons in this setting by simplifying the general ODEs and constraint in Lemma 3. We then determine the necessary conditions ensuring that solutions to these ODEs smoothly extend across the singular orbit in the Bryant–Salamon (BS), BGGG and Bogoyavlenskaya \(G_2\)manifolds in Sect. 4.2. In the final section Sect. 4.3, we explicitly describe the \(G_2\)instantons which exist near the singular orbit. This leads to a stronger classification result in the BS case, and existence and nonexistence results for global \(G_2\)instantons in the BGGG case.
4.1 The \(\mathrm{SU}(2)^2\times U(1)\)invariant ODEs
Proposition 6
Proof
4.2 Initial conditions
To investigate \(\mathrm{SU}(2)^2 \times U(1)\)invariant \(G_2\)instantons A on the BGGG \(G_2\)manifold \(\mathbb {R}^4\times S^3\), as well as the BS and Bogoyolavenskaya cases, we study the conditions for A to extend smoothly over the singular orbit \(\mathrm{SU}(2)^2 / \Delta \mathrm{SU}(2) \cong \{0\}\times S^3 \).
Lemma 5
 eitherin which case A extends smoothly as a connection on \(P_{1}\);$$\begin{aligned} f^{}&=f_2^ t^2 + O(t^4),\quad g^= g_2^ t^2 + O(t^4),\\ f^{+}&= f_1^+ t + O(t^3),\quad g^+= g_1^+ t + O(t^3), \end{aligned}$$
 orin which case A extends smoothly as a connection on \(P_{{{\mathrm{id}}}}\).$$\begin{aligned} f^{}&= b_0^ + b_2^ t^2 + O(t^4) ,\quad g^= b_0^ + b_2^ t^2 + O(t^4), \\ f^{+}&= \frac{2}{t} + \left( b_2^+4C_1(0) \right) t + O(t^3) ,\quad g^+= \frac{2}{t} + \left( b_2^+4C_2(0)\right) t + O(t^3), \end{aligned}$$
Proof
We now turn to \(P_1\). Here we instead apply Lemma 10 to the 1form A itself and conclude that \(A_1 f^{+}\), \(A_2 g^+\), \(B_1 f^{}\), \(B_2 g^\) must all be even and vanish at \(t=0\). Hence, by (4.13), we see that \(f^+,g^+\) are odd and \(f^,g^\) are even such that\(f^(0)=g^(0)=0\). \(\square \)
4.3 Solutions
We now investigate existence of solutions of the \(\mathrm{SU}(2)^2\times U(1)\)invariant \(G_2\)instanton equations with gauge group \(\mathrm{SU}(2)\) on the BS, BGGG and Bogoyavlenskaya \(G_2\)manifolds \(\mathbb {R}^4\times S^3\). There are two cases: when the bundle is \(P_1\) or \(P_{{{\mathrm{id}}}}\), in the notation of the previous subsection. In both cases we explicitly classify the invariant \(G_2\)instantons defined near the singular orbit which extend smoothly and, as a consequence, extend our uniqueness result for \(G_2\)instantons on the BS \(\mathbb {R}^4\times S^3\) to the case of \(\mathrm{SU}(2)^2 \times U(1)\)symmetry, and obtain both existence and nonexistence results for \(G_2\)instantons with decaying curvature on the BGGG \(\mathbb {R}^4\times S^3\).
4.3.1 Solutions smoothly extending on \(P_{1}\)
We shall now investigate the existence of solutions that smoothly extend over the singular orbit \(S^3 = \mathrm{SU}(2)^2/\Delta \mathrm{SU}(2)\) on the bundle \(P_1\). The main results are Proposition 7 and Theorems 7–9. Proposition 7 shows the existence of a 2parameter family of \(G_2\)instantons in a neighbourhood of the singular orbit, so there is at most a 2parameter family of \(\mathrm{SU}(2)^2\times U(1)\)invariant \(G_2\)instantons on \(P_1\) on the BS, BGGG and Bogoyavlenskaya \(G_2\)manifolds. Theorem 7 shows that in the BS case, just a 1parameter family of these local instantons extends, and these are either given by Clarke’s \(\mathrm{SU}(2)^3\)invariant examples from Theorem 4 or are abelian. Theorems 8 and 9 show that, unlike the BS case, there is a 2parameter family of local \(G_2\)instantons which extend to the whole BGGG \(\mathbb {R}^4\times S^3\) so that their curvature is bounded, as well as a 2parameter family which do not extend so as to have bounded curvature.
Proposition 7
Let \(X \subset \mathbb {R}^4\times S^3 \) contain the singular orbit \(\{0\}\times S^3 \) of the \(\mathrm{SU}(2)^2 \times U(1)\) action and be equipped with an \(\mathrm{SU}(2)^2 \times U(1)\)invariant holonomy \(G_2\)metric. There is a 2parameter family of \(\mathrm{SU}(2)^2 \times U(1)\)invariant \(G_2\)instantons A with gauge group \(\mathrm{SU}(2)\) in a neighbourhood of the singular orbit in X smoothly extending over \(P_{1}\).
Proof
Notice that all these \(G_2\)instantons have \(v_1(0)=0=v_2(0)\). Thus, setting the smaller singular initial value problem above with \(f^\) and \(g^\) both vanishing gives the same local existence and uniqueness result, and hence the uniqueness implies that in fact \(f^(t)\), \(g^(t)\) must vanish identically for any solution extending smoothly over the singular orbit. The resulting ODEs (4.14), (4.15) then follow from Proposition 6. \(\square \)
Remark 10
Recall that the BS, BGGG and Bogoyavlenskaya \(G_2\)metrics all have SU(2)\(^2 \times U(1)\)symmetry and so Proposition 7 yields \(G_2\)instantons in these cases.
Our first result shows that the sign of \(g_1^+\) determines the sign of \(g^+\).
Lemma 6
Let \((f^+,g^+)\) solve (4.14), (4.15). The sign of \(g^+\) does not change as long as \(f^+\) does not blow up, and if \(g^+(t_0)=0\) for some \(t_0> 0\) or if \(g_1^+=0\) then \(g^+\equiv 0\).
Proof
Suppose, for a contradiction, that the sign of \(g^+\) changes. Then there is \(t_0> 0\) such that \(g^+(t_0)=0\). The ODE (4.15) implies that \(\dot{g}^+(t_0)=0\) and thus \(g^+\equiv 0\) (as \(g^+\) solves a linear first order ODE), giving our contradiction. The same argument using (4.15) yields the statement. \(\square \)
Remark 11
The ODEs (4.14), (4.15) are invariant under \(g^+\mapsto g^+\). We may therefore exchange \(g^+\) with \(g^+\) and, by virtue of Lemma 6, assume that \(g_1^+\ge 0\), and thus \(g^+\ge 0\), if we wish.
We first focus on the BS \(G_2\)manifold \(\mathbb {R}^4\times S^3\). It follows from Proposition 7 that there is at most a 2parameter family of \(\mathrm{SU}(2)^2\times U(1)\)invariant \(G_2\)instantons defined globally on the BS \(G_2\)manifold. We have a 1parameter family of such instantons (with more symmetry) from Theorem 4 and a 1parameter family of abelian examples from Corollary 1. We now show that these examples provide a complete classification.
Theorem 7
Let A be a \(\mathrm{SU}(2)^2\times U(1)\)invariant \(G_2\)instanton with gauge group \(\mathrm{SU}(2)\) on the BS \(G_2\)manifold \(\mathbb {R}^4\times S^3\) which extends smoothly on \(P_1\). Either A is \(\mathrm{SU}(2)^3\)invariant, and so is given in Theorem 4; or it is reducible, in which case it has gauge group U(1) and is given in Corollary 1(a) with \(x_2=x_3=0\).
Proof
If \(c>0\) there are two types of solutions to (4.20): either \(F^2(s)=c\tanh ^2(a+\sqrt{c}s)\) or \(F^2=c\). The first solutions have \(F^2c<0\) which contradicts \(F^2c=G^2\). The second solutions force \(G\equiv 0\), which give abelian instantons as in Corollary 1.
If \(c=0\), then \(F^2=G^2\), which means \(F=\pm G\) so \(f^+=\pm g^+\). By Remark 11, we may assume that \(f^+=g^+\). In this case, A is SU(2)\(^3\)invariant and the result then follows from Theorem 4. \(\square \)
We now focus attention on the BGGG \(G_2\)manifold, though some of our results hold for the 1parameter family of Bogoyavlenskaya metrics which includes the BGGG metric. It is natural in the study of \(G_2\)instantons on noncompact \(G_2\)manifolds to assume a decay condition on the curvature of the connection at infinity. The weakest reasonable assumption we can make is the curvature is bounded. In this setting we can prove both existence and nonexistence results.
We first observe the conditions imposed on \(f^+,g^+\) for the Bogoyavlensakaya metrics when the curvature is bounded.
Lemma 7
Let A be the \(G_2\)instanton on one of the Bogoyavlenskaya \(G_2\)manifolds induced by the pair \((f^+,g^+)\) as in Proposition 7. Then \(F_A\) is bounded only if \(g^+\) is bounded, and if both \(f^+\) and \(g^+\) are bounded then \(F_A\) is bounded.
Proof
We have a 1parameter family of reducible invariant \(G_2\)instantons on the BGGG \(G_2\)manifold which have gauge group \(U(1)\subseteq \mathrm{SU}(2)\): they are given in Corollary 1(b) with \(x_2=x_3=0\) and have bounded (in fact, decaying) curvature. We start with our nonexistence result, which shows that a 2parameter family of initial conditions leads to local \(G_2\)instantons which either do not extend with bounded curvature or can only extend as one of the above abelian instantons.
Theorem 8
Let A be a \(\mathrm{SU}(2)^2\times U(1)\)invariant \(G_2\)instanton with gauge group \(\mathrm{SU}(2)\) defined in a neighbourhood of \(\{0\}\times S^3\) on the BGGG \(G_2\)manifold \(\mathbb {R}^4\times S^3\) smoothly extending over \(P_1\) as given by Proposition 7.
If \(f_1^+\le \frac{1}{2}\), or \(g_1^+\ge 0\) with \(g_1^+\ge f_1^+\), then A extends globally to \(\mathbb {R}^4\times S^3\) with bounded curvature if and only if A has gauge group U(1) and is given in Corollary 1(b) with \(x_2=x_3=0\).
Proof
If \(g^+\equiv 0\) then we obtain an abelian instanton as in Corollary 1(b) with \(x_2=x_3=0\). Suppose, for a contradiction, that \(g^+\) is not identically zero and that A is defined for all t has bounded curvature. By Lemma 6 and Remark 11 we may assume without loss of generality that \(g_1^+>0\) and thus \(g^+>0\) for all t.
Suppose first that \(f_1^+\le \frac{1}{2}\). Since \(f^+=f_1^+t+O(t^3)\) and \(A_1=t/2+O(t^3)\) by Example 2, we see that \(F(0)=2f_1^+\le 1\). Moreover, F is strictly decreasing by (4.22) as \(G>0\), so there is \(\epsilon >0\) so that \(F(s)\le 1\epsilon \) for all \(s>0\). As \(H(s)\rightarrow 1\), there exists \(s_0>0\) so that \(H(s)F(s)>\frac{\epsilon }{2}\) for all \(s\ge s_0\). We deduce from (4.22) that \(G'>\frac{\epsilon }{2}G\) for all \(s\ge s_0\) as \(G>0\), and hence \(G\ge e^{\epsilon s/2}\). Therefore, \(g^+\) grows at least exponentially, so either the solution explodes for a finite t, or \(F_A\) is unbounded by Lemma 7, giving a contradiction.
Now suppose \(g_1^+\ge f_1^+\). Then \(G(0)F(0)\ge 0\) and one sees that \(G(s)F(s)>0\) and is increasing for small \(s>0\) using \(g^+=g_1^+t+u_2(0)t^3+O(t^5)\), \(f^+=f_1^+t+u_1(0)t^3+O(t^5)\) and the formulae (4.16), (4.17), where the values \(C_1(0),C_2(0)\) for the BGGG metric are given in Example 2.
If \(F(s)\le 1\) for some s, then we are in the same situation as the previous case of \(f_1^+\le \frac{1}{2}\), which leads to a contradiction. If instead \(F(s)>1\) for all s then F is bounded below, so as F is strictly decreasing we need from (4.22) that \(\lim _{s\rightarrow \infty }G(s)=0\). Hence, as \(H(s)\rightarrow 1\), there exists \(s_0\) so that \(G'>\frac{1}{2}G\) for all \(s\ge s_0\), which implies that \(g^+\) grows at least exponentially. This again gives a contradiction by Lemma 7. \(\square \)
Remark 12
The above proof of nonexistence of irreducible instantons for \(f_1^+\le \frac{1}{2}\) immediately extends to the Bogoyavlenskaya metrics by the asymptotics in (4.21). The proof for \(g_1^+\ge 0\) and \(g_1^+\ge f_1^+\) would also extend if we knew that H given in (4.23) continued to be positive for all \(t>0\) for the Bogoyavlenskaya metrics.
We now give our existence result, which provides a full 2parameter family of irreducible \(\mathrm{SU}(2)^2\times U(1)\)invariant \(G_2\)instantons with gauge group \(\mathrm{SU}(2)\) on the BGGG \(G_2\)manifold.
Theorem 9
Let A be a \(\mathrm{SU}(2)^2\times U(1)\)invariant \(G_2\)instanton with gauge group \(\mathrm{SU}(2)\) defined in a neighbourhood of \(\{0\}\times S^3\) on the BGGG \(G_2\)manifold \(\mathbb {R}^4\times S^3\) smoothly extending over \(P_1\) as given by Proposition 7.
If \(f_1^+\ge \frac{1}{2}+g_1^+>\frac{1}{2}\), then A extends globally to \(\mathbb {R}^4\times S^3\) with bounded curvature.
Proof
By (4.22), F is decreasing and thus F is bounded as it is bounded below (by 1). We also know that \(0<G<F1\) so G is also bounded. As H is also bounded, we deduce that a long time solution to the ODEs (4.22) must exist.
Since F is bounded below by 1, decreasing and exists for all s we must have again from (4.22) that \(G(s)\rightarrow 0\) as \(s\rightarrow \infty \), and that \(\lim _{s\rightarrow \infty }F(s)\) exists and equals some constant greater or equal to 1. Hence, both \(f^+\) and \(g^+\) are bounded and so A has bounded curvature by Lemma 7. \(\square \)
Remark 13
Via the asymptotics in (4.21), we see that the function H in (4.23) for any given Bogoyavlenskaya metric is always bounded above by some \(C\ge 1\) (possibly depending on the metric, though one might hope to show that \(C=1\)). Thus, the proof of Theorem 9 extends to prove the existence of \(G_2\)instantons with bounded curvature for \(f_1^+\ge \frac{C}{2}+g_1^+>0\) in these cases.
Corollary 3
For any fixed \(g_1^+ >0\), the map (4.26) is surjective.
Proof
Remark 14
By contrast, in Proposition 5, we showed that the irreducible \(\mathrm{SU}(2)^2\times U(1)\)invariant \(G_2\)instantons for the BS metric are asymptotic to an irreducible connection and the rate of convergence is \(O(t^{3})\).
In summary, on the BGGG \(G_2\)manifold \(\mathbb {R}^4\times S^3\), we have shown nonexistence for irreducible \(\mathrm{SU}(2)^2\times U(1)\)invariant \(G_2\)instantons with gauge group \(\mathrm{SU}(2)\) and bounded curvature for \(g_1^+>0\) and \(f_1^+\le \frac{1}{2}\) or \(g_1^+\ge f_1^+\), and existence for \(f_1^+\ge \frac{1}{2}+g_1^+>\frac{1}{2}\). This currently leaves open the region where \(0<f_1^+\frac{1}{2}<g_1^+<f_1^+\). Some numerical investigation indicates that some of these initial conditions may lead to globally defined instantons with bounded curvature and some may not.
4.3.2 Solutions smoothly extending on \(P_{{{\mathrm{id}}}}\)
We now turn our attention to the more difficult case of solutions to the \(\mathrm{SU}(2)^2\times U(1)\)invariant \(G_2\)instanton equations on \(\mathbb {R}^4\times S^3\) which smoothly extend on the bundle \(P_{{{\mathrm{id}}}}\). Here the ODE system does not simplify, but we obtain a 1parameter family of local solutions in a neighbourhood of the singular orbit. Although the strategy of proof remains the same as in our earlier similar results, the analysis is more involved. In order to ease computations, we use the Taylor expansion for a smooth \(\mathrm{SU}(2)^2 \times U(1)\)symmetric \(G_2\)holonomy metric in a neighbourhood of a singular orbit \(\lbrace 0 \rbrace \times S^3\) at \(t=0\), computed in (A.4)–(A.7), which depends on constants b, c.
Proposition 8
Let \(X \subset \mathbb {R}^4\times S^3 \) contain the singular orbit \(\{0\}\times S^3 \) of the \(\mathrm{SU}(2)^2 \times U(1)\) action and be equipped with an \(\mathrm{SU}(2)^2 \times U(1)\)invariant holonomy \(G_2\)metric. There is a 1parameter family of \(\mathrm{SU}(2)^2 \times U(1)\)invariant \(G_2\)instantons A with gauge group \(\mathrm{SU}(2)\) in a neighbourhood of the singular orbit in X smoothly extending over \(P_{{{\mathrm{id}}}}\).
Proof
Remark 15
Since the BS, BGGG and Bogoyavlenskaya \(G_2\)metrics all have SU(2)\(^2 \times U(1)\)symmetry, Proposition 8 yields \(G_2\)instantons in these cases. In particular, in the BS case we have \(c=\frac{1}{24 b^2}\), \(b=\frac{1}{\sqrt{3}}\) and these \(G_2\)instantons coincide with those given in Proposition 4.
In light of the existence result in Theorem 5 and the local existence result in Proposition 8, it is certainly an interesting nontrivial question which members of the 1parameter family of local \(G_2\)instantons from Proposition 8 extend on \(P_{{{\mathrm{id}}}}\) on the BS, BGGG or Bogoyavlenskaya \(\mathbb {R}^4\times S^3\).
Another natural problem for further study is to understand the limits of the family of instantons constructed in Theorem 9, and their possible relationship to any extensions of the local instantons given in Proposition 8. We saw in Proposition 5 that global \(G_2\)instantons on the BS \(\mathbb {R}^4\times S^3\) have a limit at infinity given by a canonical connection on the link \(S^3\times S^3\) of the asymptotic cone. For the instantons constructed in Theorem 9 we know, by Remark 14, that these are asymptotic to the abelian \(G_2\)instantons with a rate depending on the asymptotic connection. It is also certainly an interesting problem to investigate the behaviour of the family of instantons from Theorem 9 when one or both of \(f_1^+\) and \(g_1^+\) go to infinity. We would expect bubbling phenomena as in the BS case in Theorem 6, with possible relationship to the ASD instantons on Taub–NUT found in [11]. The lack of an explicit formula for our instantons makes the bubbling analysis more difficult.
One other interesting problem is to investigate the behaviour of \(G_2\)instantons as the underlying metric is deformed. For instance, Remark 13 shows how to adapt the proof of existence in Theorem 9 to the Bogoyavlenskaya \(G_2\)manifolds, and we would want to analyse these instantons as the size of the circle at infinity gets very large or small. When it gets very large we expect them to resemble \(G_2\)instantons for the BS metric given in Theorem 7. When it gets very small, there may be a relation with Calabi–Yau monopoles on the deformed conifold (as in [21]).
Footnotes
 1.
For further background on \(G_2\)manifolds, the reader may wish to consult Joyce’s book [16].
 2.An SU(3)structure on an almost complex 6manifold (M, J) can be given by a pair of a real (1, 1)form \(\omega \) and a real 3form \(\Omega _2\) such thatwhere \(\Omega _1=J\Omega _2\).$$\begin{aligned} \omega \wedge \Omega _2 =0 , \quad \omega ^3 =  \frac{8}{3} \Omega _1 \wedge \Omega _2 , \end{aligned}$$
 3.
 4.
There are, in fact, distinct \(\mathrm{SU}(2)^3\)invariant torsionfree \(G_2\)structures on \(\mathbb {R}^4\times S^3\) inducing the same asymptotically conical Bryant–Salamon metric, determined by their image in \(H^3(S^3\times S^3)\).
 5.
We thank Lorenzo Foscolo and Mark Haskins for bringing the metrics in [3] to our attention.
Notes
Acknowledgements
We would like to thank Tom Beale, Robert Bryant, Lorenzo Foscolo, Derek Harland, Mark Haskins, Thomas Madsen, David Sauzin and Mark Stern for discussions. We would particularly like to thank Lorenzo Foscolo for introducing us to Eschenburg–Wang’s analysis for extending invariant tensors over singular orbits [12], and the reference [18] for singular initial value problems. The first author was partially supported by EPSRC Grant EP/K010980/1.
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