Abstract
We use the highly symmetric Stenzel Calabi–Yau structure on \(T^{\star }S^{4}\) as a testing ground for the relationship between the Spin(7) instanton and Hermitian–Yang–Mills (HYM) equations. We reduce both problems to tractable ODEs and look for invariant solutions. In the abelian case, we establish local equivalence and prove a global nonexistence result. We analyze the nonabelian equations with structure group SO(3) and construct the moduli space of invariant Spin(7) instantons in this setting. This is comprised of two 1parameter families—one of them explicit—of irreducible Spin(7) instantons. Each carries a unique HYM connection. We thus negatively resolve the question regarding the equivalence of the two gauge theoretic PDEs. The HYM connections play a role in the compactification of this moduli space, exhibiting a removable singularity phenomenon that we aim to further examine in future work.
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1 Introduction
1.1 Summary and main results
The inclusion of SU(4) in Spin(7) demonstrates that a Calabi–Yau (CY) fourfold is in a natural way a Spin(7) manifold. The SU(4) and Spin(7) structures give rise to associated generalised instanton equations: the Hermitian Yang–Mills (HYM) equations and the Spin(7) instanton equations respectively. It is natural to inquire about the relationship of these two gauge theoretic problems. One immediately observes that HYM is a stronger condition. In the compact case, it is known that as long as an HYM connection exists, the two types of instantons coincide ([9], p. 25 Theorem 3.1). Consequently, if one hopes to display a compact counterexample to equivalence, there must not be any HYM connections at all. Furthermore, we have a general existence theorem for HYM connections over stable holomorphic bundles [17]. This restricts the choices of bundles one could look at. Finally, one could attempt to simplify the prohibitively complicated analysis by dimensional reduction. Unfortunately, compact, irreducible special holonomy manifolds admit no continuous symmetries [7]. This precludes the use of symmetry techniques. We are thus motivated to look for a noncompact counterexample. Since Lewis’s argument is essentially an energy estimate, it does not apply to the noncompact setting.
We work over a noncompact CY fourfold: the cotangent bundle of the 4sphere equipped with the Stenzel metric [16]. We use the cohomogeneity one isometric SO(5)action to reduce the instanton equations to tractable ODEs and proceed to classify the SO(5)invariant solutions. In Sect. 2, we study the abelian equations. The two problems turn out to be locally equivalent (Theorem 2.1). We classify all solutions (Theorem 2.2) and prove a global nonexistence result (Theorem 2.3). In Sect. 3, we initiate our study of the nonabelian equations corresponding to structure group SO(3). We classify the relevant cohomogeneity one bundles and their invariant connections and solve the equations on the trivial bundle (Theorem 3.2). On this topology, all nontrivial instantons have holonomy U(1) and have thus already been encountered in Sect. 2. Section 4 is concerned with the construction of the moduli space of SO(5)invariant Spin(7) instantons with structure group SO(3) over the Stenzel space. This contains solutions living on two distinct bundles which agree outside of a codimension 4 Cayley submanifold. On one of them, we are able to solve the equations explicitly (Theorem 4.8). We find a noncompact 1parameter family of Spin(7) instantons with a unique HYM connection embedded in its interior. This negatively resolves the question regarding the equivalence of the two gauge theoretic PDEs (Theorem 4.9). The equations on the other bundle cannot be integrated explicitly. We study them using dynamical systems analysis. We find a compact 1parameter family of solutions carrying a unique HYM connection in its interior. This HYM connection is identified as the missing endpoint of the explicit family from before. Attaching it to the noncompact end, we obtain a natural compactification of the moduli space. This removable singularity phenomenon suggests that the HYM connections might play a role in the compactification of general Spin(7) instanton moduli spaces (over noncompact CY fourfolds), an issue we intend to further examine in future work.
1.2 The Stenzel manifold
In this section we provide a brief introduction to the Stenzel CY fourfold \(X^{8}\). For details, we refer to the articles [13, 16]. The latter carries out the corresponding calculations in complex dimension 3. The overall technique for studying invariant objects in cohomogeneity one is essentially the same as in the article [10].
1.2.1 The underlying manifold and the SO(5)action
The underlying space of \(X^{8}\) is the cotangent bundle of the 4sphere. The natural SO(5) action on \(S^{4}\) extends to \(T^{\star }S^{4}\) by pullback. The singular orbit is the zero section \(S^{4}\). Its stabiliser is the group SO(4). The principal orbits are the positive radius sphere bundles. Their stabiliser is the group SO(3). They are 7dimensional Stiefel manifolds.
The cotangent bundle of \({\mathbb {R}}^{5}\) can be identified with the following space:
Here, we adopt the convention that given a choice of basis elements of a vector space are represented by column vectors, whereas linear forms lying in its dual are represented by row vectors. This allows us to identify the natural duality pairing with matrix multiplication of the coordinate representations.
We obtain a model of \(T^{\star }S^{4}\) as an embedded submanifold of \(T^{\star }{\mathbb {R}}^{5}\) by including \(S^{4}\hookrightarrow {\mathbb {R}}^{5}\) as the unit radius round sphere. In this model, the SO(5) action takes the following form:
An alternative model can be obtained by embedding \(T^{\star }S^{4}\) in \({\mathbb {C}}^{5}\) as a complex quadric. It thus inherits a natural complex structure. This is done as follows. Consider the degree 2 homogeneous polynomial:
We split the complex coordinates of \({\mathbb {C}}^{5}\) into their real and imaginary parts \(z_{j}=x_{j}+iy_{j}\) and introduce the functions:
Define the map:
It may be easily seen that this restricts to a diffeomorphism:
The following relations hold over the level set \(X^{8}\cong F^{1}(1)\subset {\mathbb {C}}^{5}\):
The minimum value of r on \(X^{8}\) is \(r=1\) and the associated level set corresponds to the singular orbit. The latter sits inside \(X^{8}\) as an embedded totally real submanifold [14]. We will denote the principal orbit at radius \(r>1\) as \({\mathcal {O}}_{r}\). We define the reference points:
They form a ray from \(p_{1}\in S^{4}\) to infinity. This choice fixes the embeddings of the principal and singular stabiliser groups in SO(5). They are the copies of SO(4) and SO(3) occupying the lower right \(4\times 4\) and \(3\times 3\) blocks respectively. Furthermore, the choice (1.5) induces a projection map exhibiting \({\mathcal {O}}_{r}\) as a coset manifold:
The complement of the singular orbit splits as:
We study the adjoint action of SO(5) to obtain a natural frame on \(T_{p_{r}}{\mathcal {O}}_{r}\). The Lie algebra \(\mathfrak {so}(5)\) consists of all \(5\times 5\) antisymmetric matrices under the commutator bracket. It is given by:
where \(C_{ij}=e_{ij}e_{ji}\) and \(e_{ij}\) is the matrix with ij entry equal to 1 and all other entries vanishing. The bracket is characterized by the relations:
We introduce the following notation:
Let \(\rho \) be the restriction of \(\text {Ad}_{\text {SO(5)}}\) to Stab(\(p_{r}\)). An element \(g\in \text {SO(3)}\) acts on \(A\in \mathfrak {so}(5)\) by conjugation. We split this representation into irreducibles:
The first summand is trivial. The other three are isomorphic to the vector representation of SO(3). The order in which the \(X_{i}\) appear corresponds to the standard ordered basis of \({\mathbb {R}}^{3}\). The Lie algebra of the stabiliser is given by the last summand. We define the natural reductive complement:
Owing to (1.10), it is stable under \(\rho \) yielding the isotropy representation. Using (1.2), (1.3) and (1.6) we find that:
Here \(c_{1}(\cdot )\) denotes the operation of taking the first column and \(r_{2}(\cdot )\) denotes the operation of taking the second row. Using (1.12) we obtain the equations:
It is evident from (1.13)–(1.15) that (for \(r=1\)) \(X_{1},X_{2},X_{3},X_{4}\) correspond to infinitesimal motions in the horizontal direction along the base \(S^{4}\). Similarly, (1.17)–(1.19) demonstrate that (for \(r>1\)) \(X_{5},X_{6},X_{7}\) correspond to infinitesimal vertical motions along the fiber \(S^{3}\) of the sphere bundle.
The vectors \((X_{1},\ldots ,X_{10})\) provide a basis for \(\mathfrak {so}(5)\). We denote the dual basis by \((\theta ^{1},\ldots ,\theta ^{10})\) so that:
We have constructed appropriate frames (and dual coframes) for the tangent spaces to the orbits \({\mathcal {O}}_{r}\) along the reference ray (1.5). We now wish to complete these frames so that they span the full tangent space \(T_{p_{r}}X^{8}\):
Proposition 1.1
There exists a unique smooth vector field \(\partial _{r}\) on \(X^{8}{\setminus } S^{4}\) characterised by the following properties:

1.
The vector field \(\partial _{r}\) is tangent to \((0,\infty )\) in the splitting (1.7).

2.
\(dr\left( \partial _{r}\right) =1\).
Let \((x,y)\in X^{8}\subset {\mathbb {C}}^{5}\). The vector field \(\partial _{r}\) can be expressed as follows in terms of the standard coordinate vector fields on \({\mathbb {C}}^{5}\):
Evaluating the expression (1.21) at \(p_{r}\) we obtain:
Over \(p_{r}\), tensors can be written as linear combinations of tensor products of \(X_{i}\), \(\partial _{r}\) and the dual coframe \(\theta ^{i},\; dr\). The tensor in question is invariant if and only if it is stabilised by the isotropy action. In that case, the basis expansion at \(p_{r}\) is well defined over \(X^{8}{\setminus } S^{4}\). Of all the vectors in our frame, only \(\partial _{r}\) and \(X_{1}\) satisfy this condition. From here on, we will suppress application of \(d\pi \) and evaluation at \(p_{r}\). Using (1.13)–(1.22) we conclude that:
Using equations (1.23)–(1.27) we obtain:
1.2.2 The Stenzel metric
Since the second cohomology group of \(T^{\star }S^{4}\) vanishes, any Kähler structure comes from a global Kähler potential \({\mathcal {F}}(r^{2})\). The associated Kähler form will then be:
We introduce the functions:
Differentiating in the ambient \({\mathbb {C}}^{5}\) and changing frame using (1.28)–(1.32), we obtain:
The volume form of the resulting metric looks like:
The complex structure J can be described by the following equations:
Formulae (1.35), (1.37)–(1.40) can be used to obtain the associated Riemannian metric:
We wish to refine the invariant Kähler structures discussed so far by imposing the Calabi–Yau condition. This requires the canonical bundle of the underlying complex manifold to be trivial:
Proposition 1.2
The canonical bundle \(K_{X^{8}}\) is holomorphically trivial.
Proof
Let \(S_{i}\subset {\mathbb {C}}^{5}\) be the open subset where \(z_{i}\ne 0\). Introduce the following (4, 0)form on \(S_{i}\):
It is easily checked that the forms \(\iota _{X^{8}}^{\star }\Omega _{i}\) glue to a global holomorphic volume form on \(X^{8}\). \(\quad \square \)
Using (1.42) and (1.28)–(1.32), the trivialization \(\Omega \) constructed in the preceding proposition is easily written in terms of invariant forms:
Writing \(n=\text {dim}_{{\mathbb {C}}}\left( X^{8}\right) =4\), we have that the associated volume form is given by:
The Calabi–Yau equation is equivalent to volume compatibility:
Using (1.36) and (1.45), we discover that it amounts to an ODE for \({\mathcal {F}}'(r^{2})\):
Equation (1.47) can be solved explicitly. It admits a unique solution extending smoothly across the singular orbit \(S^{4}\). The resulting Riemannian metric has holonomy equal to SU(4). It is known as the Stenzel metric. We obtain:
Using (1.41), (1.48) and (1.49) we compute the pointwise norms of the vectors in the standard framing:
As \(r\rightarrow 1\), \(\partial _{r}^{2}\) blows up monotonically, \(X_{1}^{2}\), \(X_{2}^{2}\), \(X_{3}^{2},X_{4}^{2}\) approach 1 and \(X_{5}^{2},X_{6}^{2},X_{7}^{2}\) tend to 0. We can pull g back to the singular orbit \(S^{4}\) to find that it is round of unit radius.
The Cayley calibration of a CY fourfold is given by ([15], p. 81):
Using (1.54), we obtain:
When we pull \(\Phi \) back to the singular orbit \(S^{4}\), only the \(\theta ^{1234}\) term survives. On \(S^{4}\) we have \(r=1\). We therefore find that:
We conclude that the singular orbit is calibrated for \(\Phi \) and is therefore a Cayley submanifold of the Spin(7) manifold \((X^{8},\Phi )\). As such, it is volume minimizing in its homology class [7].
Modifying the radial coordinate so that the corresponding radial vector is asymptotically of unit length and expanding the Stenzel metric at infinity, one finds that it is asymptotically conical of rate \(\mu =\frac{8}{3}\). In fact, this is the optimal rate [1]. In particular, on setting:
one finds that:
Here, the conical metric:
is defined over \(X^{8}{\setminus } S^{4}\) and it is built on the homogeneous link \(\left( \frac{\mathrm {SO(5)}}{\mathrm {SO(3)}},{\overline{h}}\right) \), where:
We briefly consider uniqueness issues for the above construction. The form \(\Omega \) is not the only possible choice of holomorphic trivialization for \({\mathcal {K}}_{X^{8}}\). Others may be constructed through multiplication by a nowherevanishing holomorphic function. Scaling by nontrivial SO(5)invariant functions will yield new SO(5)invariant holomorphic trivializations. Changing \(\Omega \) can affect the resulting metric and Cayley calibration through (1.46) and (1.54). An obvious modification is to scale by nonzero complex numbers. Scaling by nonnegative reals modifies the Calabi–Yau equation, which ultimately results in rescaling the metric. Scaling by elements of \(\mathrm {U(1)}\) does not affect (1.46) and hence leaves the metric unaltered. It does however alter the Cayley calibration. The trivialization \(\Omega \) has been chosen in hindsight for the special geometric features it entails: it makes the radius of the round \(S^{4}\) equal to 1 (which serves as a natural normalization condition for the rescaling freedom) and furthermore guarantees that the latter is calibrated for the induced Spin(7) structure.
2 SO(5)Invariant Instantons with Structure Group U(1)
2.1 Cohomogeneity one bundles with structure group U(1)
Let \(r>1\). The homogeneous U(1) bundles over the orbit \({\mathcal {O}}_{r}\) correspond to elementconjugacy (i.e. conjugation by a fixed element in the target) classes of Lie group homomorphisms:
For the classification of homogeneous bundles and invariant connections see ([13, 18] Section 3.1, [10] Section 2.4). Our notation and conventions agree with the latter.
Since the target is abelian, the elementconjugacy relation is trivial: the classes are singletons. The only map of type (2.1) is \(\lambda =1\). Consequently, the only homogeneous U(1) bundle over \({\mathcal {O}}_{r}\) up to equivariant princpal bundle isomorphism is the trivial one:
SO(5)invariant U(1)connections on \(P_{1}\) are parameterised by representation morphisms:
Recalling the decomposition (1.11) and applying Schur’s lemma, we obtain that:
Here, the imaginary number \(i\alpha \) corresponds to:
The cohomogeneity one bundle over \(X^{8}{\setminus } S^{4}\) associated to \(P_{1}\) is obtained by pulling back along the map:
We slightly abuse notation by suppressing the pullback symbol and denoting the resulting bundle by \(P_{1}\). It is trivial and it admits a unique extension across the singular orbit given by \(X^{8}\times \text {U}(1)\).
Connections over \(X^{8}{\setminus } S^{4}\) can be put in temporal gauge (vanishing dr component) through an equivariant gauge transformation ([10], p. 973, Remark 3). Consequently, each invariant connection on \(P_{1}\) is equivariantly gauge equivalent to one lying in the space:
Such connections can only be related by an rindependent gauge transformation. If such a gauge transformation is equivariant, it is given by a fixed element of U(1) and it stabilises all connections. It follows that no two distinct elements of \({\mathcal {A}}_{\text {inv}}\left( P_{1}\right) \) are equivariantly gauge equivalent.
We compute the curvature of \(A\in {\mathcal {A}}\left( P_{1}\right) \):
To simplify the second term we use the MaurerCartan relations ([8], p. 41). For this calculation we require the structure constants of \(\mathfrak {so}(5)\). They can be computed using (1.8) and (1.9). Carrying out the calculation gives:
Incorporating this into (2.7), we obtain:
The AmbroseSinger holonomy theorem implies that any nonflat U(1) connection is irreducible. Consequently, all elements of \({\mathcal {A}}_{\text {inv}}\left( P_{1}\right) \) excluding the trivial connection are irreducible.
2.2 The SO(5)invariant ODEs
The Spin(7) instanton equation reads:
Since the metric diagonalises we have:
where just for this equation we have denoted dr by \(\theta ^{0}\). The sign in (2.10) is positive if and only if \((\theta ^{i_{0}},\ldots ,\theta ^{i_{7}})\) forms an oriented coframe. Using (2.10) and (1.41) we compute:
Incorporating these in (2.8), we obtain:
We now use (1.55) and (2.8) to compute:
Imposing (2.9) and comparing coefficients gives two equations. These are the same and read:
The Hermitian Yang–Mills equations are:
We find that (2.15) holds identically. This can be seen by direct computation using (1.43), (1.44) and (2.8). It follows that an SO(5)invariant U(1)connection A is HYM if and only if (2.14) holds.
Over a Hermitian manifold of complex dimension n, we have:
Using (1.35), we compute:
Using (2.16), (2.17) and (2.8) we calculate:
It follows that an SO(5)invariant U(1)connection is HYM if and only if:
We observe that this equation is the same as (2.13).
Using the uniqueness part of the standard Picard theorem, we obtain:
Theorem 2.1
An SO(5)invariant U(1)connection over \(X^{8}{\setminus } S^{4}\) is a Spin(7) instanton if and only if it is Hermitian–Yang–Mills.
2.3 Explicit solution and breakdown near the singular orbit
We study the ODE (2.18). Using (1.48) and (1.49) we write it as:
We integrate (2.19) directly to see that the solution takes the following form for some \(K\in {\mathbb {R}}\):
An elementary calculation yields:
Recalling the formulae (2.6) and (2.8) and incorporating (2.20) and (2.21), we obtain:
Theorem 2.2
Let \(P_{1}\) be the unique cohomogeneity one U(1)bundle over \(X^{8}{\setminus } S^{4}\) (i.e. the trivial bundle). There exists a oneparameter family of smooth SO(5)invariant Spin(7) instantons \(A_{K}\in {\mathcal {A}}_{\text {inv}}\left( P_{1}\right) \):
All elements of this family are HYM. They are all irreducible apart from the trivial connection and no two of them are gauge equivalent. Furthermore, these are all the invariant Spin(7) instantons on \(P_{1}\).
The curvature of \(A_{K}\) is given by:
We need to justify our claim that these solutions lie in distinct gauge equivalence classes. We have seen that this holds when we quotient by the group of equivariant gauge transformations. The claim follows from the fact that gauge equivalent, irreducible, invariant connections are automatically equivariantly gauge equivalent ([12], Corollary 4.5).
Using (1.50)–(1.53), we find that:
In particular:
Since the metric extends smoothly to the singular orbit, this behaviour is precluded for connections that are smooth over the whole space. We therefore obtain the following global nonexistence result:
Theorem 2.3
There exist no global, abelian, SO(5)invariant Spin(7) instantons/ HYM connections on the Stenzel manifold \(X^{8}\) apart from the trivial connection \(A=0\) (corresponding to \(K=0\)).
As a closing remark, we note that breakdown around Cayley fourfolds is an interesting feature of the Spin(7)instanton equation. It is related to the noncompactness of the moduli space. In Donaldson theory, noncompactness occurs in the form of a sequence of ASD instantons failing to have a limit due to finitely many point singularities. In the 8dimensional Spin(7) setting, points are typically replaced by fourdimensional Cayley submanifolds [3, 4].
The nonexistence result we encountered has to do with abelian gauge theory being too coarse to capture the behaviour we would like to see. In the following section we study the nonabelian equations associated to the structure group SO(3). The nonlinearity induced by the noncommutativity of the group smoothes the equations and we are able to obtain solutions that extend over the singular orbit \(S^4\).
3 SO(5)Invariant Instantons with Structure Group SO(3): Preliminaries and Analysis on the Trivial Bundle
3.1 Cohomogeneity one bundles with structure group SO(3)
3.1.1 Bundles and bundle extensions
Let \(r>1\). The homogeneous SO(3) bundles over the orbit \({\mathcal {O}}_{r}\) correspond to elementconjugacy classes of Lie group homomorphisms:
There are two such classes. They are represented by the trivial map and the identity respectively. Consequently, there are precisely two homogeneous principal SO(3) bundles over \({\mathcal {O}}_{r}\) up to equivariant principal bundle isomorphism. We denote these by \(P_{1}\) and \(P_{\text {Id}}\). Slightly abusing notation, we also denote by \(P_{1}\) and \(P_{\text {Id}}\) the pullbacks of the respective bundles along the map:
We now classify smooth homogeneous extensions of \(P_{1}\) and \(P_{\text {Id}}\) across the singular orbit \(S^{4}\). These correspond to elementconjugacy classes of Lie group homomorphisms:
Once such a map is chosen, one uses it to form the associated homogeneous bundle \(P_{\mu }\) over \(S^{4}\). The extension is then determined by pulling \(P_{\mu }\) back over \(X^{8}\) through the natural projection:
The elementconjugacy class of the restriction of \(\mu \) to the lower right copy of SO(3) determines which bundle is being extended.
We are therefore required to classify elementconjugacy classes of homomorphisms of type (3.2). Natural representatives are described by passing through the respective universal covers. We have the twosheeted covering maps:
where:
Considering (3.3) and (3.4), we obtain:
There are precisely three elementconjugacy classes of homomorphisms of type (3.2), one of them being that of the trivial map. The two nontrivial classes are represented by the two projections
Hence, there are precisely three principal SO(3)bundles of cohomogeneityone over \(X^{8}\). We denote these as \(P_{1}\), \(P_{\pi _{1}}\) and \(P_{\pi _{2}}\). The first is the trivial bundle. It extends the trivial bundle on \(X^{8}{\setminus } S^{4}\). The other two bundles are nontrivial (see Sect. 4.1.2). They provide distinct extensions of \(P_{\text {Id}}\).
3.1.2 Invariant connections on the complement of the singular orbit
We now classify the invariant connections on the bundles \(P_{1}\) and \(P_{\text {Id}}\) over \(X^{8}{\setminus } S^{4}\). For each of these connections we compute the associated curvature tensor in terms of the standard framing.
We introduce the following basis for \(\mathfrak {so}(3)\):
Here the matrices \(C_{ij}\) are defined as in section 1.2.1 and obey the commutation relations (1.8) and (1.9). The adjoint representation of SO(3) takes the form:
Here SO(3) acts by its natural irreducible vector representation on \({\mathbb {R}}^{3}\) and we maintain the convention that the ordering in the bracket reflects the associated identification \(\mathfrak {so}(3)\cong {\mathbb {R}}^{3}\).
In general, an invariant connection \(A\in {\mathcal {A}}_{\text {inv}}\left( P_{\lambda }\right) \) corresponds to a map of representations:
Given such a map, we use the canonical invariant connection \(d\lambda \) as a reference and write:
We first deal with \(P_{1}\). In this case \(\lambda =1\) and the target representation is trivial. Recalling the splitting (1.11) and applying Schur’s lemma, we see that \(\Lambda \) must take the form:
The canonical invariant connection \(A^{\mathrm {can}}_{1}\) is represented by \(d\;1=0\). Evidently, it is flat.
As in Sect. 2.1, any connection over \(X^{8}{\setminus } S^{4}\) can be brought to temporal gauge by an equivariant gauge transformation. It follows that any invariant connection on \(P_{1}\) is equivariantly gauge equivalent to one lying in the space:
A gauge transformation relating two elements of \({\mathcal {A}}_{\text {inv}}\left( P_{1}\right) \) must be rindependent. If it is equivariant, it is given by a fixed element of SO(3) acting on \({\mathcal {A}}_{\text {inv}}\left( P_{1}\right) \) by conjugation. It follows that the elements of \({\mathcal {A}}_{\text {inv}}\left( P_{1}\right) \) need not lie in distinct equivariant gauge equivalence classes.
A calculation analogous to the one in section 2.1 yields:
The AmbroseSinger holonomy theorem implies that the elements of \({\mathcal {A}}\left( P_{1}\right) \) need not be irreducible. This happens for instance if one of the components \(a^{i}\) vanishes identically, in which case \(e_{i}\) does not lie in the holonomy algebra.
We now work on \(P_{\text {Id}}\). In this case, the target representation is the adjoint representation of SO(3). Recalling the decomposition (1.11) and applying Schur’s lemma, we see that equivariant maps of type (3.7) always vanish on the first summand and either restrict to isomorphisms or the zero map on the second and third summands. The automorphisms of \(\text {Ad}_{\text {SO}(3)}\) are given by multiplication by fixed scalars. We conclude that for \(\lambda =\text {Id}\), maps of type (3.7) look like:
Written over the symmetry group SO(5), the canonical invariant connection \(A^{\text {can}}_{\text {Id}}=d\;\text {Id}_{\text {SO}(3)}\) on \(P_{\text {id}}\) takes the form:
It is not flat. Its curvature is given by:
The radial component of an invariant tensorial 1form is an invariant section of the adjoint bundle. In this context, these objects correspond to fixed points of \(\text {Ad}_{\text {SO(3)}}\). This representation has no fixed points, implying that all invariant connections are already in temporal gauge. Consequently, the space of invariant connections is given by:
Equivariant gauge transformations correspond to central elements of SO(3). Since SO(3) is centerless, the only possibility is the identity. Consequently, each invariant connection constitutes its own equivariant gauge equivalence class.
To compute the curvature of a general element \(A=A^{\text {can}}_{\text {Id}}+\Lambda \in {\mathcal {A}}_{\text {inv}}(P_{\text {Id}})\), we use the formula:
Routine calculation yields:
The final summand is also easily seen to take the form:
Overall, we obtain the following expression for the curvature:
The AmbroseSinger holonomy theorem implies that all elements of \({\mathcal {A}}_{\text {inv}}\left( P_{\text {Id}}\right) \) are irreducible. Since gauge equivalent, irreducible, invariant connections are equivariantly gauge equivalent ([12], Corollary 4.5), the elements of \({\mathcal {A}}_{\text {inv}}\left( P_{\text {Id}}\right) \) all lie in distinct gauge equivalence classes.
3.1.3 Invariant connections on the extended bundles
It remains to understand how to describe invariant connections on the extensions of \(P_{1}\) and \(P_{\text {Id}}\) over \({S^{4}}\). For \(P_{1}\) this is easy. The unique extension is given by the trivial bundle. The canonical invariant reference connection is still equal to zero. It follows that the \(\text {ad}(P_{1})\)valued forms (3.10) are still meaningful over the extended bundle and describe the relevant invariant connections with this choice of reference.
The situation is slightly more subtle for \(P_{\text {Id}}\). The canonical invariant connection of \(P_{\text {Id}}\) disagrees with those of \(P_{\pi _{1}}\) and \(P_{\pi _{2}}\). In fact, \(A^{\text {can}}_{\text {Id}}\) does not smoothly extend on either bundle extension. To see this, we compute the canonical invariant connections of \(P_{\pi _{1}}\), \(P_{\pi _{2}}\). These are given by:
Now, an invariant connection \(A\in {\mathcal {A}}_{\text {inv}}(P_{\text {Id}})\) over \(X^{8}{\setminus } S^{4}\) takes the form:
We rewrite it using \(A_{\pi _{1}}^{\text {can}}\) and \(A_{\pi _{2}}^{\text {can}}\) as the reference. This yields:
The forms \(\theta ^{5}\), \(\theta ^{6}\) and \(\theta ^{7}\) blow up as \(r\rightarrow 1\). We conclude that a necessary condition for A to extend to \(P_{\pi _{1}}\) is:
Similarly, if A extends to \(P_{\pi _{2}}\) we have:
The connection \(A^{\text {can}}_{\text {Id}}\) corresponds to \(a=b=0\). Both conditions (3.21) and (3.22) fail. Consequently \(A^{\text {can}}_{\text {Id}}\) does not extend to either \(P_{\pi _{1}}\) or \(P_{\pi _{2}}\).
3.2 The SO(5)invariant ODEs on \(P_{1}\): derivation and explicit solution
We proceed to study the Spin(7) instanton and Hermitian–Yang–Mills equations on the bundle \(P_{1}\). A general invariant connection A is equivariantly gauge equivalent to one of the form (3.10). The associated curvature tensor \(F_{A}\) is given by (3.11). These expressions are manifestly similar to (2.6) and (2.8). An identical computation to the one carried out in the abelian case gives:
Consequently, the invariant Hermitian Yang–Mills equations take the form:
Using (1.54) and computing as in the abelian case we obtain:
Equating these expressions yields the invariant Spin(7) instanton ODEs. They are identical to (3.23). We thus obtain the following local equivalence result:
Theorem 3.1
An SO(5)invariant SO(3)connection \(A\in \mathcal {A_{\mathrm {inv}}}\left( P_{1}\right) \) over \(X^{8}{\setminus } S^{4}\) is a Spin(7) instanton if and only if it is Hermitian–Yang–Mills.
The ODE (3.23) has already been studied in the context of the abelian equations. We immediately obtain an analogous existence/classification result:
Theorem 3.2
Let \(P_{1}\) be the trivial cohomogeneity one SO(3)bundle over \(X^{8}{\setminus } S^{4}\). There is a 3parameter family of invariant Spin(7) instantons \(A_{K^{1},K^{2},K^{3}}\in {\mathcal {A}}_{\text {inv}}\left( P_{1}\right) \):
They are all HYM. Any invariant Spin(7) instanton on \(P_{1}\) is equivariantly gauge equivalent to some element of this family.
The curvature of \(A_{K^{1},K^{2},K^{3}}\) is given by:
The AmbroseSinger holonomy theorem implies that all the instantons of theorem 3.2 are reducible. In particular, the holonomy algebra of \(A_{K_{1},K_{2},K_{3}}\) is given by:
When \(A_{K_{1},K_{2},K_{3}}\ne 0\), at least one of the parameters \(K^{i}\) does not vanish and \(\mathfrak {hol}\left( A_{K_{1},K_{2},K_{3}}\right) \) is onedimensional. The holonomy group \(\text {Hol}\left( A_{K_{1},K_{2},K_{3}}\right) \) is the associated oneparameter subgroup. Recalling the elementary fact that the oneparameter subgroups of SO(3) are embedded circles, we conclude that (when \(A_{K^{1},K^{2},K^{3}}\ne 0\)) the holonomy group is a copy of U(1) in SO(3). Let \(Q\subset P_{1}\) denote the trivial U(1)subbundle with fiber \(\text {Hol}\left( A_{K_{1},K_{2},K_{3}}\right) \). The reduction theorem implies that \(A_{K^{1},K^{2},K^{3}}\) restricts to an irreducible connection on \(Q_{_{X^{8}{\setminus } S^{4}}}\). The resulting instanton is one of those promised by Theorem 2.2. This makes rigorous the apparent similarities with the situation in section 2.
Unless \(K^{1}=K^{2}=K^{3}=0\), the curvature norm of \(A_{K^{1},K^{2},K^{3}}\) is unbounded as \(r\rightarrow 1\). We thus obtain the following global nonexistence result:
Theorem 3.3
There are no global SO(5)invariant Spin(7) instantons/ HYM connections on the trivial SO(3)bundle \(P_{1}\) over \(X^{8}\) apart from the trivial connection \(A=0\).
3.3 Extendibility of connections across the singular orbit
It remains to study the equations on \(P_{\text {Id}}\). This is the content of Sect. 4. As we shall see, \(P_{\text {Id}}\) admits solutions that are well behaved near \(S^{4}\). As a preliminary step, we need to understand when a general \(A\in {\mathcal {A}}_{\text {inv}}\left( P_{\text {Id}}\right) \) arises as the restriction of a global connection (either on \(P_{\pi _{1}}\) or \(P_{\pi _{2}}\)). Our task is to formulate the criterion of Eschenburg and Wang ([6], Lemma 1.1, p. 113) in the context of gauge theory on \(X^{8}\). A similar analysis is carried out in the foundational article of Lotay–Oliveira ([10], Appendix A.2).
3.3.1 Extendibility of tensorial forms: the criterion of Eschenburg and Wang
Let S be a Lie group and let \(\mu \) be a Lie group homomorphism:
Denote by \(\lambda \) the restriction of \(\mu \) to the bottom right copy of SO(3). Let P be the cohomogeneity one principal Sbundle over \(X^{8}\) whose restrictions over each orbit \({\mathcal {O}}_{r}\) (\(r\ge 1\)) are given by:
Let \((V,\rho )\) be a representation of the group S. We can form the associated vector bundle over \(X^{8}\):
We consider the problem of extending SO(5)invariant \(\rho \left( P\right) \)valued kforms across the singular orbit \(S^{4}\).
Eschenburg and Wang give necessary and sufficient conditions for extending invariant linear tensors across the singular orbit of a cohomogeneity one space. Since we are interested in bundlevalued forms, their technique does not apply directly. We resolve this issue by passing to the total space P and working with Vvalued forms instead. In this section we set up the requisite framework to implement this idea.
The manifold P is a cohomogeneity one space for the group \(\text {SO(5)}\times S\). Its principal orbits are isomorphic to \(P_{\lambda }\) and its singular orbit is the bundle \(P_{\mu }\).
Define the reference points:
With this definition, the point \(x_{r}\) lies in the fiber above \(p_{r}\in {\mathcal {O}}_{r}\) for all \(r\ge 1\).
Using these reference points, the isotropy subgroups corresponding to the principal and singular orbits are respectively given by:
In formulae (3.24) and (3.25), SO(4) and SO(3) denote the bottom right inclusions of these groups in SO(5). In what follows, when we consider the action of SO(4) on P, it will be through its embedding in \(\text {SO}(5)\times S\) as the singular isotropy group (3.25).
Let \(\omega \) be an invariant, tensorial form of type \(\rho \). Its extendibility can be decided by studying the restriction \(\omega _{_{W}}\) along a particularly simple embedded submanifold \(W\subset P\). This will make the problem tractable. Let W be the union of the SO(4)orbits of all points \(x_{r}\) in P:
This is a 4dimensional linear SO(4)representation. The SO(4)action is obvious. The linear structure is inherited from \(T^{\star }S^{4}_{p_{1}}\) through the projection map:
In particular, the inverse function theorem implies that \(\pi \) restricts to a diffeomorphism:
The latter is a smoothly embedded submanifold of \(X^{8}\) stable under the action of SO(4). The equivariance of \(\pi \) implies that W and \(T^{\star }_{p_{1}}S^{4}\) are isomorphic SO(4)representations. Since \(T^{\star }_{p_{1}}S^{4}\) is a vector space, it can be naturally identified with the tangent space at its origin (e.g. by the exponential map of the underlying additive group). Endowing the latter with the isotropy action, this identification becomes equivariant. These considerations allow us to view W as the vector representation of SO(4):
The extendibility problem for invariant tensors is addressed by examining their restrictions along W. We are thus interested in finding a useful way to describe these restrictions. Pull the bundle
back to W using the inclusion map (here we denote by \({\underline{V}}\) the trivial vector bundle with fiber V). Since W is linear, the pullback is trivial. We will now give a particular trivialization that elucidates the action of SO(4). Using the canonical invariant connection to decompose TP into vertical and horizontal distributions, we obtain an equivariant identification:
Furthermore, there is an obvious SO(4)equivariant trivialization:
Putting these together we have:
Here, the action of SO(4) is as follows: The action on \({\mathfrak {s}}\) is trivial. The action on V is obtained by composing \(\mu \) and \(\rho \). Finally, the spaces \(\left<X_{1},\ldots ,X_{4}\right>^{\star }\) and \(\left<\partial _{y^{2}},\partial _{y^{3}},\partial _{y^{4}},\partial _{y^{5}}\right>^{\star }\) are vector representations.
We study the restriction of \(\omega \) along \(W_{0}\): the vector space W punctured at its origin
Since tensorial forms vanish on vertical vectors, \(\omega _{_{W_{0}}}\) is a section of the trivial bundle with fiber equal to:
Due to the triviality of the bundle, the form \(\omega _{W_{0}}\) amounts to an SO(4)equivariant function:
The invariance of \(\omega \) implies that no information is lost in passing from \(\omega \) to f. In fact, \(\omega \) is determined by the values of f on the reference points \(x_{r}\) forming a ray from the origin of W to infinity. This recovers our usual description of invariant forms as curves in a group representation:
Eschenburg and Wang prove that the extendibility of \(\omega \) is contingent to a representationtheoretic condition on the formal Taylor series expansion of an appropriate reparameterization of \(\omega _{r}\). This condition reflects the behaviour of f near \(x_{1}\).
The requisite reparameterization is obtained as follows. Using (3.28), the Euclidean metric on \({\mathbb {R}}^{10}\) induces an inner product on W. We consider the radial function of the associated norm. Concretely, we set:
We thus obtain a curve:
It is clear that f determines \(\gamma \) and viceversa:
The result of Eschenburg and Wang ([6], Lemma 1.1, p. 113) asserts that \(\omega \) extends smoothly over the singular orbit if and only if the following hold:

The curve \(\gamma \) is smooth from the right at \(t=0\)

The formal Taylor series of \(\gamma \) at \(t=0\) can be written as:
$$\begin{aligned} \gamma \sim \sum _{k\ge 0}u_{k}\left( x_{_{t=1}}\right) t^{k} \end{aligned}$$where:
$$\begin{aligned} u_{k}:W\rightarrow E \end{aligned}$$is a homogeneous equivariant polynomial of degree k.
Note that we have provided explicit descriptions of the SO(4)actions on W and E. These descriptions facilitate the computations required for applications.
3.3.2 Application: extendibility of connections
We are interested in studying the extendibility of tensorial forms \(\omega \) describing connections on P (relative to the canonical invariant connection). Therefore in the context of our application we have:
Given our setup, \(\omega \) will usually be available in the form (3.31). Given this data, we need to pass to the associated curve \(\gamma (t)\) and express it in a basis of E coming from evaluation of homogeneous equivariant polynomials at \(x_{_{t=1}}\in W\). To achieve this, we need to be able to find appropriate equivariant polynomials. This task can be simplified if we understand the relevant representations in terms of quaternions. To this end, we identify the spaces W and \(\left<X_{1},\ldots ,X_{4}\right>\) with \({\mathbb {H}}\) by:
Furthermore, we lift the action of SO(4) to \(\text {Sp}(1)^{2}\) using the covering map \(\pi _{\text {Spin}(4)}\). Under these identifications, the SO(4)action is captured by the usual spinor representation of \(\text {Sp}(1)^{2}\) on \({\mathbb {H}}\).
General points \(p\in W\) and \(q\in \left<X_{1},\ldots ,X_{4}\right>\) can be written as:
With this choice of coordinates we have:
The Lie algebra \(\mathfrak {so}(3)\) can be naturally identified with \(\mathfrak {sp}(1)=\mathfrak {Im}\left( {\mathbb {H}}\right) \) using the differential of the covering map \(\pi _{\text {Spin}(3)}\) (defined in (3.4)). This identification is Adequivariant. Explicitly, it takes the following form:
These considerations demonstrate that we require homogeneous \(\text {Sp}(1)^{2}\)equivariant polynomials:
with prescribed value at \(x=1\). Separating the components in the target, such maps take the form:
The \(\text {Sp}(1)^{2}\)equivariance condition for \(u:W\rightarrow E\) translates to the following:
3.3.3 The Case of \(P_{\pi _{1}}\)
In this case \(\mu =\pi _{1}\). The action of \(\text {Sp}(1)^{2}\) on \(\mathfrak {Im}\left( {\mathbb {H}}\right) \) is given by projecting the group element to the first factor and conjugating by the result. Conditions (3.34), (3.35) become:
Using (3.19), we write a general invariant connection over \(X^{8}{\setminus } S^{4}\) as \(A^{\text {can}}_{\pi _{1}}+\omega \), where:
Using (1.24)–(1.27) and (3.33) we find that the form \(\theta ^{2}\otimes e_{3}\theta ^{3}\otimes e_{2}+\theta ^{4}\otimes e_{1}\) corresponds to:
Similarly, using (1.24)–(1.27), (3.32) and (3.33), we find that the form \(\theta ^{5}\otimes e_{3}\theta ^{6}\otimes e_{2}+\theta ^{7}\otimes e_{1}\) corresponds to:
Any equivariant polynomial u satisfying:
has the following restriction on \(S^{3}\subset {\mathbb {H}}\):
Similarly, any equivariant polynomial v satisfying:
has the following restriction on \(S^{3}\subset {\mathbb {H}}\):
As soon as u and v are specified on the unit sphere, they are extended to \({\mathbb {H}}\) by homogeneity. The extensions depend on the degree d, which is yet unspecified. Given d, we define:
The admissible values of d are constrained: not all choices yield polynomial extensions. Equations (3.41) and (3.43) demonstrate that \(u_{0}\) and \(v_{0}\) are not constant functions. If they were polynomials they would have homogeneous degree \(d=0\) and would thus be constant. We conclude that the choice \(d=0\) is not admissible. However, it is clear that we could choose \(d=1\). This would correspond to defining the extensions by the formulae (3.41) and (3.43) on the whole of \({\mathbb {H}}\). Now, if two homogeneous polynomials agree on the unit sphere, they are related through multiplication by the homogeneous degree 2k polynomial \(x^{2k}\). Consequently, all other admissible choices of d are obtained by adding even integers to \(d=1\). We conclude that for each odd positive integer:
we have precisely one homogeneous equivariant polynomial \(u_{d}\) of degree d satisfying (3.40) and precisely one homogeneous equivariant polynomial \(v_{d}\) of degree d satisfying (3.42).
We rewrite the form \(\omega \) as:
Applying the criterion of Eschenburg and Wang we obtain:
Proposition 3.4
Let \(A\in {\mathcal {A}}_{\mathrm {inv}}\left( P_{\mathrm {Id}}\right) \) be an invariant connection defined over \(X^{8}{\setminus } S^{4}\). Let
be the tensorial form expressing A with respect to the canonical invariant connection of \(P_{\pi _{1}}\). Then A extends over the singular orbit on \(P_{\pi _{1}}\) if and only if the following hold:

The function a(r(t)) is smooth from the right at \(t=0\), odd and \(O\left( t\right) \).

The function \(b(r(t))1\) is smooth from the right at \(t=0\), even and \(O\left( t^{2}\right) \).
3.3.4 The Case of \(P_{\pi _{2}}\)
In this case \(\mu =\pi _{2}\). The action of \(\text {Sp}(1)^{2}\) on \(\mathfrak {Im}\left( {\mathbb {H}}\right) \) is given by projecting the group element to the second factor and conjugating by the result. Conditions (3.34), (3.35) become:
Using (3.20), we write a general invariant connection over \(X^{8}{\setminus } S^{4}\) as \(A^{\text {can}}_{\pi _{2}}+\omega \), where:
Recalling (3.38) and (3.39) we seek homogeneous equivariant polynomials u and v satisfying (3.40) and (3.42) respectively. Equivariance specifies their restrictions on \(S^{3}\subset {\mathbb {H}}\):
Arguing as in the case of \(P_{\pi _{1}}\) we find that there is precisely one homogeneous equivariant polynomial \(u_{d}\) and precisely one homogeneous equivariant polynomial \(v_{d}\) satisfying the requisite conditions in each odd degree \(d=1+2k\).
We rewrite the form \(\omega \) as:
Applying the criterion of Eschenburg and Wang we obtain:
Proposition 3.5
Let \(A\in {\mathcal {A}}_{\mathrm {inv}}\left( P_{\mathrm {Id}}\right) \) be an invariant connection defined over \(X^{8}{\setminus } S^{4}\). Let
be the tensorial form expressing A with respect to the canonical invariant connection of \(P_{\pi _{2}}\). Then A extends over the singular orbit on \(P_{\pi _{2}}\) if and only if the following hold:

The function a(r(t)) is smooth from the right at \(t=0\), odd and \(O\left( t\right) \).

The function \(b(r(t))+1\) is smooth from the right at \(t=0\), even and \(O\left( t^{2}\right) \).
4 SO(5)Invariant Instantons with Structure Group SO(3): Analysis on \(P_{\text {Id}}\)
In the remainder, we study the invariant equations on the bundle \(P_{\text {Id}}\). The relevant analysis constitutes the heart of the present article.
4.1 The SO(5)invariant HYM ODEs on \(P_{\text {Id}}\)
We begin with the HYM equations for which the situation is significantly simpler.
4.1.1 Derivation
A general invariant connection \(A\in {\mathcal {A}}_{\text {inv}}\left( P_{\text {Id}}\right) \) takes the form (3.12). The associated curvature tensor \(F_{A}=F_{A}^{j}\otimes e_{j}\) is given in (3.15). The Hodge dual of the Kähler form has been computed in (2.17). We observe that:
Consequently, the HYM equations reduce to (2.15). We write:
Using (1.43) and (1.44) we obtain the results:
Observe that there are similarities among the various components. In particular the vanishing of any one of them is equivalent to the full HYM system. Setting any one of the components to be zero gives the invariant equations. They are as follows:
Consequently, the invariant HYM connections over \(P_{\text {Id}}\) obey the differential equations (4.8), (4.9) and satisfy the order 0 constraints (4.10), (4.11). Observe that (2.15) involves only the holomorphic volume form of \(X^{8}\). As a result, the coefficients of the Stenzel metric do not appear in (4.8)–(4.11).
4.1.2 Explicit solution and asymptotics of the curvature tensor
The Eqs. (4.8)–(4.11) can be solved explicitly. We obtain precisely two solutions:
The notation has been chosen in hindsight to reflect the bundle on which these connections extend. In particular, using Propositions 3.4 and 3.5, we find that \(A_{\text {HYM}_{\pi _{1}}}\) extends to \(P_{\pi _{1}}\), while \(A_{\text {HYM}_{\pi _{2}}}\) extends to \(P_{\pi _{2}}\). We treat \(A_{\text {HYM}_{\pi _{1}}}\) in detail. Recalling (3.32) we find that in this case:
The conditions on a(r(t)) are trivially satisfied. The function \(b(r(t))1\) is obviously smooth from the right at \(t=0\). It is even, since t only appears in power 2. One easily computes that both \(b(r(t))1\) and its first derivative vanish at \(t=0\). Consequently, \(b(r(t))1=O(t^{2})\).
Using (3.15) we compute the associated curvature tensors:
Since both solutions smoothly extend to the singular orbit, we can study their pullbacks. The connection \(A_{\text {HYM}_{\pi _{1}}}\) pulls back to the canonical invariant connection of \(P_{\pi _{1}}\) over \(S^{4}\):
Its curvature is given by:
An explicit calculation demonstrates that \(A_{\text {SD}}\) is a selfdual instanton [2] on \(S^{4}\). This justifies our choice of notation.
The connection \(A_{\text {HYM}_{\pi _{2}}}\) pulls back to the canonical invariant connection of \(P_{\pi _{2}}\):
Its curvature is given by:
An explicit calculation demonstrates that \(A_{\text {ASD}}\) is an antiselfdual instanton [2] on \(S^{4}\). This justifies our choice of notation.
We wish to endow \(\text {ad}\left( P_{\pi _{1}}\right) \) with a fiber metric. To this end, it suffices to choose an Adinvariant inner product on \(\mathfrak {so}(3)\). In general, the choice of such an inner product is free. However, we shall choose \(\langle \cdot ,\cdot \rangle \) so that:
This is the unique inner product on \(\mathfrak {so}(3)\) satisfying:
This identity is required to relate the Yang–Mills energy of instantons to characteristic classes of the underlying bundle. Using the Stenzel metric and the fiber metric (4.20) we see that the curvature norms of the two connections have the same constant value on all points of \(S^{4}\):
Since the restriction of the Stenzel metric on the singular orbit is round of unit radius, we have:
Owing to (4.21), any SO(3)connection satisfies:
By ChernWeil theory, the first Pontryagin class of the underlying SO(3)bundle is the cohomology class of the 4form:
Using (4.23) and (4.24), we conclude that the selfdual SO(3)instantons on \(S^{4}\) have Yang–Mills energy equal to \(8\pi ^{2}\) times the integral of the first Pontryagin class of the underlying bundle. Similarly, the antiselfdual instantons have Yang–Mills energy equal to \(8\pi ^{2}\) times the integral of the first Pontryagin class. Using (4.22) we obtain:
We already knew that \(P_{\pi _{1}}\) and \(P_{\pi _{2}}\) are not equivariantly trivial nor equivariantly isomorphic to each other. The above calculation demonstrates that they are genuinely nontrivial and nonisomorphic (even if we drop the requirement that the identification be equivariant).
Over a manifold with vanishing second cohomology, each SO(3)bundle is associated to a unique \(\mathrm {SU(2)}\)bundle through the natural two sheeted covering projection. The second Chern class of the lift is related to the first Pontryagin class of the original bundle by the equation ([2], pp. 41–42):
Consequently, the SU(2)lift of \(A_{\text {ASD}}\) has topological charge equal to 1. The compactified moduli space of chargeone antiselfdual instantons with structure group \(\mathrm {SU(2)}\) over \(S^{4}\) can be identified with the closed 5ball \(B^{5}\) ([2], p. 126). The lift of \(A_{\text {ASD}}\) coincides with the center of \(B^{5}\). It is the only SO(5)invariant instanton in this moduli space. Analogous remarks hold for \(A_{\text {SD}}\). This can be seen by reversing orientation to make it antiselfdual.
Finally, we study the asymptotic behaviour of the curvature tensors corresponding to the two HYM solutions. Since (4.14) and (4.15) only differ by certain signs, it suffices to treat \(A_{\text {HYM}_{\pi _{1}}}\). Using (4.20), (1.50)–(1.53) and (4.14), we find that:
In particular, we see that as \(r\rightarrow \infty \):
Here we have used the radial coordinate s(r) introduced in (1.57) to exhibit the asymptotically conical growth of the Stenzel metric. This decay rate is not sufficient for the Yang–Mills energy to be finite: on an asymptotically conical nmanifold, a function is integrable when it decays faster than \(O(s^{n})\).
In fact, the decay rate of the curvature norm could have been inferred directly from (4.12) and (4.13). Both HYM solutions decay to the translation invariant limit \(A^{\text {can}}_{\text {Id}}\). When a connection defined over an asymptotically conical manifold decays to a translation invariant limit, it is itself termed asymptotically conical ([5], p. 38). Such connections always have curvature tensors decaying like \(O(s^{2})\).
An important (gaugeinvariant) attribute of an asymptotically conical connection is the rate at which the associated curvature tensor approaches its limit. In this case, we have:
with the same being true of \(A_{\text {HYM}_{\pi _{2}}}\).
4.2 The SO(5)invariant Spin(7) instanton ODEs on \(P_{\text {Id}}\)
4.2.1 Derivation
We consider a general invariant connection \(A\in {\mathcal {A}}_{\text {inv}}\left( P_{\text {Id}}\right) \) with associated curvature field:
We remind the reader that these take the form (3.12), (3.15). Using (1.54) we compute:
We now wish to calculate the Hodge dual of the curvature. We will require the Hodge duals of all 2forms \(\theta ^{ij}\). These can be computed using (1.41) and (2.10). Carrying out this calculation yields:
The Spin(7) instanton equations are given by:
The set of equations obtained by imposing (4.35) is the same for each \(i=1,2,3\). It is as follows:
This system is overdetermined unless the metric coefficients satisfy the condition:
We recognize this as the SO(5)invariant Monge–Ampère equation (1.47) distinguishing the Stenzel metric among the Kähler metrics induced from SO(5)invariant potentials.
It is useful to work in coordinates compatible with the EschenburgWang analysis. We therefore switch to the variable \(t=R_{}\). An elementary calculation shows that the system takes the form:
where we have introduced the functions \({\mathcal {P}},{\mathcal {Q}}\in C^{\infty }\left[ 0,\infty \right) \) defined by:
4.2.2 Elementary observations
We begin our analysis of the system (4.41). In this section we make a few elementary observations about the dynamics. First, we have the following:
Proposition 4.1
The dynamics (4.41) preserve the vanishing of a and correspondingly if \(a(t)\ne 0\) for some \(t> 0\), then \(a(t)\ne 0\) for all \(t> 0\).
Proof
The first statement is trivial. The second follows by the uniqueness part of the standard Picard theorem. \(\quad \square \)
Next we observe a symmetry in the solution space:
Proposition 4.2
Suppose that the pair (a, b) solves the system (4.41). Then so does \((a,b)\).
Proof
This follows from a trivial calculation. \(\quad \square \)
We conclude the following: either \(a=0\) for all time, or a has a fixed sign throughout its lifespan. Furthermore it suffices to study the case \(a>0\) as owing to the above observation all solutions (a, b) with \(a<0\) can be obtained by considering a solution where \(a>0\) and reversing its sign.
The next proposition establishes that if one solution lies above another at some instant \(t^{\star }\), the inequality persists for all time. Here, ‘lying above’ is interpreted componentwise.
Proposition 4.3
Suppose that (a, b), \(({\widetilde{a}},{\widetilde{b}})\) are two solutions to the system (4.41) such that \(a(t)>0\) and \({\widetilde{a}}(t)>0\) for all \(t>0\). Suppose further that for some time \(t^{\star }> 0\) we have \(a(t^{\star })>{\widetilde{a}}(t^{\star })\) and \(b(t^{\star })>{\widetilde{b}}(t^{\star })\). These inequalities remain true for all \(t\ge t^{\star }\) for which both solutions exist.
Proof
Suppose not. Let \(t_{f}\) be the first time for which the inequality fails. There are three cases:

1.
\({\widetilde{a}}(t_{f})=a(t_{f})\) and \({\widetilde{b}}(t_{f})=b(t_{f})\),

2.
\({\widetilde{a}}(t_{f})=a(t_{f})\) and \({\widetilde{b}}(t_{f})<b(t_{f})\),

3.
\({\widetilde{a}}(t_{f})<a(t_{f})\) and \({\widetilde{b}}(t_{f})=b(t_{f})\).
Case 1 contradicts the uniqueness part of the standard Picard theorem.
Suppose case 2 holds. Consider the evolution of \(a{\widetilde{a}}\):
At \(t=t_{f}\) we have \({\widetilde{a}}(t_{f})=a(t_{f})=s>0\) and \({\widetilde{b}}(t_{f})<b(t_{f})\). Consequently:
It follows that \(a(t)<{\widetilde{a}}(t)\) for some time \(t<t_{f}\) and the intermediate value theorem contradicts the fact that \(t_{f}\) is the first time for which the inequalities fail.
Suppose case 3 holds. Consider the evolution of \(b{\widetilde{b}}\):
At \(t=t_{f}\) we have \({\widetilde{b}}(t_{f})=b(t_{f})\) and \({\widetilde{a}}(t_{f})<a(t_{f})\). Consequently:
which leads to a contradiction as above. \(\quad \square \)
Proposition 4.4
Suppose that (a, b) is a solution of (4.41) defined in a neighbourhood of \(t_{0}>0\). Take initial data at \(t_{0}\) satisfying \(a(t_{0})>0\), \(b(t_{0})<0\) and flow backwards. Then \(a\rightarrow +\infty \) as \(t\rightarrow t_{\text {blowup}}\ge 0\).
Proof
We will bound a from below by a function v satisfying \(v\rightarrow +\infty \) as \(t\rightarrow 0\).
Consider the evolution of the product ab. Using the Eq. (4.41), compute:
where in the last line we used the fact that \(a>0\) for all time. By comparison, flowing backwards in time, ab stays below the solution of the I.V.P:
By assumption, the initial data satisfy:
Consequently, \(u<0\) for all \(0<t<t_{0}\) and we conclude that the same is true of ab.
This allows us to estimate:
Consequently, a lies above the solution to the following I.V.P backwards of \(t_{0}\):
This is easily solved explicitly and we obtain the inequality:
\(\quad \square \)
Corollary 4.5
Let \(T>0\) and let \((a,b)\in C^{1}[0,T]\) be a solution of (4.41) satisfying \(a\ne 0\). We have that \(b(t)>0\) for all \(t\ge 0\) for which the solution exists.
Proof
Trivially, \(b(0)=\pm 1\). If not, then blows up as \(t\rightarrow 0\). Hence it suffices to prove the result for \(t>0\). If we achieve this, the possibility that \(b(0)=1\) is excluded by continuity and thus we have that \(b(0)=1\).
Suppose that for some \(t_{0}>0\), \(b(t_{0})<0\). We have that \(a(t_{0})\ne 0\) by assumption. If \(a(t_{0})>0\), the above proposition implies that a blows up to \(+\infty \) near \(t=0\) contradicting the boundedness of the solution. If \(a(t_{0})<0\), then \(a(t_{0})>0\). Since \((a,b)\) is a solution, \(a\) blows up to \(+\infty \) near \(t=0\). Hence, a blows up to \(\infty \) near \(t=0\).
Suppose that \(b(t_{0})=0\) for some \(t_{0}>0\). At such a point we have:
It follows that \(b(t)<0\) for some \(0<t<t_{0}\) and this brings us to the previous case. \(\quad \square \)
Putting the above together: if (a, b) is a global solution of (4.41), either \(a=0\) identically or the sign of a is fixed and \(b>0\).
The final proposition in this section asserts that when \(a\ne 0\) the longterm behaviour of (4.41) is essentially driven by a.
Proposition 4.6
Let \((a,b)\in C^{1}[0,T]\) be a solution of (4.41). Suppose that \(a\ne 0\). Either (a, b) survives for all \(t>0\), or there exists a finite blowup time \(0<t_{\text {blowup}}<\infty \) such that:
Proof
By the preceding results of this section, we have that \(b>0\) and the sign of a is fixed. Proposition 4.2 permits us to assume \(a>0\) without loss of generality. By the standard escape lemma, either the solution is global, or its phase space norm blows up to \(+\infty \) at some finite time \(0<t_{*}<\infty \).
The task is to prove that when the latter occurs, we have that \(a\rightarrow +\infty \) as \(t\rightarrow t_{*}^{}\). Suppose not. Then \(b\rightarrow +\infty \) as \(t\rightarrow t_{*}^{}\). Let \(\epsilon _{1}>0\) be such that:
The equation governing a implies that it is monotonically increasing past \(t_{*}\epsilon _{1}\). Since it doesn’t converge to \(+\infty \), it must be bounded above. We then have that:
Let \(0<\epsilon _{2}<\epsilon _{1}\) be such that:
Using the equation governing b, we find that for times past \(t_{*}\epsilon _{2}\):
It follows that b is bounded above. This yields the requisite contradiction, establishing the claim with \(t_{\text {blowup}}=t_{*}\). \(\quad \square \)
4.2.3 Solutions extending on \(P_{\pi _{2}}\): an explicit family of Spin(7)instantons containing a unique HYM connection
We classify the solutions that extend on \(P_{\pi _{2}}\). In the division suggested in the preceding subsection, this will be seen to correspond to the case \(a=0\). The system (4.41) reduces to a single nonlinear ODE that we can solve explicitly. We thus exhibit an explicit 1parameter family of Spin(7) instantons only one of which is HYM. This resolves (negatively) the question regarding the equivalence of the two gauge theoretic problems.
Owing to Proposition 3.5, solutions extending to \(P_{\pi _{2}}\) must satisfy \(b(0)=1\). Due to Corollary 4.5, the acomponent of such a solution must vanish identically. The system (4.41) reduces to the following ODE:
This can be solved explicitly. We fix a positive reference time and parameterize solutions by their value at that time. We choose to work with \(t_{\text {ref}}=\frac{\sqrt{6}}{2}\) (corresponding to \(r_{\text {ref}}=2\)). This choice is arbitrary. Note that our approach excludes solutions blowing up at \(t_{\text {ref}}\). This is not an issue as we are only interested in global instantons. Writing: \(\nu =b\left( t_{\text {ref}}\right) \), the associated solution to (4.48) takes the form:
Corresponding to \(b_{\nu }\) there is a local Spin(7) instanton (3.20) on the restriction of \(P_{\text {Id}}\) over an open submanifold of the form:
An elementary calculation yields the values of \(\nu \) for which there exists a finite blowup time:
Proposition 4.7
Let \(\nu \in (\infty ,\frac{2\sqrt{5}}{5})\cup (\frac{\sqrt{10}}{5},\infty )\). The connection \(A_{\nu }\) blows up (as witnessed for instance by a blowup of the pointwise curvature norm) at time \(t_{\text {blowup}}(\nu )\) given by:
For \(\nu \) outside of the range considered in the proposition, the solutions stay bounded for all time. These considerations lead to the following existence/classification result:
Theorem 4.8
Let \(\nu \in [\frac{2\sqrt{5}}{5},\frac{\sqrt{10}}{5})\). The connection \(A_{\nu }\) is a smooth Spin(7) instanton on the extended bundle \(P_{\pi _{2}}\). Furthermore, these are all the invariant Spin(7) instantons on \(P_{\pi _{2}}\).
Proof
For \(\nu \in [\frac{2\sqrt{5}}{5},\frac{\sqrt{10}}{5}]\), the function \(b_{\nu }\) is of class \(C^{\infty }[0,\infty )\). We need to verify the extension conditions of Proposition 3.5. In particular we need to prove that \(b_{\nu }(t)+1\) is even and \(O(t^2)\) at \(t=0\). We immediately exclude \(\nu =\frac{\sqrt{10}}{5}\) as the associated solution satisfies \(b(0)=1\) and consequently fails the second extension condition. Now work for \(\nu \in [\frac{2\sqrt{5}}{5},\frac{\sqrt{10}}{5})\). The first condition is clear as t always appears squared in (4.49). The second condition is easily established by computing that:
For uniqueness, we note that any invariant Spin(7) instanton on \(P_{\pi _{2}}\) obeys (4.48) and all other solutions of this equation blow up. \(\quad \square \)
The HYM connection \(A_{\text {HYM}_{\pi _{2}}}\) lies in the interior of this family and corresponds to the choice \(\nu =\frac{\sqrt{10}}{5}\). It is the only HYM connection in the family. The boundary point \(\nu _{\partial }=\frac{2\sqrt{5}}{5}\) corresponds to the solution:
The associated Spin(7) instanton \(A_{\nu _{\partial }}\) differs from the others in that it yields a different limiting connection on \(P_{\text {Id}}\) over the Stiefel manifold \(\frac{\text {SO}(5)}{\text {SO}(3)}\) at infinity. In particular, for \(\nu \in (\frac{2\sqrt{5}}{5},\frac{\sqrt{10}}{5})\) it is easily seen that:
The associated limiting connection is therefore equal to \(A^{\text {can}}_{P_{\text {Id}}}\). However, for \(\nu =\nu _{\partial }\), we have:
and the associated connection at infinity is given by:
In the study of gauge theoretic moduli problems over noncompact manifolds, it is common practice to fix the limiting connection at infinity. With this perspective in mind, we could view the family of Theorem 4.8 as having a missing endpoint in both directions. The noncompactness at the left endpoint is then mended by adding in the instanton \(A_{\nu _{\partial }}\) (the latter lying in another moduli space because of its different asymptotic behaviour). Perhaps, similar noncompactness phenomena occur in general Spin(7) instanton moduli spaces.
When a principal Gbundle over a compact CY fourfold admits HYM connections, the HYM equations determine the minima of the YangMills energy functional. We may thus (naively) expect the HYM solutions to be distinguished by a “smaller” Yang–Mills energy density in an appropriately defined sense. Over an asymptotically conical CY fourfold, a natural (yet crude) formulation of such a statement would be in terms of the radial decay rate of the pointwise curvature norm. Our examples suggest that this attribute fails to distinguish the HYM solutions among the Spin(7) instantons. We have already computed the pointwise curvature norm of \(A_{\text {HYM}_{\pi _{2}}}\). It is of order \(O(s^{2})\) at infinity. It is clear from (4.54) and (4.55) that all elements of the family are asymptotically conical (i.e. they decay to translation invariant limits). Consequently, all the associated curvature tensors decay like \(O(s^{2})\).
We collect our observations regarding the relationship of the two equation systems in the following result:
Theorem 4.9
Let \((M^{8},g,J,\Omega )\) be a geodesically complete Calabi–Yau fourfold (i.e. Hol(g)= SU(4)). It is possible for Spin(7) instantons on \(M^{8}\) not to satisfy the HYM equation. We shall refer to such connections as pure Spin(7) instantons. When \(M^{8}\) is noncompact in direct contrast to the compact case pure Spin(7) instantons can live on a principal Gbundle admitting HYM solutions. When \(M^{8}\) is asymptotically conical, the HYM solutions cannot in general be distinguished by a faster decay rate. That is, the pointwise curvature norm of a pure Spin(7) instanton can decay with the same rate as that of an HYM connection on the same bundle.
4.2.4 Solutions extending on \(P_{\pi _{1}}\)
We now wish to classify solutions that smoothly extend over \(P_{\pi _{1}}\). In the previous section we found all solutions where \(a=0\). The only one satisfying \(b(0)=1\) corresponds to \(\nu =\frac{\sqrt{10}}{5}\). The associated instanton is \(A_{\text {HYM}_{\pi _{1}}}\). Any other solution would have nonvanishing acomponent. Consequently, we have to deal with the full system (4.41). The first step is to obtain short time existence and uniqueness near the pole of the ODE. In doing so, we impose the EschenburgWang extension conditions of Proposition 3.4. Subsequently, the task is to characterize which of these local solutions survive for all time to yield global Spin(7) instantons.
Short Time Existence and Uniqueness The analysis in this section relies on the method of Eschenburg and Wang ([6], section 6). We have adapted their existence result to our equation system and refined it to include continuous dependence on initial data. This does not follow from the standard Grönwall estimate as the I.V.P under consideration is singular. The continuity proof is based on the technique employed by McLeod et al. ([11], p. 147).
Theorem 4.10
Let \(a_{0}\) be a fixed real number. There exists a unique solution:
to the system (4.41) such that a is odd at \(t=0\), b is even at \(t=0\) and furthermore:
This solution satisfies the extension conditions of Proposition 3.4 and thus yields a Spin(7) instanton on the restriction of \(P_{\pi _{1}}\) over the open submanifold defined by \(0\le t<t_{\max }(a_{0})\).
Furthermore, we have that for any \(K>0\):
and the following mapping is continuous:
We will prove this result in four stages. The first step is to study the formal Taylor series of smooth solutions at \(t=0\). The second step is to derive and analyze ODEs governing perturbations of high order polynomial truncations of the series. The idea is to show that, if the order is high enough, the resulting ODEs are uniquely soluble for sufficiently short time in suitable Banach spaces. The third step is to argue that the solutions so obtained are smooth and have the correct formal series at \(t=0\). The final step is to understand how this existence/ uniqueness argument behaves under change of initial data. This involves proving that the estimates can be made to be uniform in \(a_{0}\) for \(a_{0}\) in compact sets and establishing the desired continuity result.
Proposition 4.11
Fix \(a_{0}\in {\mathbb {R}}\). There exists a unique \((a,b)_{a_{0}}\in {\mathbb {R}}[[t]]^{2}\) solving the system (4.41) and satisfying the conditions (4.56)–(4.58). Here, differentiation is understood in the formal sense (as a derivation of the formal power series ring).
Proof
Considering our assumptions on a and b, we introduce their formal Taylor series at \(t=0\):
Using the parity of a, b and the coefficient functions, we introduce the series:
The ODE for a translates to the condition:
We compute \(c_{k}\) in terms of \(a_{0},\ldots ,a_{k},b_{0},\ldots ,b_{k}\). This yields:
Here \({\mathcal {G}}\) denotes some function of coefficients of lower order. We will slightly abuse notation and maintain use of the symbol \({\mathcal {G}}\) in subsequent calculations even though the particular function may not be the same. Using (4.62) we obtain:
This determines \(a_{k}\) in terms of coefficients of lower order provided that \(k\ne 0\). We conclude that we are allowed to choose \(a_{0}\) freely.
We perform a similar calculation for b. The second equation in (4.41) translates to the relation:
We note that \(e_{k}\) and \(f_{k}\) only involve terms depending on \(a_{0},\ldots ,a_{k},b_{0},\ldots ,b_{k}\) and it is thus unnecessary to compute them. We compute \(d_{k+1}\) in terms of \(a_{0},\ldots ,a_{k},b_{0},\ldots ,b_{k+1}\):
Using (4.64), we obtain:
It follows that \(b_{k+1}\) is determined by lower order coefficients for each \(k\ge 0\).
The above calculations demonstrate that the formal Taylor series at 0 is uniquely determined by induction given a choice of \(a_{0}\in {\mathbb {R}}\). \(\quad \square \)
Although the content of the preceding proposition is enough for the purposes of our existence theorem, continuity requires more refined knowledge of the formal Taylor series. In particular, we are interested in the dependence of its coefficients on \(a_{0}\). We explicitly calculate the first few terms of the series associated to some fixed choice of \(a_{0}\):
In fact, we are able to obtain the following:
Proposition 4.12
The coefficients of the formal Taylor series \((a,b)_{a_{0}}\) are polynomials (possibly of order 0) in \(a_{0}\).
Proof
This is certainly true for \(a_{0},b_{0}\) and \(b_{1}\). Repeating the calculations of the preceding proposition, but keeping track of the lower order terms yields the following recurrence relations for the coefficients:
The result follows by induction. \(\quad \square \)
We now discuss how to use this formal series in order to obtain an honest solution of the system (4.41). For ease of exposition, we introduce the following functions:
We rewrite the ODE system (4.41) as:
Further, we let \(p^{a}_{m}(t,a_{0})\), \(p^{b}_{m}(t,a_{0})\) denote the order m Taylor polynomials corresponding to the initial data \(a_{0}\). These are obtained by truncating the respective series. We also introduce the following error functions capturing the failure of the Taylor polynomials to solve (4.41):
They are smooth and \(O(t^{m})\) at \(t=0\). To see this, recall that the full formal series was constructed by matching derivatives at the origin. Consequently, the first \(m1\) derivatives of the error functions vanish at \(t=0\).
We now introduce the Banach spaces we will be working with. For any real \(T>0\) and integer \(m\ge 0\) we define:
Fix \(T>0\). We immediately observe that the error functions \(E^{a}_{m}, E^{b}_{m}\) lie in \({\mathcal {O}}_{T}(m)\) (they are \(O(t^{m})\) at \(t=0\)). Furthermore, the functions \(p^{a}_{m},\;p^{b}_{m}1\) lie in \({\mathcal {O}}_{T}(1)\). In fact in light of Proposition 4.12 we have:
Corollary 4.13
\(E^{a}_{m}(t,\cdot ),\; E^{b}_{m}(t,\cdot )\) define continuous mappings from the space of initial data (the real numbers) into \({\mathcal {O}}_{T}(m)\). Similarly, \(p^{a}_{m}(t,\cdot ),\;p^{b}_{m}(t,\cdot )1\) define continuous mappings from the space of initial data into \({\mathcal {O}}_{T}(1)\).
We finally recast the problem as an integral equation for a perturbation of the polynomials \((p_{m}^{a},p_{m}^{b})\). Given a pair of functions \((u,v)\in {\mathcal {O}}^{\oplus 2}_{T}(m)\) we define:
It can be easily checked (by expanding out the integrands, counting order of vanishing and noting that integration raises this by one) that we obtain a nonlinear integral operator:
The following proposition is the heart of the matter:
Proposition 4.14
Let \(a_{0}\) be fixed. Fix \(R>0\). For sufficiently large m (depending on \(F_{1},F_{2},F_{3},R\)) and sufficiently small T (depending on m and \(a_{0}\)), the operator \(\Theta _{m,a_{0}}\) has a unique fixed point (u, v) in \({\overline{B}}_{R}(0)\subset {\mathcal {O}}^{\oplus 2}_{T}(m)\). Furthermore, this fixed point is smooth on [0, T] and the associated solution
to the system (4.41) satisfies (4.56)–(4.58).
Proof
In what follows, our notation suppresses dependence on \(a_{0}\). Fix \(R>0\). We will select m and T such that \(\Theta _{m}\) is a contraction on \({\overline{B}}_{R}(0)\subset {\mathcal {O}}^{\oplus 2}_{T}(m)\).
Consider the domain:
Let \(L>0\) be a Lipschitz constant in the (u, v) variables for the restrictions of \(F_{1},F_{2},F_{3}\) on \(D_{R}\). The constant L is controlled by \(L^{\infty }\) bounds on the restrictions of the derivatives of the \(F_{i}\) on \(D_{R}\). Choose:
Pick T such that:
Clearly, for \(t\in [0,T]\) we have:
We claim that we also have:
To see this, we estimate as follows:
We now prove contraction estimates for \(\Theta ^{1}_{m}\) and \(\Theta ^{2}_{m}\). Fix \(0\le r\le T\) and compute:
In this calculation, the LLipschitz estimate is valid due to (4.71) and the fact that the uniform norm is controlled by the \({\mathcal {O}}_{T}(m)\) norm when \(0<T<1\). We conclude that:
A similar calculation yields:
Due to (4.72), (4.74) and (4.75), the closed Rball in \({\mathcal {O}}^{\oplus 2}_{T}(m)\) is stable under \(\Theta \). The contraction mapping theorem (CMT) yields a unique fixed point (u, v) in this ball.
This fixed point is necessarily of class \(C^{1}[0,T]\) (by the fundamental theorem of calculus). Consequently (a, b) is \(C^{1}\) and it therefore constitutes an honest solution of (4.41) on [0, T]. Considering the order of vanishing of u at 0 and looking at the equations, we observe that . Conditions (4.56)–(4.58) follow.
Full regularity follows by a simple bootstrap procedure. Since flows of smooth (nonautonomous) vector fields are smooth, (u, v) is smooth on (0, T]. The task is to establish smoothness at 0. Smoothness on (0, T] legitimizes differentiation of the equations for \(t>0\). This gives an expression for the second derivatives of u and v involving terms in \(\frac{u}{t^{2}}\), \(\frac{v}{t^{2}}\), and . It is thus clear that \(u^{(2)}(t),\;v^{(2)}(t)\rightarrow 0\) as \(t\rightarrow 0\). Hence u, v are of class \(C^{2}[0,T]\) with vanishing second derivative at 0. We can iterate this argument to conclude that u, v are of class \(C^{m1}[0,T]\) with vanishing derivatives at 0 up to order \(m1\). The only constraint on m required for the contraction argument to run is \(m>\max \left\{ 2L,1\right\} \). It follows that the operator \(\Theta _{l}\) is a contraction for arbitrarily large \(l>m\) (perhaps for shorter time T). Fixing \(l>m\), we let \((u_{l},v_{l})\) be the associated fixed point. Repeating the argument above, it lies in \(C^{l1}[0,T]\) with vanishing derivatives up to order \(l1\). It is thus \(O(t^{m+1})\). It follows that \((u_{l}+p_{l}^{a}p_{m}^{a},v_{l}+p_{l}^{b}p_{m}^{b})\) is also \(O(t^{m+1})\). Consequently by further decreasing T as necessary we can arrange that the latter has as small \({\mathcal {O}}^{\oplus 2}_{T}(m)\) norm as we like. In particular, we take this to be less than R. Furthermore, \((u_{l}+p_{l}^{a}p_{m}^{a},v_{l}+p_{l}^{b}p_{m}^{b})\) is a fixed point of \(\Theta _{m}\). But \(\Theta _{m}\) has a unique fixed point in the closed Rball. It follows that:
and hence that u, v lie in \(C^{l1}\). Since l was arbitrary, the proof is complete. \(\quad \square \)
We now have enough for the first part of Theorem 4.10. The preceding proposition guarantees the existence of a smooth solution (a, b) satisfying (4.56)–(4.58). The algebraic calculation in the start of this section uniquely specifies its full formal Taylor series at \(t=0\) so that it passes the extension criterion in Proposition 3.4. Finally, suppose that there is another smooth solution \(({\widetilde{a}},{\widetilde{b}})\) satisfying (4.56)–(4.58). Arguing as above, we find that the two solutions share the same formal Taylor series at 0 (the series discovered in Proposition 4.11). Let m be as in Proposition 4.14. We have that \((ap^{a}_{m},bp^{b}_{m})\), \(({\widetilde{a}}p^{a}_{m},{\widetilde{b}}p^{b}_{m})\) are \(O(t^{m+1})\). For short enough time T, the \({\mathcal {O}}^{\oplus 2}_{T}(m)\) norms of these functions are less than R. Since both functions are fixed points of \(\Theta _{m}\) and lie in the closed Rball, they are equal. Hence \((a,b)=({\widetilde{a}},{\widetilde{b}})\).
It remains to study the dependence of solutions on variations of the initial data \(a_{0}\). We immediately obtain:
Proposition 4.15
Fix \(K>0\). We have:
Proof
In our existence proof, once, R, L, m are fixed, T needs to be controlled from above by quantities decreasing with the \({\mathcal {O}}_{1}(m)\) norms of the error functions and the \({\mathcal {O}}_{1}(1)\) norms of \(p^{a}_{m}, \;p^{b}_{m}1\). By Corollary 4.13, these norms depend continuously on \(a_{0}\) and are hence bounded for \(a_{0}\) in a compact set. It follows that we can choose T small enough so that the contraction argument works for all \(a_{0}\in [K,K]\). \(\quad \square \)
Note that the contraction constant can be taken to be the same across all \(a_{0}\in [K,K]\). This is vital for the continuity proof, which we now discuss.
Proposition 4.16
The mapping defined by:
is continuous.
Proof
Consider the trivial (infiniterank) vector bundle over \([K,K]\):
The following map is fiberpreserving and continuous:
There is a unique section s of E that is fixed by S (the one assigning to each choice of initial data the associated fixed point of \(\Theta _{a_{0},m}\)). The task is to prove that s is continuous. To this end, we fix \(x\in [K,K]\) and prove that s is continuous at x. Fix \(\epsilon >0\) and define the following (continuous) section of E:
We will run the CMT iteration on each fiber with initial condition determined by \(u_{x}\). Letting \(0<C<1\) be the contraction constant of \(\Theta _{a_{0},m}\) and using the convergence rate estimate of the CMT we have:
Fix N large enough so that this quantity is controlled by \(\frac{\epsilon }{2}\). Since S and u are continuous, we have:
Consequently, for \(a_{0}\) sufficiently close to x, we can achieve:
Using (4.78), (4.79) and the triangle inequality completes the proof. \(\quad \square \)
Uniqueness implies that the solution associated to \(a_{0}=0\) corresponds to \(A_{\text {HYM}_{\pi _{1}}}\). This instanton will play a central role in the analysis of the global properties of the system.
Global Existence for Small Initial Data The previous section yields a characterization of shorttime solutions near the pole. We are now tasked with understanding which of these solutions are global. In this section we establish that:
Theorem 4.17
There exists an \(\epsilon >0\) such that for \(a_{0}<\epsilon \), the shorttime solutions of Theorem 4.10 are global.
The heart of the argument lies in the following proposition. Its conditions are subsequently easily verified (for small initial data) by a continuity argument.
Proposition 4.18
Suppose that \(a_{0}>0\) and let \((a,b)_{a_{0}}\) be a solution to the system (4.41) such that a attains a critical point in the spacetime region:
Then \(t_{\text {max}}(a_{0})=+\infty \).
Proof
By Proposition 4.1, \(a(t)>0\) for all \(0<t<t_{\text {max}}(a_{0})\). Looking at the ODE for a, we conclude that the critical points of a are precisely the points where \(b={\mathcal {P}}^{1}\). We seek an expression for the second derivative of a at a critical point occurring at time \(t=t_{\text {crit}}>0\). Differentiating the ODE for a and setting \(b={\mathcal {P}}^{1}\), we obtain:
The first factor is strictly positive. Consequently, the nature of the critical point depends on the sign of:
at \(\left( t_{\text {crit}},a\left( t_{\text {crit}}\right) \right) \). For (t, a) in the spacetime region (4.80), we have \(F(t,a)<0\). Hence, any critical point occurring in the region is a maximum.
Suppose that a maximum does occur inside the region (4.80). For a short amount of time thereafter a is decreasing. The only way that a can ever increase again is if it reaches a minimum. A minimum can only occur if (t, a(t)) exits the spacetime region (4.80). For this to occur, a has to increase. It follows that a decreases for as long as the solution survives. Consequently a is bounded from above. Since \(a>0\), it follows that a is also bounded from below. Since a consistently decreases after the maximum point, we have that \(b(t)<{\mathcal {P}}(t)^{1}\) for \(t>t_{\text {crit}}\). By Corollary 4.5, \(b>0\) for all time. Hence both a and b are bounded and thus survive for all time \(t\ge 0\). \(\quad \square \)
Proposition 4.18 applies provided that the initial data is small enough:
Proposition 4.19
There exists \(\epsilon >0\) such that if \(0\le a_{0}<\epsilon \), then \(a_{a_{0}}\) attains a critical point in the spacetime region (4.80).
Proof
The idea is to use a continuity argument and compare with the solution corresponding to \(a_{0}=0\):
Note that \({\mathcal {P}}(0)=2\) and \(b(0)=1\) (independently of the choice of \(a_{0}\)). Hence b always starts above \({\mathcal {P}}^{1}\). For \(a_{0}=0\), the solution \(b_{\text {HYM}}\) crosses \({\mathcal {P}}^{1}\) at the time: \(t=\frac{3\sqrt{2}}{2}\). For \(a_{0}>0\), formulae (4.65) and (4.66) show that at least for a very short time to the right of \(t=0\) we have
where the inequality is understood componentwise. By Proposition 4.3, this inequality persists for as long as the solutions exist. Consequently, if \(a_{0}>0\), \(b_{a_{0}}\) can only cross \({\mathcal {P}}^{1}\) strictly after \(t=\frac{3\sqrt{2}}{2}\).
Consider only \(a_{0}\le 1\). By the second assertion of Theorem 4.10, the maximal existence time of the resulting solutions is bounded below by a positive number \(T_{1}\). Furthermore, these solutions depend continuously on \(a_{0}\) (in the \(C^{0}[0,T_{1}]\) norm). Composing with the local flow associated to taking initial conditions at \(t=T_{1}\), we see that the maximal existence time is lower semicontinuous in \(a_{0}\). Furthermore, we see that if a particular choice of \(a_{0}\) yields a solution surviving past some time \(t=T\), the mapping sending initial conditions to their associated solutions is continuous from an open neighbourhood of \(a_{0}\) into \(C^{0}[0,T]\).
Since \((a_{\text {HYM}},b_{\text {HYM}})\) (associated to \(a_{0}=0\)) is global, initial data close to 0 lead to solutions that survive arbitrarily long. In particular, we can choose \(\epsilon >0\) to be small enough so that solutions associated to \(0<a_{0}<\epsilon \) survive past \(t=4\). Furthermore at the expense of taking \(\epsilon \) to be even smaller we can appeal to continuity to arrange that:
Condition (4.85) implies that for any \(\sqrt{3}< t\le 4\) the point (t, a(t)) lies in the spacetime region (4.80). Condition (4.86) implies that for any \(3\le t\le 4\) we have:
By the intermediate value theorem, there exists a \(0<t_{\text {crit}}<3\) where \(b_{a_{0}}\) crosses \({\mathcal {P}}^{1}\). But we have seen that this time must be after \(t=\frac{3\sqrt{2}}{2}\) and consequently after \(t=\sqrt{3}\). Hence, the critical point at \(t=t_{\text {crit}}\) occurs in the spacetime region (4.80). \(\quad \square \)
Theorem 4.17 easily follows from the preceding two propositions and the symmetry of the system (4.41) as formulated in Proposition 4.2.
Proof of Theorem 4.17
Proposition 4.19 yields a threshold \(\epsilon >0\) such that for any \(0\le a_{0}<\epsilon \), the a component of the associated solution attains a critical point in the region (4.80). Proposition 4.18 then implies that \((a,b)_{a_{0}}\) is global. Finally, Proposition 4.2 proves that solutions associated to \(\epsilon <a_{0}\le 0\) are global too. \(\quad \square \)
Finite Time Blowup for Large Initial Data We now wish to study the development of large initial data. We will obtain the following:
Theorem 4.20
Suppose that:
Then \((a,b)_{a_{0}}\) blows up in finite time at most equal to:
Furthermore, the blowup set:
is of the form:
for some \(0<x\le \frac{1}{2{{\,\mathrm{arctanh}\,}}\left( \frac{1}{2}\right) }\).
Our analysis relies on an apriori bound for \(\frac{b}{a}\):
Proposition 4.21
For any \(a_{0}\ge 0\), the solution \((a,b)_{a_{0}}\) satisfies the following inequality for all \(0\le t<t_{\text {max}}(a_{0})\):
Proof
Estimate (4.90) is clearly satisfied at \(t=0\). To show that it persists for as long as solutions survive, we let \(t_{\star }>0\) be any time such that:
and we compute:
\(\square \)
Proposition 4.21 allows us to estimate:
Fix a positive reference time \(t_{0}>0\). Estimate (4.92) implies that past \(t_{0}\) a is bounded below by the solution of the following I.V.P. of Riccati type:
Setting \(x_{0}{\mathop {=}\limits ^{\text {def}}}a(t_{0})>0\), Eq. (4.93) can be solved explicitly to give:
The task is now to determine conditions on \(t_{0}>0,x_{0}>0\) such that the lower bound \(u_{t_{0},x_{0}}\) blows up to \(+\infty \) in finite time past \(t_{0}\).
We introduce the threshold function:
An elementary calculation demonstrates that the denominator of (4.94) has a zero in the nonnegative real line if and only if:
Furthermore, when (4.96) is satisfied, this zero is unique and located at the time given by:
Consequently, the lower bound \(u_{t_{0},x_{0}}\) will exhibit the requisite behaviour provided that:
As it turns out, the second condition is vacuous:
Proposition 4.22
Fix \(t_{0}>0\). We have:
and \({\mathcal {T}}(t_{0},x_{0})\) decreases monotonically to \(t_{0}\) (as a function of \(x_{0}\)) for \(x_{0}>{\mathcal {R}}(t_{0})\). In particular, we have the following pointwise limit:
Proof
The proof is an elementary explicit calculation which we omit. \(\quad \square \)
Incorporating the content of Proposition 4.22 to our earlier discussion, we arrive at the following conclusion. For each time \(t_{0}>0\), the value \({\mathcal {R}}(t_{0})\) provides a threshold (justifying our terminology), such that if u solves (4.93) and satisfies
then it blows up in finite time equal to \({\mathcal {T}}(t_{0},u(t_{0}))>t_{0}\). Fix \(t_{0}>0\). For \(u(t_{0})\) close to (but above) the threshold, the blowup time can be arbitrarily large. As \(u(t_{0})\rightarrow \infty \), the blowup time approaches \(t_{0}\) from above. Consequently, for very large initial data, the solution survives for arbitrarily short time past \(t_{0}\).
Since \(u_{t_{0},x_{0}}\) bounds a from below, we obtain:
Proposition 4.23
Suppose that \((a,b)_{a_{0}}\) is a solution of the system (4.41) satisfying:
Then a blows up to \(+\infty \) in finite time at most equal to \({\mathcal {T}}(t_{0},a(t_{0}))\).
The task is then to verify that for large initial data \(a_{0}\), the acomponent of the solution eventually crosses the threshold \({\mathcal {R}}\), depicted below (Fig. 1).
We will use the reference time \(t_{0}=\frac{3\sqrt{2}}{2}\). This is the time where \(b_{\text {HYM}}\) introduced in (4.83) crosses \({\mathcal {P}}^{1}\) introduced in (4.42). We are able to obtain the following bound:
Proposition 4.24
Fix \(a_{0}>0\). Let \((a,b)_{a_{0}}\) be the development of the initial data \(a_{0}\). We have:
Proof
Fix \(a_{0}>0\). Formula (4.66) demonstrates that at least for a short time, \(b_{a_{0}}\) exceeds \(b_{\text {HYM}}\) to the right of \(t=0\). Proposition 4.3 establishes that \(b_{a_{0}}>b_{\text {HYM}}\) until \(t=t_{\text {max}}(a_{0})\). Incorporating this bound with the ODE governing a, we estimate (for \(t>0\)):
Motivated by this computation, we introduce the following (singular) I.V.P:
The problem is wellposed (solutions exist and are uniquely determined by the prescribed initial data) and v takes the form:
The function \(v_{a_{0}}\) has a global maximum at time \(t=t_{0}=\frac{3\sqrt{2}}{2}\) with value \(\frac{2\sqrt{2}}{3}a_{0}\).
If the I.V.P. (4.101) were not singular, we could invoke estimate (4.100) to conclude that \(v_{a_{0}}\) bounds \(a_{a_{0}}\) from below for as long as the latter survives. By virtue of the above remarks, this would complete the proof. However, the standard ODE comparison argument (for nonsingular I.V.Ps) does not apply directly.
We sketch this wellknown argument to identify where things go wrong. Suppose that we have two initial value problems on the real line associated to two smooth (potentially nonautonomous) vector fields \(F_{1}\) and \(F_{2}\) such that:
Let \(s_{1}(t)\) and \(s_{2}(t)\) be solutions satisfying the same initial condition:
Since \(F_{1}(0)>F_{2}(0)\) there is an open interval to the right of \(t=0\) where \(s_{1}>s_{2}\). Suppose, for a contradiction that this inequality does not persist for all time. At the first future crossing the difference \(s_{1}s_{2}\) is increasing, which yields the requisite contradiction.
Attempting to run this argument for I.V.P’s with a first order singularity at \(t=0\), we observe that the initial derivatives of \(s_{1}\) and \(s_{2}\) match. We thus run into trouble getting \(s_{1}\) to exceed \(s_{2}\) in the immediate future of the initial time. The rest of the argument carries through successfully.
Fortunately, the situation can be mended by comparing higher order data at the pole. The function \(v_{a_{0}}\) has the following formal Taylor series at \(t=0\):
Comparing this with (4.65), and recalling that \(a_{0}>0\) we conclude that \(a_{a_{0}}\) exceeds \(v_{a_{0}}\) in the immediate future of the initial time. The rest of the standard comparison argument carries through, ultimately establishing the requisite estimate. \(\quad \square \)
The upshot is that by choosing \(a_{0}\) to be sufficiently large, we can arrange that \(a(t_{0})\) exceeds any number we like. In particular, we can arrange that \(a(t_{0})\) exceeds the threshold \({\mathcal {R}}(t_{0})\).
We now have enough to complete the proof of Theorem 4.20.
Proof of Theorem 4.20
Evaluating (4.95) and (4.97) at the reference time \(t_{0}=\frac{3\sqrt{2}}{2}\) we obtain:
Let (a, b) be a solution of the system (4.41) satisfying:
Using (4.105) and Proposition 4.23, we conclude that the solution blows up to \(+\infty \) in finite time at most equal to \({\mathcal {T}}\left( t_{0},a\left( t_{0}\right) \right) \). Proposition 4.24 guarantees that (4.106) is satisfied provided that we take:
By Proposition 4.22, when \(x>{\mathcal {R}}(t_{0})\) we have that the function \({\mathcal {T}}(t_{0},x)\) is monotonic in x. Condition (4.107) guarantees that the right hand side of (4.99) exceeds \({\mathcal {R}}\left( t_{0}\right) \) and is thus large enough for the monotonicity statement to apply. We obtain:
Defining \(t_{\text {blowup}}(a_{0})\) to be equal to the right hand side of this inequality, we have established the first assertion of Theorem 4.20.
Define the positive and negative blowup sets as:
Proposition 4.6 implies that:
The last assertion of Theorem 4.20 will follow from Proposition 4.2 if we establish the existence of \(x>0\) such that:
We first prove that the positive blowup set is open. Let \(a_{0}\in {\mathcal {S}}^{+}_{\text {blowup}}\) and let \(t_{\star }\) be the blowup time of the associated solution. By definition:
Consequently, there is a time \(T\in [\frac{t_{\star }}{2},t_{\star })\) such that:
By continuity with respect to variation of the initial data, we obtain a \(\delta >0\) such that for \(\widetilde{a_{0}}\in (a_{0}\delta ,a_{0}+\delta )\):
Consequently:
By Proposition 4.22, the initial data \(\widetilde{a_{0}}\) lead to finitetime blowup and \({\mathcal {S}}^{+}_{\text {blowup}}\) is indeed open.
Finally, by Proposition 4.3, if a certain choice of \(a_{0}>0\) leads to finitetime blowup, so do all \(\widetilde{a_{0}}>a_{0}\). Together with openness, this property yields (4.108) for some \(x\ge 0\). Theorem 4.17 implies that \(x>0\). \(\quad \square \)
4.2.5 The Moduli space
The results of the preceding sections are sufficient to obtain a complete description of the moduli space of SO(5) invariant Spin(7) instantons with structure group SO(3) on the Stenzel manifold. We denote this object as \({\mathcal {M}}^{\text {Spin}(7)}_{\text {inv}}\left( X^{8}\right) \). The trivial bundle \(P_{1}\) doesn’t contribute to this moduli space. This is due to the nonexistence Theorem 3.3. Here we ignore the trivial solution \(A=0\).
Let P be a Ghomogeneous (or cohomogeneity one) principal Sbundle. There are two natural ways to set up a moduli space of Ginvariant solutions to a gaugetheoretic problem on P. One is to quotient the set of invariant solutions by the group of equivariant gauge transformations. The other is carried out in two steps. Initially one quotients the set of all (not necessarily invariant) solutions by the set of all (not necessarily equivariant) gauge transformations. The action of G on the total space P induces an action on the set of all connections. This action restricts to the set of solutions and passes to the quotient. The moduli space is then defined to be the Ginvariant locus. There is an obvious map from the first construction to the second construction. If the fiber S is semisimple and we restrict attention to irreducible connections, this map is an isomorphism ([12], Corollary 4.5).
In our setting, the structure group is SO(3) (which is indeed semisimple) and furthermore, all solutions are irreducible. It follows that the two constructions coincide. We will follow the first. Recall that each invariant connection constitutes its own equivariant gauge equivalence class. Consequently, the moduli spaces on the individual bundles are:
Due to the results of Sect. 4.2.4, we have that \({\mathcal {M}}\left( P_{\pi _{1}}\right) \) is a compact interval. It can be parameterized by initial conditions leading to global solutions. Using this parameterization, Theorem 4.20 gives us a number \(x>0\) such that:
The space \({\mathcal {M}}\left( P_{\pi _{1}}\right) \) contains a unique HYM connection \(A_{\text {HYM}_{\pi _{1}}}\) corresponding to \(a_{0}=0\). It is represented by the red dot in the following diagram (Fig. 2). The black dots represent the boundary points \(\pm x\).
Due to the results of Sect. 4.2.3, we have that \({\mathcal {M}}\left( P_{\pi _{2}}\right) \) is a halfopen halfclosed interval. We can parameterize it by the value \(\nu =b(t_{0})\) at time \(t_{0}=\frac{\sqrt{6}}{2}\). Using this parameterization and setting:
we have that:
The space \({\mathcal {M}}\left( P_{\pi _{2}}\right) \) contains a unique HYM connection \(A_{\text {HYM}_{\pi _{2}}}\) corresponding to \(\nu =\frac{\sqrt{10}}{5}\). It is represented by the green dot in the following diagram (Fig. 3). The black dot represents the boundary point \(\nu _{\partial }\).
We observe that \({\mathcal {M}}\left( P_{\pi _{2}}\right) \) is not compact. Interestingly, it admits a natural compactification. To understand the noncompactness phenomenon, we study the (missing) limit \(\nu \rightarrow \frac{\sqrt{10}}{5}\). To identify what the limit should be we work on \(X^{8}{\setminus } S^{4}\). Using the explicit formula (4.49) with \(\nu =\frac{\sqrt{10}}{5}\) yields the HYM connection \(A_{\text {HYM}_{\pi _{1}}}\). We conclude that (over \(X^{8}{\setminus } S^{4}\)):
This can be understood pointwise with a choice of some background reference connection or in a suitable weighted norm.
We conclude that the Spin(7) instantons in \({\mathcal {M}}\left( P_{\pi _{2}}\right) \) are trying to converge to the (unique) HYM connection of \({\mathcal {M}}\left( P_{\pi _{1}}\right) \), but fail to do so as this connection does not smoothly extend to the bundle on which they live. Notably the singularity happens around a codimension 4 Cayley submanifold (the singular orbit \(S^{4}\)). This reasoning motivates us to glue in \({\mathcal {M}}\left( P_{\pi _{1}}\right) \), by forcing the point \(a_{0}=0\) to be the missing endpoint of \({\mathcal {M}}\left( P_{\pi _{2}}\right) \). This leads to the following picture (Fig. 4) of \({\mathcal {M}}^{\text {Spin}(7)}_{\text {inv}}\left( X^{8}\right) \):
The notation in this diagram is consistent with Figs. 2 and 3. Crucially, the space \({\mathcal {M}}^{\text {Spin}(7)}_{\text {inv}}\left( X^{8}\right) \) is compact.
This suggests a potential relationship between Spin(7) instantons and HYM connections. Indeed, they are not equivalent in general; but furthermore, the structure of \({\mathcal {M}}^{\text {Spin}(7)}_{\text {inv}}\left( X^{8}\right) \) hints that the latter might play a role in the compactification of Spin(7) instanton moduli spaces (over noncompact Calabi–Yau fourfolds). It may be a general phenomenon that certain sequences of Spin(7) instantons fail to converge because the limit lives on a different bundle. Furthermore this new bundle should agree with the original one outside of a (codimension 4) Cayley submanifold. The missing limit should be HYM.
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Acknowledgements
The author would like to thank his PhD advisors Andrew Dancer and Jason Lotay for suggesting the problem discussed in this article and providing numerous helpful comments and recommendations. He would also like to thank Gonçalo Oliveira for an elucidating discussion regarding the issue of gauging in the equivariant setting. Finally, he would like to thank the anonymous referees whose suggestions improved this article considerably. This work was supported by the EPSRC Centre for Doctoral Training in Partial Differential Equations: Analysis and Applications (Grant Number EP/L015811/1). The author is supported by the Onassis Foundation  Scholarship ID: F ZQ 0501/20202021.
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Papoulias, V.E. Spin(7) Instantons and Hermitian Yang–Mills Connections for the Stenzel Metric. Commun. Math. Phys. 384, 2009–2066 (2021). https://doi.org/10.1007/s00220021040555
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DOI: https://doi.org/10.1007/s00220021040555