$\mathrm{G}_2$-instantons on the 7-sphere

We study the deformation theory of $\mathrm{G}_2$-instantons on the 7-sphere, specifically those obtained from instantons on the 4-sphere via the quaternionic Hopf fibration. We find that the pullback of the standard ASD instanton lies in a smooth, complete, 15-dimensional family of $\mathrm{G}_2$-instantons. In general, the space of infinitesimal $\mathrm{G}_2$-instanton deformations on $S^7$ is identified with three copies of the space of ASD deformations on $S^4.$

1. Introduction 1.1. Background. Let M be an oriented 7-manifold with a G 2 -structure, defined by a positive 3-form φ. Endow M with the corresponding Riemannian metric g = g φ and Hodge star operator * = * g (see §3.1 below).
A connection A on a vector bundle E → M is said to be a G 2 -instanton if its curvature This equation appeared in the physics literature during the early 1980s [12]. In the mid-1990s, not long after Joyce's construction [24] of compact Riemannian manifolds with Hol(g) = G 2 , Donaldson and Thomas [16] proposed to define an invariant of (torsion-free) G 2 -structures by "counting" G 2 -instantons in a manner analogous to the Casson invariant in dimension three.
A fruitful alternative is to consider metrics with less-than-full holonomy or G 2 -structures with certain types of torsion. The instantons that appear in these contexts can demonstrate geometric phenomena that we ultimately hope to observe in the torsion-free case.
Here ψ = * g φ φ is the dual 4-form and τ 0 is a constant. Nearly parallel G 2 -structures have much in common with the torsion-free case: their associated metrics are Einstein and they enjoy a natural spinorial description [18]. More importantly for this work, instantons on nearly parallel G 2 -manifolds are critical points of the Yang-Mills functional (see [21] or (3.12) below). This stands in contrast with general co-closed G 2 -structures. Most famously, the 7-sphere carries a "standard" G 2 -structure induced by the octonionic structure of R 8 ; for elementary reasons, this turns out to be nearly parallel. (In fact, S 7 also carries a second, "squashed," nearly parallel G 2 -structure-see Example 3.3 below and Appendix A-but we shall be concerned primarily with the standard structure.) There are three main reasons why the standard S 7 is an especially appealing arena for gauge theory.
First, determining the complete moduli space of instantons on S 7 presents a clear challenge for higher-dimensional gauge theory. This problem is at least as difficult as the corresponding problem on S 4 , solved by the spectacular theorem of Atiyah-Drinfeld-Hitchin-Manin [2]. At present, we are limited to studying the deformations of a given G 2 -instanton on S 7 , in the spirit of Atiyah-Hitchin-Singer's classic work on deformations of ASD instantons on S 4 [4].
The second motivation comes from another well-known difficulty in higher-dimensional gauge theory: the appearance of instantons with essential singularities. Given a Spin(7)instanton with an isolated singularity on a Spin(7)-manifold, the cross-section of the tangent cone is a G 2 -instanton on the standard S 7 . Hence, these are the "building blocks" for the simplest non-removable singularities in dimension eight.
The third motivation is the relationship with gauge theory in fewer than seven dimensions. Since G 2 -holonomy metrics typically collapse at the boundary of the moduli space, it is important to understand instantons on model G 2 -manifolds coming from lower-dimensional geometries. Recently, Y. Wang [36] has shown that any G 2 -instanton on S 1 × X, for X a Calabi-Yau 3-fold, is equivalent by a broken gauge transformation to the pullback of a Hermitian-Yang-Mills connection. This is the first case where the full moduli space of G 2instantons on a given 7-manifold with holonomy contained in G 2 has been identified.
In the present case, a convenient link with 4-dimensional gauge theory is provided by the quaternionic Hopf fibration With the proper conventions, the pullback of any anti-self-dual (ASD) instanton on S 4 is a G 2 -instanton on S 7 . This provides an abundant source of examples. The resulting families of G 2 -instantons can be further enlarged by the action of the automorphism group Spin (7), which interacts with the Hopf fibration in a nontrivial way. Our main results concern the dimension and completeness properties of these families.
1.2. Summary. In §2-3, we set our conventions and derive the basic results concerning instantons on nearly parallel G 2 -manifolds, while reviewing the literature in this area. In §4, we take a concrete approach to the special case of the standard instanton. In quaternionic notation on R 8 ≅ H 2 , the pullback by the (right) Hopf fibration of the standard ASD instanton may be given by The restriction of A 0 to S 7 defines a smooth connection, dubbed the standard G 2 -instanton. Proposition 4.4 describes a 15-dimensional space of infinitesimal deformations of A 0 , generated by the deformations coming from S 4 together with the Spin(7) rotations of S 7 , modulo gauge. This will turn out to be the full space of deformations of A 0 as a G 2 -instanton.
In §5, we prove our main result, Theorem 5.11, which determines the space of infinitesimal deformations of the pullback to S 7 of an arbitrary irreducible ASD instanton on S 4 . The argument is elementary, but far from trivial. The difficulty is caused by two factors: first, according to the theorem of Bourguignon-Lawson-Simons [7,8], the Yang-Mills stability operator necessarily has negative eigenvalues. Second, the Sp(1)-action giving the quaternionic Hopf fibration does not commute with the deformation operator (see Remark 2.1 below).
To obtain the result, we first prove a vanishing theorem for the vertical component of an infinitesimal deformation in Coulomb gauge, Theorem 5.8. This follows from a delicate analysis of the Weitzenbock formula for the stability operator, in which the first-order deformation equation is used crucially. Having established that the vertical component vanishes, we use another squaring trick, Lemma 5.9, to calculate the horizontal component of an infinitesimal deformation. This leads directly to Theorem 5.11, which equates the full space of infinitesimal deformations of the pullback, as a G 2 -instanton, with three copies of the ASD deformations on S 4 .
In §6, we briefly discuss the global structure of these families of G 2 -instantons. In the case of charge κ = 1 and structure group SU(2), which includes the standard instanton, we have the following result.
Theorem 1.1. The connected component of A 0 in the moduli space of G 2 -instantons on S 7 is diffeomorphic to the tautological 5-plane bundle over the oriented real Grassmannian G or (5,7).
In this description, the base space corresponds to the orbit of the Hopf fibration (1.3) under conjugation by Spin (7), and the fiber corresponds to the pullback of the unit-charge ASD moduli space. The total space is 15-dimensional, agreeing with the dimension formula of Theorem 5.11. By contrast, in the case of higher charge, we do not expect all of the infinitesimal deformations identified by Theorem 5.11 to be integrable (see §6.3 below).
Lastly, we state Conjecture 6.3, due to Donaldson, which asserts that every G 2 -instanton on S 7 having integral Chern-Simons value should arise from the pullback construction.
The 3-dimensional space of imaginary quaternions Im H, with commutator bracket, is isomorphic to the Lie algebra su(2).
We shall identify H with R 4 as follows: Denote by H l and H r the commuting subalgebras of End R 4 corresponding to left-and rightmultiplication by H, respectively, and let Sp(1) l and Sp(1) r be the corresponding subgroups of SO(4) generated by unit quaternions. With this convention, we have where Λ 2± ⊂ Λ 2 R 4 denotes the space of (anti-)self-dual 2-forms with respect to the Euclidean metric. Choose the following standard basis for the self-dual 2-forms on R 4 ∶ Here we abbreviate dx 01 = dx 0 ∧ dx 1 , etc. For i = 1, 2, 3, define the complex structure I i on R 4 by Under the identification (2.1), I i corresponds to right-multiplication by the element e i . We define a Fueter map L ∶ R 4 → R 4 to be an endomorphism satisfying The 12-dimensional subspace of Fueter maps F ⊂ End R 4 is the direct sum: In particular, F contains the space of linear maps for the standard complex structure, I 1 . (7) and its subgroups. The standard 4-form on

Spin
The group Spin (7) consists of all linear transformations of R 8 that preserve Ψ 0 under pullback. It is a simply-connected, simple, Lie subgroup of SO(8) of dimension 21 (see e.g. Walpuski and Salamon [34, §9]).
Remark 2.1. The fact that Sp(2)Sp(1) is not a subgroup of Spin(7) causes an essential difficulty. With our convention, Sp(2) is a subroup, but the block-diagonal Sp(1), giving the Hopf fibration, is not. On the other hand, with the convention used for instance by Walpuski [33], Sp(1) is a subgroup, but the commuting Sp(2) is not. Our convention is necessary for Lemma 4.1 below.

Instantons on nearly parallel G 2 -manifolds
In this section, we establish the basic facts about instantons on nearly parallel G 2manifolds. Most of the results are known to researchers informally or by analogy with the nearly Kähler case (see Xu [37]), but some (in particular Proposition 3.8) have not appeared in their present form. For the spinorial formulation, see Harland-Nölle [21].
3.1. Nearly parallel G 2 -structures. Let M be an oriented 7-manifold. Recall that a G 2 -structure on M is defined by a global 3-form φ that is positive, in the sense that for all x ∈ M and v ≠ 0 ∈ T x M. Any such φ is pointwise equivalent to the model 3-form φ 0 , given by (2.13) above (see [34,Theorem 3.2]). A positive 3-form defines a unique Riemannian metric g φ on M by the requirement We also associate to φ the dual 4-form Recall that a G 2 -structure is said to be closed if dφ = 0, and coclosed if dψ = 0. We are concerned with G 2 -structures satisfying (1.2), where we assume The basic reference for nearly parallel structures is Friedrich-Kath-Moroianu-Semmelmann [18]. With the normalization (3.4), nearly parallel G 2 -manifolds are Einstein, with There are three further equivalent formulations of the nearly parallel condition (1.2). The first is that the induced Spin (7)-structure on the cone over M be torsion-free. The second (and most frequently used) condition is that M possess a nonzero Killing spinor. The third is as follows: Here ∇ is the Levi-Civita connection associated to the metric g φ defined by (3.2).
The group of global automorphisms of φ std is Spin (7).
To obtain a more explicit expression for φ std , we define the 3-form Define ν y and ζ y i similarly. We then have It is easy to check, using the Spin(7)-invariance, that φ std defines the round metric on S 7 .
Notice that dν x = 4Vol R 4 x and dζ x i = 2ω x i , and similarly for y, so Also note that It follows from (3.9-3.10) that φ std is a nearly parallel G 2 -structure.
The squashed G 2 -structure was discovered by Awada, Duff, and Pope [5], and has automorphism group Sp(2)Sp(1). Appendix A includes a proof that φ sq is nearly parallel.
Remark 3.4. Alexandrov and Semmelmann [1] have shown that both the standard and the squashed G 2 -structures are rigid among nearly parallel G 2 -structures. These remain the only known nearly parallel structures on the 7-sphere. We also note that the (non-)existence of a closed G 2 -structure on the 7-sphere is a well-known open problem.

G 2 -instantons.
Recall that a connection A is called a G 2 -instanton if its curvature satisfies (1.1), or equivalently If the G 2 -structure φ is nearly parallel, then from (1.1) and (1.2), we have (3.12) We have used (3.11) and the Bianchi identity in the last line. Hence, in the nearly-parallel case, any G 2 -instanton is Yang-Mills. This observation goes back to Harland and Nölle [21]. The linearization of (3.11) is The infinitesimal deformations of a G 2 -instanton A, modulo gauge, therefore correspond to the first cohomology group of the following self-dual elliptic complex: Folding (3.13) and writing d = D A , we obtain the deformation operator This is a first-order, self-adjoint, elliptic operator.
Over a compact manifold, integration by parts implies ker L A = ker L 2 A . Hence, du ≡ 0 for any infinitesimal deformation on a compact nearly parallel G 2 -manifold, and u ≡ 0 if A is irreducible.
We shall use the following interior product notation: We also take interior products between differential forms, e.g.
and similarly in general using the metric. In particular, for a 2-form b, we have For a g E -valued 1-form α, we shall write Then (3.15) becomes Lemma 3.6 ([10, (2.7-2.8)]). The following identities hold between any positive 3-form φ, the associated metric g = g φ , and the dual 4 Proof. Since these are zeroth-order identities, it suffices to check them for the standard 3and 4-form, given by (2.13-2.14), and the standard metric. This is easily accomplished using the fact that G 2 acts transitively on orthonormal pairs of vectors.
where we have used Lemma 3.1 in the last line. By Lemma 3.6, this becomes which agrees with the expression (3.18).
Proposition 3.8. For a G 2 -instanton with respect to a nearly parallel G 2 -structure φ, we have is the Yang-Mills stability operator (see ). In the case of the round 7-sphere, we have Proof. From (3.17) and Lemma 3.7, we have But since A is an instanton, we have Substituting into (3.21) and applying the Bochner formula yields (3.19). Then (3.20) is obtained by substituting τ 0 = 4 and Ric g = 6g on the round 7-sphere.
Remark 3.9. Ball and Oliveira [6] have studied instantons on the Aloff-Wallach spaces, which are nearly parallel. Singhal [30] also studies instantons on homogeneous nearly parallel G 2 -manifolds, using spinorial methods similar to those of Charbonneau and Harland [11] in the context of nearly Kähler manifolds.

Hopf fibration and standard instantons
In this section, we give an explicit description of the standard (A)SD instanton and its G 2 relative. We shall use a variant of Atiyah's quaternionic notation [3] based on the convention (2.1): Here x andx are H-valued functions, and dx and dx are H-valued differential forms on R 4 . We define y andȳ similarly, and will identify 4.1. Quaternionic Hopf fibration. The (right) Hopf fibration is given by the quotient projection under right-multiplication by H × ∶ The fibration (1.3) is obtained by restricting (4.1) to S 7 , giving the quotient projection under right-multiplication by Sp (1) To see the identification HP 1 ≅ S 4 explicitly, observe that Sp(2) acts on S 7 by isometries commuting with Sp(1) r . The stabilizer of an S 3 fiber of (1.3) is the subgroup Sp(1) l ×Sp(1) l ⊂ Sp(2). Meanwhile, Sp(2) acts by conjugation on the 5-dimensional space of 2 × 2 traceless self-adjoint quaternionic matrices: where the stabilizer of an axis is again Sp(1) l × Sp(1) l . We therefore have a map which is an isometry, up to a factor of 1 2.
The basic link between instantons on S 7 and S 4 is as follows; a more general statement appears in Proposition 6.2 below. Proof. By Sp(2)-invariance both of φ std and of the fibration, it suffices to consider the point (1, 0). From (3.8), we have The orthogonal complement of the fiber through (1, 0) is R 4 y , which is mapped conformally onto the tangent space of S 4 at p = π(1, 0). Hence, F A (1, 0) = π * F B (p) is equal to a 2-form on R 4 y . It is clear from the expression (4.4) that φ ¬ F A (1, 0) vanishes if and only if this 2-form is ASD, which is equivalent to the same statement for F B (p).
The same argument applies on φ sq .

Standard (A)SD instanton.
Let P + denote the principal Sp(1)-bundle associated to the right Hopf fibration. We define the standard self-dual instanton to be the connection on P + induced by the round metric on the total space. The corresponding connection form on P + is the su(2)-valued 1-form Im [x dx +ȳ dy] x 2 + y 2 .
Pulling back to R 4 by the map x ↦ (x, 1), we obtain the well-known connection matrix See Atiyah [3, §1] for these formulae, as well as a generalization giving the complete ADHM construction.
Similarly, we define the standard anti-self-dual (ASD) instanton P − to be the principal bundle associated to the left Hopf fibration, with connection form (4.5) Im [w dw + z dz] w 2 + z 2 .
In the stereographic chart on R 4 , this has a connection matrix and curvature the Im H-valued self-dual 2-form be the fiber product over S 4 of P − with P + , considered as a principal Sp(1)-bundle via the projection to P + = S 7 . According to §4.1 and Lemma 4.1, the pullback of the standard ASD instanton on P − is a G 2 -instanton on P → S 7 , which we call the standard G 2 -instanton, A 0 .
We may obtain a connection matrix for A 0 by pulling back the connection form of the standard ASD instanton (4.5) by the fiber-preserving map H ×2 r → H ×2 l given by

This gives
which is (1.4) above. The singularity along the x-axis can be removed by applying the gauge transformation g(x, y) = y y ∶ noting that dgg −1 = − Im ydȳ y 2 , we obtain The singularity along the y-axis can be removed similarly. However, they cannot be removed simultaneously, for according to the following proposition, the bundle P → S 7 is nontrivial. Recall that by the clutching construction, topological SU(2)-bundles on S 7 are classified by π 6 (S 3 ) = Z 12 . 4.4. Curvature calculation. We check directly that A 0 is a G 2 -instanton on S 7 . By (3.6), this is equivalent to showing that (1.4) is a Spin (7)-instanton on R 8 . We calculate This gives On the other hand, we have (4.10) Adding (4.9) and (4.10), we obtain the curvature form (4.11) The 2-forms dx ∧ dx, dy ∧ dȳ, and dx ∧ dȳ are each invariant under Sp(1) r . Therefore, the 2-form part of F A 0 lies in Lie(Sp(2)) ⊂ Lie(Spin (7)), as claimed.

Linear deformations. Let
A be a conical instanton on R 8 ∖ {0} whose curvature F A takes values in Lie(Sp(2)) ⊗ g E . Given any 8 × 8 matrix M, we associate the vector field on R 8 , as well as the g E -valued 1-form Notice that (4.12) corresponds to pushforward by the diffeomorphism generated by X M , with its horizontal lift to the bundle. For, working in a local gauge where A X M = 0, we have The action on the curvature is given by Proof. Let α = α M and F = F A . In coordinates on R 8 , we have Since the curvature takes values in Lie(Sp (2)) and M belongs to the orthogonal complement, the expression (4.15) vanishes identically. The result then follows from the formula A * α − ⟨r, ∇r (α (r))⟩ and the fact that α(r) ≡ 0 for a conical instanton.
Proof. According to (4.13), the subspace {α M M ∈ W } corresponds to pushforward by elements of SL(2, H), which preserve the algebra Lie(Sp(2)); hence, these correspond to infinitesimal deformations. By Lemma 4.3, the space (4.16) is in Coulomb gauge, so the first factor lies in the kernel of L A . The second factor may be described as follows: The matrix I 1 0 0 −I 1 is just the first element in (2.8), which belongs to the subspace Λ 2 d ⊂ Lie(Spin (7)). Since I 1 commutes with H l , we also have 0zI 1 which belongs to the subspace F x,y ⊂ Spin (7) given by (2.9). Hence, the second factor corresponds to pushforward by elements of Spin (7), which are in Coulomb gauge by Lemma 4.3. The same is true of the third factor. Therefore the space V A is contained in the kernel of the deformation operator L A .
For the second statement, if the curvature F A (x, y) spans Lie(Sp(2)), then for any nonzero M, the element α M must be nonzero. In particular, the map M ↦ α M has vanishing kernel, hence rank 15.
Remark 4.5. The master's thesis of Jurke [25] studies Spin(7)-instantons using a quaternionic approach similar to that of this section. The author thanks an anonymous referee for pointing out this reference.

Infinitesimal deformations
In this section, we calculate the space of infinitesimal deformations, as a G 2 -instanton, of the pullback to S 7 of a general irreducible ASD instanton on S 4 . We write for the Levi-Civita connection on S 7 std , which we shall couple to the connection on any auxiliary bundle. We shall also write Here, π v is the orthogonal projection to the the vertical tangent space of the Hopf fibration and π h is the complementary projection, with respect to the round metric.

Vertical and horizontal components.
Let Ω 1 h be the annihilator of vertical vector fields along the Hopf fibration, and Ω 1 v its orthogonal complement. We have Letting An element of Ω (p,q) will be referred to as a (p, q)-form. Let ν denote the (3, 0) volume form of the Hopf fibration, and let ν = * ν.
The (0, 2)-forms split as where Ω 2± h are the (anti-)self-dual components with respect to the (0, 4) volume formν. For a (0, 1)-form b, we shall write d v b for the (1, 1) part of db and d h b for the (0, 2) part. A similar notation will be used for (1, 0)-forms (see Lemma 5.3 below).
Notice that {ζ i } and {ω i } are global frames for Ω 1 v and Ω 2+ h , respectively. The vertical volume form is given by ν = ζ 1 ∧ ζ 2 ∧ ζ 3 . From (3.8) and the Sp(2)-invariance, we may re-express φ std as follows: We now derive the basic properties of these frames, and use them to decompose the Laplace operator into horizontal and vertical parts.
Proof. From the Definition 5.1 and (3.7), for X, Y ∈ T S 7 , we have

Coclosedness of ζ i and (5.3) follow directly from (5.5).
Since ζ i is coclosed and equal to the restriction of a linear form, we have as can be verified directly from (5.5). Then (5.4) follows from (5.6) and the fact that Proof. This is a standard decomposition result, which can be seen directly as follows. By Lemma 5.2, we have The result follows directly from (5.7).

Lemma 5.4.
If a = f i ζ i is (co)closed on each S 3 fiber, then (∇ * ∇ h f i ) ζ i is again fiberwise (co)closed.
Proof. Let U j be the dual vector field of ζ j , for j = 1, 2, 3. The Killing vector fields U j commute with the operators ∇ * ∇ and ∇ * ∇ v , hence also with we conclude that ∇ * ∇ h preserves (co)closedness on the fibers.
Proposition 5.5. Let α = a + b be a 1-form as above, where a = f i ζ i and b ∈ Ω 1 h . The vertical component of the Laplacian on S 7 is given by Here ∇ * ∇ v denotes the Laplacian on the S 3 fiber.
Proof. We have For the first term, we write where we have used Lemma 5.2. Then (5.9) yields which gives the case b = 0 of the formula (5.8).
Assuming now that Y is horizontal and b is of type (0, 1), we compute For U vertical, we have Hence, for X ∈ T S 7 , we have Next, let {e j } 7 j=1 be an orthonormal basis of vector fields that satisfies ∇ e j e k = 0 at a given point. We compute 0 ≡ ∇ e j ∇ e j ⟨ζ i , b⟩ −) , ∇ e j b⟩ which may be rewritten as Combining (5.10) and (5.11) yields (5.8).
Remark 5.6. The previous results carry over when ∇ is coupled to a connection that is trivial along the fibers of the Hopf fibration.

Vanishing of the vertical component. This subsection proves our vanishing theorem for the vertical component of an infinitesimal deformation in Coulomb gauge.
Lemma 5.7. Let B be an ASD instanton on S 4 , and put A = π * B. Assume that f is a section of π * g E → S 7 satisfying Proof. We have since f is a 0-form and the curvature F A is purely horizontal. By (5.12), we have Applying D A to both sides, by Proposition 5.3, we obtain (5.14) and the fact that (D v A ) 2 = 0 for a pullback connection, the first two terms vanish, yielding (5.15) [ Each side of (5.15) is of type (0, 2); however, the LHS is anti-self-dual and the RHS is self-dual. Therefore both sides of (5.15) must vanish, giving Hence, f is constant on the fibers, so f = π * h for a section h of g E . Then (5.12) implies (5.13), as desired.
Theorem 5.8. Let B be an irreducible ASD instanton on a bundle E → S 4 , and A = π * B its pullback under the Hopf fibration. If α ∈ Ω 1 (π * g E ) satisfies L A α = 0, then the vertical part of α vanishes.
Proof. We write ∇ = ∇ A and d = D A throughout the proof. Decompose α = a + b into vertical and horizontal parts as above. Let a = f i ζ i , and write the self-dual part of d h b as From Proposition 5.3 and (5.2), our assumption L A α = 0 implies Noting thatω i ¬ ¯ω i = 2, according to(5.2), the (1, 0)-part of (5.16) comes out to where U j is the dual vector field to ζ j , for j = 1, 2, 3, as above. We conclude that We rewrite this as Now, the decomposition of the Laplacian (5.8) reads where we have used (5.18). Returning to (3.20), we have Taking the vertical part and inserting (5.20), since F A is purely horizontal, we obtain Applying the Bochner formula on S 3 yields We now split a into fiberwise closed and coclosed parts a = a cl + a cocl and write a cocl = g i ζ i . Taking an inner product with a cocl in (5.22), and integrating over S 7 , yields Here we have used the orthogonality between fiberwise closed and coclosed vertical 1-forms, the fact that d * a cocl = (d v ) * a cocl = 0, and Lemma 5.4. Note that the first eigenvalue of the Hodge Laplacian on coclosed 1-forms on S 3 is 4, so Inserting this into (5.23), using Cauchy-Schwarz and (5.19), we obtain This yields ∇ h g i ≡ 0 for i = 1, 2, 3. Since B is assumed irreducible, Lemma 5.7 implies that g i ≡ 0, and hence a cocl ≡ 0. But then a = a cl is fiberwise closed, so µ(a) = 0 by (5.19). Since the RHS of (5.21) is then a positive operator, we conclude that a ≡ 0, as desired.

Space of infinitesimal deformations.
This subsection proves our main theorem, which calculates the deformations of a pulled-back instanton on S 7 in terms of the ASD deformations on S 4 .
Consider the operator Let Notice that We may therefore define a local frame for Ω 1 h near S 3 0 bȳ e j = dy j − ζ i ω ijk y k , j = 0, . . . , 3.
We calculate Then Lemma 5.9. The kernel of δ 0 consists of sections of the form where L ∈ F is a Fueter map of R 4 , according to (2.4) above.
Proof. Let α = α jē j be an arbitrary section of Ω 1 h near S 3 0 . According to (5.24) and (4.3), we have Further, we calculate Again applying (5.25), we obtain Hence the first term on the RHS of (5.26) is the Laplace-Beltrami operator on S 3 0 , applied component-wise to α. We conclude that Hence, α j lies in the first nonzero eigenspace, and is the restriction of a linear function on Substituting back into (5.25), we have which recovers (2.4), as desired.
Proof. Let α ∈ ker L A . According to Theorem 5.8, the vertical component of α vanishes, so it is a global section of Ω 1 h (g E ). The image δ(α) ∈ Ω 1 h under the horizontal component of the deformation operator must therefore vanish; by Proposition 5.10, we have α = α 0 + α 1 + α 2 according to (5.27).

Global picture
In this section, we discuss the global structure of the components of the moduli space obtained by pullback under the Hopf fibration. We first prove Theorem 1.1 on the structure of the κ = 1 component. For higher charge, the picture necessarily involves Hermitian-Yang-Mills connections on the twistor space CP 3 → S 4 .
6.1. Proof of Theorem 1.1. Let W be given by (4.2). Taking A 0 in the gauge (1.4), we may define the smooth 5-dimensional family of connections By the construction of §4.1, this family is equal to the pullback by the Hopf fibration of the 5-dimension family of unit-charge ASD instantons on S 4 . Further define a smooth map By construction, the image of (6.1) consists of G 2 -instantons.
The stabilizer of V is Sp(2) × U(1) ⊂ Spin (7), which acts on V, modulo gauge, by the 5-dimensional representation of Sp(2) and the trivial representation of U(1). Taking the quotient by the gauge group G E , the map (6.1) descends to a smooth map from the associated bundle to the space of connections modulo gauge: Notice that Spin(7) Sp(2) × U(1) ≅ SO(7) SO(5) × SO(2) = G or (5, 7).
Hence, X is equal to the vector bundle over G or (5, 7) associated to the standard representation of SO(5), i.e., the tautological 5-plane bundle. This is a 15-dimensional manifold. We claim that the above map Φ is a proper embedding. For each A ∈ V, the image of the differential of Φ is equal (modulo gauge) to the space of linear deformations V A given by (4.16). By examining (4.11), it is easy to see that F A 0 (x, y) spans Lie(Sp(2)) as (x, y) varies over S 7 ; the same is true of each element of the 5-dimensional family V. Hence, according to Proposition 4.4, dim(V A ) = 15, and the differential of Φ has full rank at each point of V. Since the construction is invariant under the action of Spin (7), the same is true at each point of X. Hence, the map Φ is an embedding.
Next, note that as x ∈ X tends to infinity in a fiber V, the curvature of Φ(x) concentrates along an associative great sphere (the preimage under the Hopf fibration of a point in S 4 ). Hence, Φ(x) tends to infinity in A E G E . Since the base space G or (5, 7) is compact, this shows that Φ is a proper map. Now, since for each x ∈ X, Φ(x) is equivalent modulo Spin (7) to a pullback from S 4 , Theorem 5.11 implies that the space of infinitesimal deformations at Φ(x) has dimension 15. Since dim(X) = 15 and Φ is a proper embedding, we conclude that Φ is in fact a diffeomorphism onto a connected component of the G 2 -instanton moduli space.
Remark 6.1. By the same construction, we may compactify X fiberwise to aB 5 -bundle, where a boundary point records bubbling along an associative great sphere.
6.2. Chern-Simons functional and Hermitian-Yang-Mills connections. Given two connections A and A 1 on a bundle E, let a = A − A 1 and define the relative Chern-Simons 3-form: This satisfies On a 7-manifold M with G 2 -structure, we may define the global relative Chern-Simons functional This normalization is chosen so that the formula in the following proposition will be integervalued. Let (6.4) Π ∶ S 7 std → CP 3 be the natural projection for the standard complex structure, I 1 .
std if and only if B is Hermitian-Yang-Mills. Given any two such connections (E, B) and Proof. By SU(4)-invariance, it suffices to consider the point p = (1, 0, . . . , 0) ∈ S 7 . We shall write Then, from the expression (2.12), we have where ω is the standard Kähler form on C 3 , given by (2.11). Letting Therefore, A is a G 2 -instanton if and only if ω.F = 0, F 2,0 = 0 which is to say, B is Hermitian-Yang-Mills on CP 3 . Next, we have Assume that both A and A 1 are pullbacks of Hermitian-Yang-Mills connections on CP 3 . Computing at p as above, we have F = F 1,1 A and are left with Since the Fubini-Study form on CP 3 satisfies Π * ω F S = ω π on S 7 , the expression (6.9) agrees globally with Integrating (6.8) over the fibers of Π, we obtain as claimed.
6.3. Discussion. Theorem 5.11 can be explained in light of Proposition 6.2, as follows.
Owing to the Ward correspondence, the space of infinitesimal deformations of the pullback of an ASD instanton to the twistor space CP 3 → S 4 , as a Hermitian-Yang-Mills connection, is the complexification of the space of ASD deformations. In fact, there is an S 1 family of complex structures cos(θ)I 1 + sin(θ)I 2 such that the quotient map Π θ satisfies Proposition 6.2 and factorizes the given Hopf fibration: The span of the pullbacks by Π θ of the deformation space over CP 3 , for θ ∈ S 1 , gives the larger space (5.28). According to this argument, the fibration to CP 3 accounts for all of the infinitesimal deformations identified by Theorem 5.11. As such, we expect that only the deformations coming from (6.4), or its 6-dimensional family of Spin (7) conjugates, are integrable. So for κ > 1, the generic dimension of the pulled-back component of the G 2 -instanton moduli space should be 1 (6.10) 2 (8κ − 3) + 6 = 16κ.
These components appear to be singular along the subset of instantons coming from S 4 (i.e., instanton bundles on CP 3 satisfying a reality condition), which are themselves manifolds of dimension 8κ − 3 + 10 = 8κ + 7.
More generally, we would like to know where the instantons obtained via pullback fit within the full moduli space of G 2 -instantons on S 7 . While examples that are unrelated to any fibration may exist, the following guess is appropriate based on [36] and the present work. For the statement, fix a reference Hermitian-Yang-Mills connection B 1 on an SU(n)-bundle E 1 → CP 3 , and let E = Π * E 1 → S 7 and A 1 = Π * B 1 . Conjecture 6.3 (Donaldson). Let A be a G 2 -instanton on E → S 7 std , for which Then A is equivalent, modulo the action of Spin (7) and G E , to the pullback of a Hermitian-Yang-Mills connection on CP 3 .
1 The κ = 1 instantons all fail to be "generic" due to the extra U(1) stabilizer generated by I 3 0 0 I 3 .
As a check on our conventions, we include the following short proof that φ sq is a nearly parallel G 2 -structure.
One can check from (3.2) that the metric and volume form defined by φ a,b are g a,b = a −1 3 a g std Tv + b g std T h , V ol a,b = a 1 3 b 2 V ol std .
In this notation, we have The associated metric and volume form are g sq = 9 5 1 5 g std Tv + g std T h , V ol sq = 3 7 5 5 V ol std . To see that φ sq is nearly parallel, we calculate as follows. According to (5.7), we have dζ i = ǫ ijk ζ j ∧ ζ k + 2ω i .
Remark A.1. Notice that although φ 1,1 and φ std both correspond to the standard metric on S 7 , they are not isomorphic, since the former is not a nearly parallel G 2 -structure. In fact, Friedrich [19] has shown that φ std is the unique nearly parallel G 2 -structure (up to rotations) compatible with the round metric on S 7 .