Deformation Theory of Asymptotically Conical Spin(7)-Instantons

We develop the deformation theory of instantons on asymptotically conical $Spin(7)$-manifolds where the instanton is asymptotic to a fixed nearly $G_2$-instanton at infinity. By relating the deformation complex with spinors, we identify the space of infinitesimal deformations with the kernel of the twisted negative Dirac operator on the asymptotically conical $Spin(7)$-manifold. Finally we apply this theory to describe the deformations of Fairlie-Nuyts-Fubini-Nicolai (FNFN) $Spin(7)$-instantons on $\mathbb{R}^8$, where $\mathbb{R}^8$ is considered to be an asymptotically conical $Spin(7)$-manifold asymptotic to the cone over $S^7$. We calculate the virtual dimension of the moduli space using Atiyah-Patodi-Singer index theorem and the spectrum of the twisted Dirac operator.


Introduction
Instantons on 4-manifolds are connections whose curvatures are anti-self-dual. Instantons solve the Yang-Mills equation and hence have always been of interest to physicists in the contexts of quantum field theory, string theory, M-theory, supergravity etc. Instantons in dimensions higher than 4 were also studied by many physicists before Donaldson-Thomas [17] and Donaldson-Segal [19] explained their importance and scope to mathematical audience. Analogous to the 4-dimensional case, their prediction of the possibility to construct invariants from the moduli space has been one of the main sources of motivation behind the research on higher dimensional gauge theory for mathematicians.
In this paper, we develop the deformation theory of instantons on a particular type of noncompact Spin(7)-manifolds known as asymptotically conical Spin(7)-manifolds. These manifolds are complete Spin(7)-manifolds asymptotic to the cone over compact nearly G 2 -manifolds. The instantons on these manifolds also exhibit the asymptotically conical behaviour. Assuming that the instanton is unobstructed, we prove that the moduli space of these instantons is a manifold, and describe a way to calculate the (virtual) dimension. In the second part of the paper, we apply the deformation theory on certain instantons on R 8 , first constructed by Fairlie-Nuyts [23] and Fubini-Nicolai [25] independently, in the context of supergravity. R 8 is indeed an asymptotically conical Spin(7)-manifold, and hence it is appropriate to study the deformations of these instantons using our theory. The main result for this part is the calculation of the virtual dimension of the moduli space of these instantons.
The study of asymptotically conical Spin(7)-manifolds goes back to 1989, when Bryant-Salamon [12] gave an example of a complete non-compact Spin(7)-manifold, namely, the negative spinor bundle over the 4-sphere. In 2014, Clarke [15] constructed a Spin(7)-instanton on this Bryant-Salamon Spin (7) Asymptotically conical manifolds have been studied by many authors, e.g., asymptotically conical G 2 manifolds by Karigiannis-Lotay [32] and recently, asymptotically conical Spin(7) manifolds were studied by Lehmann [37]. The analytic frameworks for studying asymptotically conical manifolds, namely, the weighted Sobolev theory and theory of asymptotically conical Fredholm and elliptic operators, have been developed by  and Marshall [42].
Here is a brief outline of this paper.
After we discuss the basic notations and definitions, and fix conventions related to asymptotically conical Spin(7)-instantons in Section 2, we develop the deformation theory of asymptotically conical Spin(7)-instantons in Section 3. In the first part, we discuss the analytical framework to study instantons of asymptotically conical Spin(7)-manifolds. We use Lockhart-McOwen theory, and the relation between the Dirac operator on the cone and the Dirac operator on the link to show that the Dirac operator on the asymptotically conical manifold is Fredholm only when the rate of decay is not a critical weight, and the critical weights are precisely the rates that differ from the eigenvalues of the Dirac operator on the link by a fixed constant.
In the second part of Section 3, using the analytical framework and implicit function theorem, we prove that if the rate of decay is not a critical weight, the moduli space of asymptotically conical Spin(7)-instantons is a smooth manifold, given that the deformations are unobstructed; moreover the dimension of the moduli space is precisely the index of the Dirac operator on the asymptotically conical manifold.
In Sections 4 and 5 we carry out an in-depth study of Fairlie-Nuyts-Fubini-Nicolai (FNFN) Spin(7)-instanton on R 8 and its deformation theory. We apply the deformation theory developed in Section 3 by considering R 8 to be the asymptotically conical Spin(7)-manifold asymptotic to the nearly G 2 -manifold S 7 .
In order to study the moduli space, we need to identify the critical weights, and hence need to calculate the eigenvalues of the Dirac operator on the link S 7 in a certain range determined by the fastest rate of convergence of FNFN-instanton. In Section 4 we use various techniques in representation theory and harmonic analysis, namely, the Frobenius reciprocity to decompose the space of L 2 -sections of the spinor bundle into direct sums of finite dimensional Hilbert spaces indexed by Spin(7)-representations. Moreover, we express the Dirac operator as a sum of Casimir operators. We also calculate an eigenvalue bound which yields only six representations of Spin (7) for which the eigenvalues of the Dirac operator could be in the prescribed range. Then we explicitly calculate the eigenvalues of the Dirac operator for these representations, and identify the critical rates.
In the first part of Section 5, we reconstruct the FNFN Spin(7)-instanton using algebraic techniques, by identifying S 7 with the homogeneous space Spin(7)/G 2 . In the second part we use Atiyah-Patodi-Singer theorem and the critical rates calculated in Section 4 to calculate the virtual dimension of the moduli space of the FNFN instanton. It turns out that the virtual dimensions of the moduli space are determined by precisely two known deformations of FNFN-instanton, namely dilation and translations. Acknowledgement I would like to express my heartfelt thanks to PhD supervisor Dr. Harland for his continuous guidance on this project and spending hours discussing various ideas with me. I would also like to thank University of Leeds for giving me the opportunity to carry out my research. where e ijk := e i ∧ e j ∧ e k . The group G 2 is the stabilizer group of φ restricted to each tangent space, and is a 14-dimensional simple, connected, simply connected Lie group.
For our purpose, we start by fixing a representation of the Clifford algebra Cl (7) in which the volume form Γ 7 acts as − Id.
Let / S(X) be the spinor bundle over a 7-manifold X with G 2 -structure. Let ξ ∈ Γ(/ S(X)) be a unit spinor such that ω · ξ = 0 for all ω ∈ Ω 2 14 , where · denotes Clifford multiplication. Then we have an isomorphism given by Lemma 2.3. The 3-form φ and 4-form ψ = * φ act on the subspaces Λ 0 and Λ 1 of / S(X) with eigenvalues Proof. Since Λ 0 and Λ 1 are irreducible representations of G 2 and φ is G 2 -invariant, by Schur's lemma φ preserves the decomposition. Furthermore, φ must act on each spaces as a constant and this action is traceless.
Corollary 2.5. Let f ∈ Ω 0 (X), v, u ∈ Ω 1 (X). Then Clifford multiplication of (f, v) by u is given by Hence the result follows from s being isomorphism.
Definition 2.7. Let / S(Σ) be the spinor bundle on Σ. A real spinor ξ ∈ Γ(/ S(Σ)) is called a Killing spinor if there exists δ ∈ R \ {0} such that for all X ∈ Γ(T Σ), ξ satisfies the Killing equation given by The scalar δ is called the Killing constant for the Killing spinor ξ.
We note that a unit spinor ξ on a nearly G 2 -manifold satisfying ω · ξ = 0 for all ω ∈ Ω 2 14 is a Killing spinor. Conversely, any Riemannian 7-manifold admitting a Killing spinor is a nearly G 2 -manifold. In fact, there is a one to one correspondence between nearly G 2 -structures and real Killing spinors on Σ [7].
where we have used the fact that the Killing constant δ can be written in term of τ 0 as δ = τ 0 8 . If g is the metric induced by the nearly G 2 structure φ, then the Ricci curvature is given by Ric g = 3 8 τ 2 0 g, and hence every nearly G 2 manifold is Einstein. The scalar curvature is Scal g = 21 8 τ 2 0 . We note that we can always re-scale τ 0 . If we take τ 0 = 4, then we have dφ = 4ψ. The reason for this particular choice is that the unit 7-sphere S 7 has scalar curvature 42, and so, Scal g = 21 8 τ 2 0 = 42, which implies τ 0 = 4 (where as taking τ 0 = −4 would just change the orientation of the manifold). Hence we have For a nearly G 2 -manifold (Σ, φ), we can define a 1-parameter family of affine connections on T Σ. Let t ∈ R. Then ∇ t is a 1-parameter family of connection on T Σ defined by Let T t be the torsion (1, 2)-tensor of the affine connection ∇ t . Then using the fact that Levi-Civita connection is torsion-free. Hence the torsion tensor T t is which is totally skew-symmetric, being proportional to φ. Now, ∇ t lifts to the spinor bundle / S(Σ) given by where η ∈ Γ(/ S(Σ)) and X ∈ Γ(T Σ). Then, using the eigenvalues (2.3), we find Therefore, for t = 1, the Killing spinor ξ is parallel with respect to the connection ∇ 1 . Then the connection ∇ 1 has holonomy group contained in G 2 with totally skew-symmetric torsion. This connection ∇ 1 on the nearly G 2 -manifold Σ is known as the canonical connection.
Proposition 2.8. [48] The Ricci tensor of the connection ∇ t is given by As a corollary, we have the scalar curvature of the canonical connection to be 112 3 . (7)-Manifolds and Spin(7)-Instantons Definition 2.9. Let X be an 8-dimensional Riemannian manifold equipped with a 4-form Φ ∈ Ω 4 (X) such that in local orthonormal basis e 0 , e 1 , . . . , e 7 , we have Φ = e 0 ∧ φ + ψ where φ is as in (2.1) and * (e 0 ∧ φ) = ψ. Then Φ is said to be a Spin(7)-structure on X and (X, Φ) is said to be an almost Spin(7)-manifold.
Theorem 2.10 ( [47]). There are orthogonal decompositions where Ω k d is a Spin(7)-invariant subspace of Ω k with point-wise dimension d and where L ξ is the Lie derivative with respect to ξ. 28]). If Φ is a Spin (7)-structure on a manifold X, then X is a Spin manifold. Moreover, if Φ is torsion-free, then X admits a non-trivial parallel spinor.
The canonical spin structure can be identified in the following way.
Let X be a Spin(7) manifold and P be a principal G-bundle on X. Let g P be the adjoint vector bundle. Then we have Ω 2 (g P ) = Ω 2 7 (g P ) ⊕ Ω 2 21 (g P ) Definition 2.12. Let π 2 7 : Ω 2 (g P ) → Ω 2 7 (g P ) be the projection. Then a connection A on P is said to be a Spin (7) This follows from the fact that the operator on Λ 2 defined by ω → * (Φ ∧ ω) has eigenvalues −1 and 3 with eigenspaces Λ 2 21 and Λ 2 7 respectively. Moreover, A is an instanton if and only if F A annihilates the parallel spinor, i.e., for parallel spinor ξ, we have F A · ξ = 0, where · denotes Clifford multiplication.

Asymptotically Conical Spin(7)-Manifolds
where r ∈ (0, ∞) is the coordinate. Σ is called the link of the cone. The metric g C compatible with Φ C is given by We note that condition dφ = 4ψ implies the torsion free condition dΦ C = 0, which implies that (C(Σ), g C , Φ) is a Spin(7)-manifold.
Remark 2.13. We note that a Spin(7)-cone is not complete. Hence, we consider complete Spin(7)manifolds whose geometry is asymptotic to the given (incomplete) G 2 -cone.
Definition 2.14. Let (X, g, Φ) be a non-compact Spin(7)-manifold. X is called an asymptotically conical (AC) Spin(7)-manifold with rate ν < 0 if there exists a compact subset K ⊂ X, a compact connected nearly G 2 manifold Σ, and a constant R > 1 together with a diffeomorphism for each p ∈ Σ, j ∈ Z ≥0 , r ∈ (R, ∞); where ∇ C is the Levi-Civita connection for the cone metric g C on C(Σ), and the norm is induced by the metric g C .
X \ K is called the end of X and Σ the asymptotic link of X.
Remark 2.15. For simplicity, we'll drop the points (r, p) while writing the norm, and will understand it from the context.
Remark 2.16. It can be proved that (see [32]) the metric g satisfies the same asymptotic condition is an AC manifold with any rate ν < 0.
• Some more examples of AC Spin(7)-metrics can be found in the recent work of Lehmann [36].
Let X be an AC Spin(7) manifold. In order to define "weighted Banach spaces" on X, we first define a notion of radius function.
where h : (R, ∞) × Σ → X \ K is the diffeomorphism, and r is a smooth interpolation between its definition at infinity and its definition on K, in a decreasing manner.
Let π : E → X be a vector bundle over X with a fibre-wise metric and a connection ∇ compatible with the metric. Definition 2.19. Let p ≥ 1, k ∈ Z ≥0 , ν ∈ R and C ∞ c (E) be the space of compactly supported smooth sections of E. We define the conically damped or weighted Sobolev space W k,p ν (E) of sections of E over X of weight ν as follows: For ξ ∈ C ∞ c (E), we define the weighted Sobolev norm · W k,p ν (E) as which is clearly finite and indeed a norm. Then the weighted Sobolev space W k,p ν (E) is the completion of C ∞ c (E) with respect to the norm · W k,p ν (E) . Remark 2.20. [37] • We note that W 0,2 −4 (E) = L 2 (E).
which is well defined and a norm. Then the weighted C k space C k ν (E) is the closure of C ∞ c (E) with respect to this norm. We also define C ∞ ν (E) := k≥0 C k ν (E).
is a continuous embedding.
In order to ensure that we work with continuous sections, we shall always assume k ≥ 4. This follows from the first part of the weighted Sobolev embedding theorem by putting l = 0 and p = 2.
Theorem 2.23 (Weighted Sobolev Multiplication Theorem). [20] Let ξ ∈ W k,2 µ (E), η ∈ W l,2 ν (F ). If l ≥ k > 8 2 = 4, then the multiplication is bounded. In other words, there a constant C > 0 such that From Proposition 2.24, we have the pairing This defines the isomorphism [37] W 0,2 Now, let P → X be a principal G bundle. Consider the associated vector bundle E : A is a connection on g P , then E inherits a metric from T and a connection from the Levi-Civita connection on T and the connection on g P . If ξ ∈ C ∞ c (T ⊗ g P ) ⊂ Γ(T ⊗ g P ), then the weighted Sobolev norm is given by Equation 2.31 where ∇ = ∇ LC ⊗ 1 g P + 1 T ⊗ ∇ A , ∇ A being the connection on g P .
Before moving forward let us fix few notations:

Asymptotically Conical Spin(7)-Instantons and Moduli Space
Definition 2.25. Let X be an AC Spin(7)-manifold asymptotic to the cone C(Σ). Let P → X be a principal G-bundle over X. P is called asymptotically framed if there exists a principal bundle Q → Σ such that h * P ∼ = π * Q where π : C(Σ) → Σ is the natural projection.
We note that such framing always exists [41]. So we fix a framing Q.
Definition 2.26. Let X be an AC Spin(7)-manifold asymptotic to the cone C(Σ). Let P → X be an asymptotically framed bundle. A connection A on P is called an asymptotically conical connection with rate ν if there exists a connection A Σ on Q → Σ such that for each p ∈ Σ, j ∈ Z ≥0 , ν < 0. The norm is induced by the cone metric and the metric on g.
A is called asymptotic to A Σ and ν 0 := inf{ν : A is AC with rate ν} is called the fastest rate of convergence of A.
• We have defined the rate of convergence in term of conical metric and the coordinate r on the cone. However, we could also have chosen in terms of the AC metric and the radius function . But both the cases the rate of convergence would be the same.
Let A P be the space of AC connections on P . Fix a reference connection A ∈ A P . Then for any other connection A = A + α, α ∈ Ω 1 (g P ) identifies the spaces A P and Ω 1 (g P ). Denote the space of W k,2 ν−1 -connections by and define which is the space of C ∞ ν−1 -connections. Now, a gauge transform is ϕ ∈ Aut(P ) and acts on a connection A by ϕ · A = ϕAϕ −1 − dϕϕ −1 . Let G → GL(V ) be a faithful representation of G, and consider the associated vector bundle End(V ). Then we define the weighted gauge group by (see [43]) We also define G ν := ∞ l=1 G l,ν .
Lemma 2.28. [16] The point-wise exponential map defines charts for which G k+1,ν is a Hilbert Lie group with Lie algebra modelled on Ω 0,k+1 ν (g P ) for k ≥ 3. The group G k+1,ν acts on A k,ν−1 smoothly via gauge transformations, for k ≥ 4. Definition 2.29. Let X be an AC Spin(7)-manifold asymptotic to C(Σ). Let P → X be a principal G-bundle asymptotically framed by Q → Σ. Let A Σ be an instanton on the nearly G 2 manifold Σ. Then the moduli space of Spin(7)-instantons asymptotic to A Σ with rate ν is given by

Deformation Theory of Asymptotically Conical Spin(7)-Instantons
In this section we describe the deformation theory of asymptotically conical Spin(7)-instantons.
In the first part we discuss the necessary analytic framework, following the works of Lockhart-McOwen [39], Marshall [42], Karigiannis-Lotay [32] and Driscoll [21]. In the second part we develop the general theory, where we closely follow Donaldson [18] and Driscoll [21].

Fredholm and Elliptic Asymptotically Conical Operators
We begin this section by defining the operators that will be important in developing the deformation theory.
1. Let Σ be a nearly G 2 -manifold and Q → Σ be a principal G-bundle. Let A Σ be a connection on Q. Consider the bundle / S(Σ)⊗g Q where / S(Σ) is the spinor bundle on Σ and g Q = Q× Ad g. Then we have a twisted Dirac operator (3.1) 2. Let X be an AC Spin (7)-manifold with link Σ. Let P → X be an asymptotically framed bundle. Let A ∈ A P be an AC connection asymptotic to A Σ .
Consider the bundle / S(X) ⊗ g P where / S(X) is the spinor bundle over X and g P = P × Ad g. Then we have a Dirac operator (3.2) 3. Let C(Σ) = (0, ∞) × Σ be a Spin (7)-cone over Σ and π * Q → C(Σ) be a principal bundle over C(Σ). Let A C = π * A Σ . Now consider the bundle / S(C(Σ)) × g π * Q , where / S(C(Σ)) is the spinor bundle on C(Σ) and g π * Q = π * Q × Ad g. Then we have a Dirac operator The objective behind introducing this Dirac operator / D A C is to study the Fredholm properties of the Dirac operator / D A using Lockhart-McOwen theory.
Then we have the following sets of operators: Then we consider the set of critical weights D(K C ) given by Hence, we focus our attention on finding the set of critical weights for the operators / The set of critical weights for the Laplace operator d * We want to find the set of critical weights for the Laplace operator d * . This set corresponds to a subset of the kernel of the operator containing elements of homogeneous order λ.
An easy calculation yields, Thus, ξ is in the kernel if and only if λ(λ + 6) is an eigenvalue of d * Hence, we have the following proposition.
Proposition 3.4. Let A be a AC connection over an AC Spin (7)-manifold X. If ν ∈ (−6, 0), then the coupled Laplace operator The set of critical weights for the Dirac operator / D A C Now, we want to find the set of critical weights for the Dirac operator / D A C , i.e. the set D( / D A C ). This set corresponds to a subset of the kernel of the operator containing elements of homogeneous order λ.
Recall that the volume element Γ 8 acting on / S(C) satisfies Γ 2 8 = 1 and this gives an eigen-space decomposition / S(C) = / S + (C) ⊕ / S − (C) corresponding to +1 and −1 eigenvalues respectively. We also have the decomposition of the Dirac operator: Proof. We fix the convention where the indices i, j, k run from 1 to 7 and µ, ν run from 0 to 7. Let e i be a local orthonormal frame of T Σ and using the metric g Σ , the dual e i be that of T * Σ. Then E 0 := dr and E i := re i for i = 1, . . . , 7 form a local orthonormal frame for T * C(Σ).
Moreover, let ∂ i be the differentiation with respect to e i and D µ be the differentiation with respect to E µ (vector field dual of E µ using the metric g C ). Then and Comparing (3.7) and (3.8), we get Now let Γ µ σν be the Christoffel symbols of the Levi-Civita connection on C(Σ) and γ i kj be that of on Σ. Then, Now, we consider the natural embedding of Cl (7) into Cl 0 (8) by e i → E 0 E i . Then the action of Dirac operator D Σ on η ∈ Γ(/ S(Σ)) is given by ). The action of negative Dirac operator D − C on C(Σ) is given by The result follows from (3.9) and (3.10).
The following corollary follows from the relations / . Corollary 3.6. Consider the following two twisted Dirac operators / D A Σ and Then, Finally, we have the description of the set critical weights of the twisted Dirac operator.
Then the set of critical weights is given by Proof. Consider the section σ of homogeneous degree ν − 1. i.e., σ = e (ν−1)t η, where e t = r.
The following theorem is a consequence of the fact that the Dirac operator / D A Σ is self-adjoint.
then m = 0. Hence elements of K(λ) − C have no polynomial terms.
Proof. Expanding the expression (3.14) using (3.11), we have Considering this as a polynomial in log r and comparing coefficients of (log r) m and (log r) m−1 respectively, we get Then using the self-adjoint property of / Since η m = 0, we have m = 0. Now, consider the Dirac operator (3.12) and denote its index by Index ν / D − A . Then, we have the following theorem.
From the Proposition 3.9, we conclude that K(λ) − C is precisely the λ + 5 2 eigenspace of the operator / D A Σ . Summarising, we have the following theorem.
Theorem 3.11. The Dirac operator is Fredholm if ν is not a critical weight, i.e., ν + 5 2 ∈ R \ Spec / D A Σ . Moreover, for two non-critical weights ν, ν with ν ≤ ν , the jump in the index is given by

Deformations of Asymptotically Conical Spin(7)-Instantons
Let A be an asymptotically conical reference connection that also satisfies the Spin(7)-instanton equation. Then, we have π 7 (F A ) = 0. Now, we can write any other connection in some open neighbourhood of A as A = A + α for α ∈ Ω 1 (g P ). Then, Hence the connection A is a Spin (7)-instanton if and only if π 7 (F A+α ) = 0, i.e., We also have the gauge fixing condition d * A α = 0. We consider the non-linear operator Hence, the local moduli space of Spin (7) In order to calculate the zero set of the non-linear operator, we calculate the kernel of the linearised Dirac operator, using the analytic techniques discussed in the previous subsections.
First we want to investigate the moduli space of AC Spin (7)-connections. We start with the following lemma.
As an immediate consequence, we have, Proof. Since there are no critical weights in (−6, 0), then if d * and by integration by parts.
The proof of the following lemma follows from the maximum principle.
Thus |f | 2 is a harmonic function and hence by Lemma 3.14, is zero. Thus, f = 0.
The following lemma can easily be proved using inverse mapping theorem.
Moreover, we have the following simple result from the theory of Banach spaces.
Lemma 3.17. Let X, Y be Banach spaces and T : X → Y be a bounded linear operator. Then ker T is closed and a closed subspace X 0 ⊂ X is a complement of ker T if and only if T | X 0 is injective and T (X) = T (X 0 ).
First let us show that the image of d A is closed.
is a closed subspace of Ω 1,k+1 ν−1 (g P ). Proof. Let Hence, by uniqueness of limits, we get α = d A f , and hence Im d A is closed.
Proposition 3.19. If ν ∈ (−6, 0), then we have the decomposition Proof. Let us consider the operator d * This is a bounded operator. Hence the kernel is a closed subspace. We want to show that X 0 := Im d A satisfies the conditions of Lemma 3.17. From Lemma 3.18 we note that Im d A is closed. Then for T := d * A and X := Ω 1,k+1 ν−1 (g P ) we have the result.
Then f − g is harmonic function and hence zero, which implies d A f − d A g = 0, which establishes the injectivity.
. This follows from the topological isomorphism in Lemma 3.16.
Let us consider the moduli space of connections B k+1,ν = A k+1,ν−1 /G k+2,ν . Then the infinitesimal action of the gauge group G k+2,ν is given by Thus we can view the Proposition 3.19 as a "slice theorem" which gives us the complement of the action of the gauge group.
Lemma 3.20. The action of the gauge group G k+2,ν on the space of connections A k+1,ν−1 is free.
Proof. Let us consider the stabilizer group We consider gauge transformations as sections of End(V ). Now, the connection A has holonomy contained in G and hence it preserves the inner product on V as well as on End(V ). Since, ϕ ∈ Γ A,ν is a gauge transformation, by definition 2.37, we have ϕ − I ∈ Ω 0,k+2 ν (g P ) and hence, We note that unlike the case where the manifold X is compact, when X is AC, reducible connections do not produce singularities in the space of connections modulo gauge.
Let us define the set Then T A,ν, ⊂ ker d * A models a local neighbourhood of the moduli space B k+1,ν . We note that studying the moduli space using the local model T A,ν, is basically same as solving the Coulomb gauge fixing condition d * A α = 0. This condition is local: locally, near A it selects a unique gauge equivalent class. Due to the topological obstructions, a global gauge fixing condition fails to exist.
The following lemma provides a sufficient condition for solving the gauge fixing condition. It is the weighted version of Proposition 2.3.4 of [18].
then there is a gauge transformation ϕ ∈ G ν such that ϕ(A ) is in Coulomb gauge relative to A. Now, we turn out focus to the main objective of this section: the moduli space of AC Spin(7)instantons.
Let us define the spaces, The proof of the following proposition is a weighted version of Proposition 4.2.16 of [18] and very similar to the proof for 7-dimensional case given by Driscoll [21]. Hence by Proposition 3.23 and weighted Sobolev embedding theorem, we see that M(A Σ , ν) k consists of smooth connections. We obtain the following important corollary.
Corollary 3.24. If ν ∈ (−6, 0), then the zero set of the non-linear twisted Dirac operator Finally, we have all the tools necessary to define the deformation and obstruction spaces, state and prove the main theorem of this section.

23
Then dF| (0,0) is surjective and dF| (0,0) = 0 if and only if ( / D A α, β) = (0, 0). Hence ker dF| (0,0) =: K = I(A, ν) × {0} is finite dimensional, and we have a decomposition of X as X = K ⊕ Z, where Z ⊂ X is a closed subspace. Moreover, we can write Z = Z × O(A, ν) for a closed subset Z ⊂ Ω 1,k+1 ν−1 (g P ). By implicit function theorem, we choose the open subsets U ⊂ I(A, ν), V 1 ⊂ Z and V 2 ⊂ O(A, ν), and smooth maps F i : U → V i for i = 1, 2, such that In order to study the deformations of FNFN instantons, we need to calculate the spectrum of the twisted Dirac operator on the link S 7 . We use representation theory, Frobenius reciprocity, and Casimir operators, to write the Dirac operators as a sum of Casimir operators. Then the problem of finding the spectrum of Dirac operator reduces to finding the eigenvalues of the Casimir operators. This method relies on S 7 being a homogeneous manifold and is developed based on the works of [49], [5], [6], [20]. for η ∈ L 2 (G, V ) H , and g, h ∈ G.

Dirac operators on Homogeneous Nearly
The right action of G on the space L 2 (G, V ) gives a representation ρ R , called the right regular representation defined by ρ R (h)η(g) = η(gh)

24
Then from H-equivariance, If we use the same notations for Lie algebra representations, then, Let G be the set of equivalence classes of irreducible representations of G and for γ ∈ G we have a representative (V γ , ρ γ ). Then Frobenius reciprocity implies the decomposition Now, since G/H is reductive, we have an orthogonal decomposition g = h ⊕ m induced by the Killing form K on G, defined by K(X, Y ) = Tr g (ad(X) ad(Y )).

(4.2)
We define a nearly G 2 -metric given by We note that in this framework G-invariant tensors on the tangent bundle T (G/H) correspond to H-invariant tensors on m [33]. Now, we consider the complex spinor bundle / S(Σ) = G × ρ,H ∆ where ∆ is the spinor space. From the splitting / S C (Σ) ∼ = Λ 0 C ⊕ Λ 1 C , we have ∆ ∼ = C ⊕ m * C . We now twist the spinor bundle by the associated bundle E = G × ρ V ,H V for a representation V of H. Then where e a is the basis of m * dual to I a and η ∈ L 2 (G, ∆ ⊗ V ) H . Then the Dirac operator / D 1 Then, from, (2.15) and (4.4), we have a family of Dirac operators where the action of G on V γ of the right hand side of the expression corresponds to the action of ρ L on L 2 (/ S C (Σ) ⊗ E). We note that the Dirac operator commutes with the left action of G and hence it respects the decomposition (4.6). Then, by Schur's lemma, for every t ∈ R, the Dirac operator If {I A } is an orthonormal basis of g, then the Casimir Operator Cas g ∈ Sym 2 (g) is the inverse of the metric on g, defined by Using Lichnerowicz formula, we can write the square of the Dirac operator as sum of Casimir operators. for η ∈ Γ(/ S C (Σ) ⊗ E).
Restricting the operator / D The self-adjointness of this operator implies that it is diagonalisable with real eigenvalues. Frobenius reciprocity and Proposition 4.1 implies

Eigenvalue Bounds
We have the nearly G 2 -manifold G/H which is a reductive homogeneous space. Now, the Casimir operators commute with the group action, and hence on irreducible representation, they act as a multiple of the identity. That is, where c g γ and c h γ are real numbers, called Casimir eigenvalues. Now, let V γ be an irreducible representation of G. Then we have the decomposition of V γ as where W σ are irreducible representations of H and I is a finite sequence in H which may have repeated entries. Similarly, for finite sequences J, K in H, we have the decomposition Let us assume that in the decomposition of W α ⊗ W β in to irreducible representations, W γ σ occurs with multiplicity 1. Then we consider the composition map where the first map is projection map and the third one is equivariant embedding. Since the decomposition of V γ , ∆ and V into irreducible representations of H are orthogonal, {q σ αβ } is an orthogonal basis of Hom(V γ , ∆ ⊗ V ) H . Hence, q σ αβ are eigenvectors of (4.11) and Av, v and max

The Twisted Dirac Operator on S 7
We identify S 7 with the homogeneous space Spin(7)/G 2 . Let m be the orthogonal complement of g 2 ⊂ spin(7) with respect to the Killing form (4.2) on the Lie algebra spin (7). Clearly, [g 2 , m] ⊂ m and hence the homogeneous space Spin(7)/G 2 is reductive. Consider the Maurer-Cartan form θ on Spin(7) and the splitting θ = θ g 2 ⊕ θ m induced by the decomposition spin(7) = g 2 ⊕ m. Then θ g 2 =: A Σ is canonical connection on the bundle G 2 → Spin(7) → S 7 whose curvature is given by for X, Y ∈ m. This is a G 2 -invariant element in Λ 2 m * ⊗ g 2 . The torsion is given by In (2.14) putting t = 1 for canonical connection and denoting T 1 (X, Y ) by T (X, Y ), we have The nearly G 2 metric, normalised such that the scalar curvature of the canonical connection is 112 3 , can be written as (4.3) where K is the Killing form (4.2) and c is a constant to be determined. Let {I A : A = 1, . . . , 21} be an orthonormal basis for spin(7), {I a : a = 1, . . . , 7} be a basis for m and {I i : i = 8, . . . , 21} be a basis for g 2 . Let f C AB be the structure constants defined by We lower the indices as f ABC := f D AB δ DC . Then, Then for I a , I b ∈ m, (4.14) and (4.15) imply A simple calculation involving the relations shows that, Tr spin(7) (ad(X) ad(Y )).
( 4.17) is a nearly G 2 -metric, normalised so that the scalar curvature of the canonical connection is 112 3 . Let us consider Cl (7), the Clifford algebra over R 7 . Let ∆ be a 8-dimensional representation and ρ ∆ : spin(7) → End(∆) be the restriction to spin (7). From (4.6), recall the identification We want to compare the operator (4.18) with the Dirac operator (4.4). That is, compare ρ ∆ (I a ) with the Clifford multiplication by I a .
Then from (4.28) we have Now, we want to calculate the eigenvalues of the Casimir operators appearing in the expression of the Dirac operator. Let V (a,b,c) be an irreducible representation of spin(7) with highest weight (a, b, c) and V (a,b) be an irreducible representation of g 2 with highest weight (a, b). The Casimir operators can be written as The Casimir eigenvalues are given by, (a 2 + 3b 2 + 3ab + 5a + 9b), (4.33) c spin (7) (a,b,c) = − 1 3 (4a 2 + 8b 2 + 3c 2 + 8ab + 8bc + 4ca + 20a + 32b + 18c). (4.34) These expressions differ from that of [48], because we use a different normalisation of the Casimir operator and an opposite convention for the order of a, b, c.
Corollary 4.5. Consider the irreducible representations of Spin (7) given by If V γ is not one of these irreducible representations, then the operator has no eigenvalues in the interval − 5 2 , 5 2 .

Calculation of Eigenvalues of the Twisted Dirac Operator
In this section we explicitly calculate the eigenvalues of the twisted Dirac operator corresponding to the representations mentioned in Corollary 4.5. Let us describe the outline of the method.
Let V γ be an irreducible representation of Spin (7). We want to find the matrix of the operator given in (4.32).
• From the explicit expressions of q-basis and p-basis elements, we write p (i,j,k) (m,n) in terms of q (i,j)(k,l) (m,n) and the change of basis matrix.
• Next, we calculate the matrix of M γ in the q-basis. From (4.31), we see that it is a diagonal matrix with entries either 1 or −1, since q i factors through either V (0,0) ∼ = Λ 0 ⊂ ∆, or in the q-basis (4.30).
• In the q-basis, we have φ acting as a diagonal matrix with entries either 7 or −1, by Lemma 2.3, since q i factors through either Consequently, using (4.32), we calculate the matrix of / D t A Σ γ in the q-basis.

FNFN Spin(7)-Instanton
In this subsection, we derive FNFN-instanton using homogeneous space techniques. The exact same result and similar approach can also be found in [45].
We want to find all Spin(7)-invariant connections on Q. From Wang's theorem [52], we know that this corresponds to all G 2 -equivariant linear maps m → spin (7). That is, the set and hence, restricting to m, we have the decomposition Recall Schur's lemma: Thus, we have all the G 2 -equivariant linear maps Λ : m → spin (7), explicitly given by where the complex number ϕ is necessarily real because Λ is (the complexification of) a map between real vector spaces m → m.
Now the basis I A for spin (7) can be represented by left invariant vector fields E A on Spin(7) and also by the dual basisê A of left invariant 1-forms. Denote the natural projection map π : Spin(7) → Spin(7)/G 2 g → gG 2 of the principal bundle. Let U be a contractible open subset of Spin(7)/G 2 . Then we define the map L : U → Spin(7) such that π•L = Id U , i.e., L is a local section of the bundle Spin(7) → Spin(7)/G 2 . We put e A := L * êA . Then {e a : a = 1, . . . , 7} form an orthonormal frame for T * (Spin(7)/G 2 ) over U .
For e A , we have the Maurer-Cartan equations In local coordinates the connection on the nearly G 2 -manifold Spin(7)/G 2 can be written as A = e i I i + ϕe a I a where e i I i , is the canonical connection A Σ . Now consider the 8-dimensional manifold R × Spin(7)/G 2 . We choose the metric g 8 = (e 0 ) 2 + g 7 where e 0 = dt for t the coordinate of R, and g 7 is the metric on Spin(7)/G 2 . This metric is conformal to the flat metric on R 8 . The connection 1-form is given by A = A 0 e 0 + A a e a which gives the Spin(7)-invariant connection Here without loss of generality, we take A 0 = 0 (called temporal gauge), since we can always choose such a gauge. We note that the connection A on R × S 7 can be identified with a family {A t : t ∈ R} of connections on S 7 . The curvature of this connection is given by Now, the ASD instanton equation Φ ∧ F A = − * F A can be written as Now, we have φ = 1 6 φ abc e a ∧ e b ∧ e c , * φ = ψ = 1 24 ψ abcd e a ∧ e b ∧ e c ∧ e d , for a, b, c = 1, . . . , 7, where φ abc are structure constants of the octonions and ψ pqrs = abcpqrs φ abc . We have already seen that we can write the structure constants of the octonions φ abc in terms of the structure constants f abc as f abc = − 2 3 φ abc . Then, we get Then, from (5.4) we have two equations, F 0a = 3 4 f abc F bc and F bc = −F 0a φ abc − 1 2 F ad ψ abcd , but the first one implies the second. Hence, the ASD instanton equation 5.4 reduces to Applying the Maurer-Cartan equations 5.1, we calculate dA t , as well as [A t ∧A t ], and the curvature is given by Then, the ASD instanton equation 5.5 is equivalent tȯ Simplifying, we have the differential equationφ = 2(ϕ 2 − ϕ). Solving, we get ϕ = 1 1 + e 2t+2C 1 = 1 Cr 2 + 1 (5.7) for C > 0, using the substitution r = e t .
From the calculations above, and from the fact that the ASD instanton equations are conformally invariant, it follows that the connection A defined in (5.2), where ϕ is given in (5.7), is in fact an instanton on R 8 . We call this the FNFN Spin(7)-instanton. Clearly FNFN Spin(7)-instanton A is asymptotic to the canonical connection A Σ with fastest rate of convergence −2, since ϕ = O(r −2 ) as r → ∞.

Index of the Twisted Dirac Operator
We want to calculate the index of the Dirac operator / D − A on / S(R 8 ) twisted by the trivial bundle g P := spin(7) × R 8 over R 8 . We use the Atiyah-Patodi-Singer Index Theorem for manifolds with boundaries, by relating the index of the Dirac operator / D − A on R 8 with the index of the Dirac operator on a closed ball B 8 R of large enough radius R. Moreover, we consider the FNFN instanton to be an instanton on R 8 and, for the purposes of calculating the eta-invariant appearing in the index theorem, on R × S 7 .

Atiyah-Patodi-Singer Index Theorem
The Atiyah-Patodi-Singer index theorem is applicable when the manifold has non-empty boundary (for more details see [2], [3], [4], [22], [26]). Let M be a 8-manifold with non-empty boundary ∂M . Let us define a function ϕ : R → R by where α is a smooth interpolation between its definition at T 2 and T and α is that of between its definition at −T and − T 2 . Then we have a connection A = A Σ + ϕ(t)e a I a (5.11) We note that A − A = (ϕ(t) − ϕ(t))e a I a ∈ Ω 1 (g P ) (5.12) where g is the cylindrical metric g = dt 2 + g S 7 .
Then, we have the Fourier expansion of η given by η = n∈Z e λn(t−ln R) η n where η n ∈ ker / D A Σ − λ n .
Hence, η ∈ L 2 implies η n = 0 when λ n > 0. So η can be written as a sum of eigenvectors η n of Dirac operator on the boundary with negative eigenvalues. Hence η solves Atiyah-Patodi-Singer boundary condition.

Eta Invariant of the Boundary
We recall that we can identify the family of Dirac operators / D A t,Σ t∈R on S 7 with a Dirac operator / D − A on the cylinder R × S 7 , where the identification is given by Then, the index of the Dirac operator / D − A on the cylinder R × S 7 is precisely the negative of the spectral flow of the operator sf / D A t,Σ t∈R (see [34] proposition 14.2.1). This follows from the fact that d dt and / D A t,Σ have opposite signs, and Clifford multiplication by E 0 is an isomorphism that does not affect index. Hence, from (5.16), we have Moreover, since ∂(R × S 7 ) = S 7 S 7 , where S 7 is S 7 with opposite orientation, we have η(∂(R × S 7 )) = η(D Σ , S 7 ) + η( / D A Σ , S 7 ) = η( / D A Σ , S 7 ) − η(D Σ , S 7 ) = η( / D A Σ , S 7 ), since, eta-invariant of D Σ is zero, which follows from the fact that the metric and Levi-Civita connection of S 7 are invariant under an orientation-reversing isometry. We note that the orientation of S 7 corresponding to the operator / D A Σ is the same as the boundary S 7 of R 8 .
So, finally, we have Since, B 8 R is a compact manifold with boundary, applying Atiyah-Patodi-Singer index formula,

The Main Result
Finally, we have the main result on the deformations of FNFN Spin(7)-instanton. Proof. From (5.18) we have that the index of the Dirac operator / D − A corresponding to the rate −5/2 is zero. Moreover, from Corollary 4.7, we see that the only critical rates greater that −5/2 are −2 and −1, corresponding to the eigenvalues 1/2 and 3/2 respectively. Then, from the facts that the eigenspace of the eigenvalue 1/2 is 1-dimensional and the eigenspace of the eigenvalue 3/2 is 8-dimensional, the result follows from Theorem 3.11. Now, the two known deformations of the FNFN instanton on R 8 are the translation and the dilation. It is clear that translation being 8-dimensional, should come from spin representation, whereas dilation being one dimensional, should come from the trivial representation.
From the fact that the eigenvalues of the twisted Dirac operator in the range [1/2, 5/2] are 1/2 and 3/2, corresponding to the trivial and spin representations respectively, we should expect that the rate of dilation should be 1/2 − 5/2 = −2 and that of translation should be 3/2 − 5/2 = −1. This can be easily verified from the fact that the two deformations translation and dilation are given by ι ∂ ∂x i F A and ι x i ∂ ∂x i F A respectively. Moreover, if dt ∧ b + w ∈ Λ 3 48 (C 8 ), where b ∈ Λ 2 (C 7 ) and w ∈ Λ 3 (C 7 ), since, (dt ∧ b + w) ⌟ Φ = 0, we have b ⌟ φ = −w ⌟ ψ.