Deformations of $\mathrm{G}_2$-instantons on nearly $\mathrm{G}_2$ manifolds

We study the deformation theory of $\mathrm{G}_2$-instantons on nearly $\mathrm{G}_2$ manifolds. There is a one-to-one correspondence between nearly parallel $\mathrm{G}_2$ structures and real Killing spinors, thus the deformation theory can be formulated in terms of spinors and Dirac operators. We prove that the space of infinitesimal deformations of an instanton is isomorphic to the kernel of an elliptic operator. Using this formulation we prove that abelian instantons are rigid. Then we apply our results to describe the deformation space of the canonical connection on the four normal homogeneous nearly $\mathrm{G}_2$ manifolds.

by ∇ g the Levi-Civita connection and its lift on the spinor bundle. The G 2 -structure ϕ is nearly parallel if for some τ 0 = 0 dϕ = τ 0 * ϕ ϕ, or equivalently if there exists a real Killing spinor η such that Nearly G 2 manifolds were introduced as manifolds with weak holonomy G 2 by Gray in [Gra71]. Some examples of such manifolds are the round and squashed 7-spheres, the Aloff-Wallach spaces, and the Berger space SO(5)/SO(3). The inclusion of the exceptional Lie group G 2 as a possible holonomy group for Riemannian manifolds in Berger's list [Ber55] led mathematicians to look for examples of manifolds with holonomy G 2 . In [Wan89] Wang established the first correspondence between parallel spinors and integrable geometries. Later the classification of manifolds with real Killing spinors in [Hij86,FK89,FK90,Gru90,Bär93] established a link between manifolds with weak holonomy and manifolds with real Killing spinors. These manifolds are Einstein with positive scalar curvature. Except for the round 7-sphere, the dimension of the space of Killing spinors on a nearly G 2 manifold is 1,2 or 3 (see [FKMS97]) giving rise to three types: proper, Sasaki-Einstein and 3-Sasakian respectively. The cones over these manifolds have holonomy contained in Spin(7) which makes these spaces particularly important in the construction and understanding of manifolds with torsion free Spin(7)-structures.
The correspondence between nearly parallel G 2 structures and Killing spinors has been extensively used to produce many results on nearly G 2 manifolds. The infinitesimal deformation space of nearly G 2 -structures was explicitly described as an eigenspace of a Dirac operator in [AS12]. In the homogeneous setting, non-trivial deformations were only found for the Aloff-Wallach space and which in [DS20] were proved to be obstructed.
The spinorial approach can also be used to study gauge theory on manifolds with weak holonomy. A connection A on M is a G 2 -instanton if its curvature F satisfies the algebraic condition F ∧ ϕ = * ϕ F, or equivalently F · η = 0. In this article we describe the infinitesimal deformation space of instantons on nearly G 2 manifolds as the eigenspaces of the Dirac operators associated to the one parameter family of connections with skew-symmetric torsion described in [AF04, ABBK13, AF14, ACFH15, AH15]. At t = −1, the connection ∇ −1 is the characteristic connection which is a G 2 -instanton. We explicitly describe the infinitesimal deformation space of the characteristic connections for the normal homogeneous nearly G 2 manifolds classified in [FKMS97]. In [CH16] an analogous description for the infinitesimal deformation space of instantons on nearly Kähler 6-manifolds is given. On an oriented manifold with real Killing spinor η the volume form vol defines a Killing spinor vol ·η. On a nearly Kähler 6-manifold {η, vol ·η} defines a 2 dimensional space of Killing spinors whereas on a nearly G 2 manifold η and vol ·η are linearly dependent. This prevents us from having a relation like in [CH16,Proposition 4(iii)] which makes the computation of the infinitesimal deformation space much more convenient (See 4.3). In fact we show in §4 that such a relation does not exist in the nearly G 2 case by explicitly computing the kernel of the elliptic operator for the homogeneous nearly G 2 manifolds.
In [Dri20] the author uses the spinorial approach to describe the deformation space of instantons on asymptotically conical G 2 manifolds.
Let M 7 be a manifold with a G 2 structure ϕ and let η be the Killing spinor associated to ϕ. A connection A on M is a G 2 -instanton if its curvature F A satisfies the algebraic condition The above condition is equivalent to F A · η = 0 as shown in §3. When the G 2 structure is parallel (the case when the constant τ 0 = 0) these instantons clearly solve the Yang-Mills equation d * ∇ F = 0. The analogous result was proved in the nearly G 2 case by Harland-Nölle [HN12]. They showed that the instantons on manifolds with real Killing spinors solve the Yang-Mills equation which makes the study of instantons on nearly G 2 manifolds important from the point of view of gauge theory in higher dimensions. However G 2 -instantons in the parallel case are the minimizers of the Yang-Mills functional which is not necessarily true for the nearly parallel case, as proved by Ball-Oliveira in [BO19]. The first examples of G 2 -instantons on parallel G 2 manifolds were constructed in [Cla14], [Wal16] and [SEW15]. In [BO19] the authors proved the existence of nearly G 2 -instantons on certain Aloff-Wallach spaces and classified invariant G 2 -instantons on these spaces with gauge group U(1) and SO(3). Recently, Waldron [Wal20] proved that the pullback of the standard instanton on S 7 obtained from ASD instantons on the 4-sphere via the quaternionic Hopf fibration lies in a smooth, complete, 15-dimensional family of G 2 -instantons.
In §2 we describe a 1-parameter family of connections on the spinor bundle / S over nearly G 2 manifolds and the associated Dirac operators. In [DS20] the authors introduced a Dirac type operator and used it to completely describe the cohomology of nearly G 2 manifolds and proved the obstructedness of infinitesimal deformations of the nearly G 2 structure on the Aloff-Wallach space. We remark that the Dirac type operator introduced there is not associated to any connection in the 1-parameter family.
In §3 we describe the deformation space of a nearly G 2 instanton A as an eigenspace of a Dirac operator associated to A and the characteristic connection (Theorem 3.2). Using this description, we show that on a compact nearly G 2 manifold the G 2 -instanton A is rigid if the structure group is abelian (cf. Theorem 3.7(i)) or if all the eigenvalues of a linear operator L A are greater than − 28 5 (Theorem 3.7(ii)) . The instanton A is also rigid if all the eigenvalues of L A are less than 6, as shown in [BO19,Proposition 8] where authors used a Weitzenböck formula, while the proof of Theorem 3.7(ii) uses the Schrödinger-Lichnerowicz formula for the family of Dirac operators associated to ∇ t and A.
In §4 we describe the infinitesimal deformation space of the characteristic connection on all the homogeneous nearly G 2 manifolds whose nearly G 2 metric is normal. By considering the actions of the Lie groups H and G 2 on G/H we can view the characteristic connection as an H-connection or a G 2 -connection. We compute its infinitesimal deformation spaces in both of these cases. The results are recorded in Theorem 4.6. The deformations are shown to be genuine in all cases except that of the Aloff-Wallach space SU(3)×SU(2) SU(2)×U(1) . In the latter case the author is currently unaware of any known family of nearly G 2 -instantons for which the infinitesimal deformations are the ones found in Theorem 4.6.
Agricola for her suggestions which improved the article vastly. The author is also thankful to Shubham Dwivedi for many useful interactions regarding the project.

Nearly parallel G 2 structures
Let M be a 7-dimensional Riemannian manifold equipped with a positive 3-form ϕ ∈ Ω 3 + (M ). The 3-form ϕ induces an orientation and a metric on M and thus a Hodge star operator * ϕ on the space of differential forms (see [Bry87]). The G 2 structure ϕ is called a nearly parallel G 2 structure on M if it satisfies the following differential equation for some non-zero τ 0 ∈ R, dϕ = τ 0 * ϕ ϕ. (2.1) We denote the 4-form * ϕ ϕ by ψ in the remainder of this article. The condition dϕ = τ 0 ψ implies dψ = 0, thus the nearly parallel G 2 structure ϕ is co-closed.
Every manifold with a G 2 structure is orientable and spin, and thus admits a spinor bundle / S. Let ∇ LC be the Levi-Civita connection of the induced metric on M . A spinor η ∈ Γ(/ S) is a real Killing spinor if for some non-zero δ ∈ R, There is a one-to-one correspondence between nearly parallel G 2 structures and real Killing spinors on M . Given a nearly parallel G 2 structure ϕ that satisfies (2.1) there exists a real Killing spinor η that satisfies (2.2) with δ = − 1 8 τ 0 and vice-versa. Switching − τ 0 8 to τ 0 8 corresponds to changing the orientation of the cone M × r 2 R + . See [BFGK91] and [Bär93] for more details.
The constant τ 0 can be altered by rescaling the metric and readjusting the orientation. In this article we use τ 0 = 4. With this choice of τ 0 our nearly G 2 structure ϕ and Killing spinor η satisfies the following equations respectively dϕ = 4ψ, Manifolds with nearly parallel G 2 structures have several nice properties which can be found in detail in [BFGK91]. In particular they are positive Einstein. Let g be the metric induced by ϕ, then the Ricci curvature Ric g = 3 8 τ 2 0 g and the scalar curvature Scal g = 7Ric g = 21 8 τ 2 0 . A G 2 structure on M induces a splitting of the spaces of differential forms on M into irreducible G 2 representations. The space of 2-forms Λ 2 (M ) decomposes as where Λ 2 l has pointwise dimension l. More precisely, we have the following description of the space of forms : Note that we are using the convention of [Kar09] which is opposite to that of [Joy00] and [Bry06].
The space Λ 2 14 is isomorphic to the Lie algebra of G 2 denoted by g 2 . Since the group G 2 preserves the G 2 structure ϕ, it preserves the real Killing spinor η induced by ϕ. The space Λ 2 14 can be equivalently defined as We make use of this identification when defining the instanton condition on M in §3.

The spinor bundle
For a 7-dimensional Riemannian manifold M with a nearly parallel G 2 structure ϕ, the spinor bundle / S is a rank-8 real vector bundle over M and is isomorphic to the bundle R⊕T M = Λ 0 ⊕Λ 1 . At each point p ∈ M , we can identify the fiber of If η is the real Killing spinor on M induced by ϕ then we have the isomorphism Under this isomorphism any spinor s = (f · η, α · η) ∈ / S can be written as s = (f, α) ∈ Λ 0 ⊕ Λ 1 . The 3-form ϕ induces a cross product × ϕ on vector fields X, Y ∈ Γ(T M ). Throughout this article we use e i to denote both tangent vectors and 1-forms, identified using the metric. All the computations are done in a local orthonormal frame {e 1 , . . . , e 7 } and any repeated indices are summed over all possible values. With respect to this local orthonormal frame, we have The octonionic product of two octonions (f 1 , X 1 ) and (f 2 , X 2 ) is given by, As shown in [Kar10] the Clifford multiplication of a 1-form Y and a spinor (f, Z) is the octonionic product of an imaginary octonion and an octonion and is thus given by (2.5) Note that the product defined above differs from [Kar10] by a negative sign due to our choice of the representation of Cl 7 on / S [LM89, Chapter 1.8]. We define the Clifford multiplication of any p-form β = β i 1 ...ip e i 1 ∧ e 2 ∧ · · · ∧ e ip with a spinor by, β · (f, X) = β i 1 ...ip (e i 1 · (e i 2 · . . . · (e ip · (f, X)) . . .)).
We record an identity for Clifford algebras for later use and refer the reader to [LM89, Proposition 3.8, Ch1] for the proof.
The Clifford multiplication between a p-form α and a 1-form v can be written as [LM89, The vector bundle / S is a G 2 -representation and since G 2 is the isotropy group of the 3-form ϕ the map µ → ϕ · µ from the bundle of spinors / S to itself is an isomorphism. The same argument holds for the 4-form ψ. The following formulae described in [ACFH15,FI02] will prove useful in later computations.
A common feature between nearly Kähler 6-manifolds and manifolds with nearly parallel G 2 structures is the presence of a unique canonical connection ∇ can with totally skew-symmetric torsion defined below. The Killing spinor η is parallel with respect to this connection and thus we have Hol(∇ can ) ⊂ G 2 . It was proved by Cleyton-Swann in [CS04,Theorem 6.3] that a G-irreducible Riemannian manifold (M, g) with an invariant skew-symmetric non-vanishing intrinsic torsion falls in one of the following categories: 1. it is locally isometric to a non-symmetric isotropy irreducible homogeneous space, or, 2. it is a nearly Kähler 6-manifold, or, 3. it admits a nearly parallel G 2 structure.
For the nearly G 2 manifold (M, ϕ) we define a 1-parameter family of connections on T M that include the canonical connection ∇ can . Let t ∈ R and let ∇ t be the 1-parameter family of connections on T M defined for all X, Y, Z ∈ Γ(T M ) by Let T t be the torsion (1, 2)-tensor of ∇ t . Since the connection ∇ LC is torsion free Therefore the torsion tensor T t is given by which is proportional to ϕ and is thus totally skew-symmetric.
By [LM89,Theorem 4.14] the lift of the connection ∇ t on the spinor bundle which is also denoted by ∇ t acts on sections µ of / S as (2.8) The space of real Killing spinors is isomorphic to Λ 0 thus for a Killing spinor η it follows from (2.3) and Lemma 2.2 that for any vector field X since X · ϕ + ϕ · X = −2 i X ϕ, Therefore η is parallel with respect to the connection ∇ −1 . The connection ∇ −1 thus has holonomy group contained in G 2 with totally skew-symmetric torsion and is therefore the canonical connection on the nearly G 2 manifold M described in [CS04].
Proposition 2.3. The Ricci tensor Ric t of the connection ∇ t is given by Proof. By using the expression of the Ricci tensor for a connection with a totally skew-symmetric torsion from [FI02], we have The Ricci tensor for the Levi-Civita connection is given by Ric 0 = 6g. Since dψ = 0, ϕ is co-closed and the second term in the above expression vanishes. The third term can be calculated in a local orthonormal frame e 1 , . . . , e 7 using the contraction identity ϕ ijk ϕ ijl = 6δ kl as follows Summing up all the terms together give the desired identity for Ric t .

Deformation theory of instantons
Let P → M be a principal K-bundle. We denote by Ad P the adjoint bundle associated to P. Let A be a connection 1-form on P and F A ∈ Γ(Λ 2 T * M ⊗ Ad P ) be the curvature 2-form associated to A given by There are many ways to define the instanton condition on A. If (M, g) is equipped with a Gstructure such that G ⊂ O(n), there is a subbundle g(T * M ) ⊂ Λ 2 T * M whose fibre is isomorphic to g = Lie(G). The connection A is an instanton if the 2-form part of F A belongs to g(T * M ).

In global terms, A is an instanton if
Note that in dimension 7 if M is equipped with a G 2 structure then this condition implies that A is an instanton if the 2-form part of F A ∈ g 2 (T * M ) = Γ(Λ 2 14 ). The second definition of an instanton is a special case of the first when the Lie algebra g is simple. Its quadratic Casimir is a G-invariant element of g ⊗ g which may be identified with a section of Λ 2 ⊗ Λ 2 and thus to a 4-form Q by taking a wedge product. Since this Q is G-invariant the operator u → * ( * Q ∧ u) acting on 2-forms commutes with the action of G and hence by Schur's Lemma the irreducible representations of G in Λ 2 are eigenspaces of the operator. Then for some ν ∈ R. In dimension 7 it turns out that Q = ψ (see [HN12]) and the above condition is equivalent to F A ∈ Γ(Λ 2 14 ) when ν = 1. Furthermore if M is a spin manifold, and the spinor bundle admits one or more non-vanishing spinors η, then A is an instanton if When M has a G 2 structure and η is the corresponding spinor, (2.4) implies that a the above condition is satisfied if and only if A is a G 2 -instanton. An interested reader can read further on these definitions and their relations in [HN12].
We remark that for an instanton A on a manifold with a G 2 structure ϕ all the above definitions are equivalent. They all imply that the curvature F A associated to A lies in Γ(Λ 2 14 ) and thus satisfies all of these equivalent conditions: (3.1) From now on in this article we use these instanton conditions interchangeably according to the context without further specification. Note that the above definitions are valid for any general G 2 structure and not only for nearly parallel ones.
On a manifold with real Killing spinors it was shown in [HN12] that instantons solve the Yang-Mills equation. In the case of a nearly G 2 -instanton we can prove this fact by direct computation. For an instanton A, (3.1) and the second Bianchi identity imply

Infinitesimal deformation of instantons
Let M 7 be a nearly G 2 manifold. We are interested in studying the infinitesimal deformations of nearly G 2 -instantons on M . An infinitesimal deformation of a connection A represents an infinitesimal change in A and thus, is a section of T * M ⊗ Ad P . If ǫ ∈ Γ(T * M ⊗ Ad P ) is an infinitesimal deformation of A, the corresponding change in the curvature F A up to first order is given by d A ǫ. A standard gauge fixing condition on this perturbation is given by (d A ) * ǫ = 0. So in total the pair of equations whose solutions define an infinitesimal deformation of an instanton A is given by On a nearly G 2 manifold we can define a 1-parameter family of Dirac operators The 1-parameter family of connections on the spinor bundle / S defined in (2.8) and the connection A on P can be used to construct a 1-parameter family of connections on the associated vector bundle / S ⊗ Ad P . We denote by ∇ t,A , the connection induced by ∇ t and A for all t ∈ R respectively. We denote by D t,A the Dirac operator associated to ∇ t,A . The following proposition associates the solutions to (3.2) to a particular eigenspace of D t,A for each t. The proposition was proved in [Fri12] for t = 0.
Proposition 3.1. Let ǫ be a section of T * M ⊗Ad P , and let D t,A be the Dirac operator constructed from the connections ∇ t,A for t ∈ R. Then ǫ solves (3.2) if and only if Proof. Let {e a , a = 1 . . . 7} be a local orthonormal frame for T * M . Then Applying Proposition 2.1 to the 1-form part of ǫ we get e a · ǫ · e a · η = 5ǫ · η. So if η is a real Killing spinor then (2.3) together with the above identity imply It follows from (2.8) and the identity a e a · i a ϕ = 3ϕ that Since ǫ · η ∈ Λ 1 · η, by Lemma 2.2 we have The equation D t,A (ǫ · η) = − t+5 2 ǫ · η is thus equivalent to (d A ǫ + (d A ) * ǫ) · η = 0, which in turn is equivalent to the pair of equations (d A ǫ) · η = 0, (d A ) * ǫ = 0 since these two components live in complementary subspaces.
Since η is parallel with respect to ∇ −1 we can view D −1,A as an operator on Λ 1 ⊗ Ad P defined by D −1,A (ǫ · η) = (D −1,A ǫ) · η. The following theorem is an immediate consequence of the above proposition.
Theorem 3.2. The space of infinitesimal deformations of a G 2 -instanton A on a principal bundle P over a nearly G 2 manifold M is isomorphic to the kernel of the operator (3.4) Remark 3.3. By Proposition 3.1, the − t+5 2 eigenspace of the operator D t,A on Λ 1 · η ⊗ Ad P is isomorphic to the infinitesimal deformation space of the instanton A for all t ∈ R and all these eigensapces are thus isomorphic to each other. In particular (3.5) The deformation space found above can be further analysed as an eigenspace of the square of the Dirac operator. In [AF04] the authors obtained a Schrödinger-Lichnerowicz type formula relating the square of the Dirac operator with torsion T to the connection with torsion 3T . Such a rescaling was earlier used in [Goe99] for η-invariant homogeneous spaces and in [Bis89] for Hermitian manifolds. The proof adapted to our setting is presented to keep the discussion self contained.
Proposition 3.4. Let EM be a vector bundle associated to P and µ ∈ Γ(/ S ⊗ EM ). Let A be any connection on P. Then for all t ∈ R, (3.6) Proof. Let {e 1 , . . . , e 7 } be an orthonormal frame for the tangent bundle. As before we obtain Squaring both sides we obtain, The first term of the above expression is given by the Schrödinger-Lichnerowicz formula The anti-commutator in the second term is given by but since M is nearly G 2 , ϕ is coclosed, therefore At the center of a normal frame, Again using the fact that d * ϕ = 0 we get Substituting the three terms in the expression of (D t/3,A ) 2 µ using (E1), (E2), (E3), (E4) we get the result.
When the connection A is an instanton on a nearly G 2 manifold the expression for (D t/3,A ) 2 can be simplified further. For the G 2 structure ϕ, ϕ 2 = 7 and under our choice of convention dϕ = 4ψ and Scal g = 42. Thus we can calculate the action of (D t/3,A ) 2 on spinors in Λ 0 η and Λ 1 · η as follows.
Let η ∈ Γ(Λ 0 M ⊗ EM ) be a real Killing spinor then Lemma 2.2 implies ψ · η = 7η and F A · η = 0 by (3.1). Thus by above proposition we obtain, Thus by above proposition In the special case when the bundle EM is equal to Ad P , the holonomy group H ⊂ G of the connection A acts on the Lie algebra g of G. Let us denote by g 0 ⊂ g the subspace on which H acts trivially. Let g 1 be the orthogonal subspace of g 0 with respect to the Killing form of G. The corresponding splitting of the adjoint bundle is given by Ad P = L 0 ⊕ L 1 . By Proposition 3.4 (D −1/3,A ) 2 is self adjoint and hence respects the decomposition We use the shorthand Λ i L j for Λ i M ⊗ L j where i, j = 0, 1. For compact M we have the following proposition.
Proposition 3.5. Let A be a G 2 -instanton on a principal G-bundle P with holonomy group H and suppose Ad P splits as above. Then But since the action of the holonomy group of A fixes no non-trivial elements in g 1 and the holonomy group of ∇ −1 acts trivially on Λ 0 we get µ = 0.
By the definition of L 0 the curvature F A acts trivially on ǫ · η in (3.8) and we get, For proving (ii) we already know that ker The reverse inclusion can be seen using the fact that since D −1/3,A and (D −1/3,A ) 2 commute they have the same eigenvectors.
Remark 3.6. Note that part (i) for the above proposition holds only for D −1/3,A and not for any other D t,A where t = −1/3 since the proof explicitly uses the fact that η is parallel with respect to ∇ −1 . But since D t,A is self adjoint for all t ∈ R, for any λ ∈ R we have the following decomposition The above proposition has the following important consequence. If the structure group G is abelian H acts as identity on the whole of g which means g 1 = 0 and L 1 is trivial. Thus by Remark 3.3 the space of infinitesimal deformations of the G 2 -instanton A which is isomorphic to In [BO19, Proposition 24] the authors prove that the G 2 -instanton A is rigid if all the eigenvalues of the operator are smaller than 6. We prove the lower bound for the eigenvalue as follows. Let λ be the smallest eigenvalue of L A . If ǫ ∈ Γ(T * M ⊗ Ad P ) is an infinitesimal deformation of A then from (3.8) and Theorem 3.2 we know that Taking the inner product with ǫ · η on both sides we get that if λ > min 5t 2 +18t−51 12 | t ∈ R = − 28 5 then ǫ = 0 is the only solution. Thus we get the following result. Theorem 3.7. Any G 2 -instanton A on a principal G-bundle over a compact nearly G 2 manifold M is rigid if (i) the structure group G is abelian, or (ii) the eigenvalues of the operator L A are either all greater than − 28 5 or all smaller than 6. Some immediate consequences of Theorem 3.7 are that the flat instantons are rigid. Also if all the eigenvalues of L A are equal then A has to be rigid.
4 Instantons on homogeneous nearly G 2 manifolds 4.1 Classification of homogeneous nearly G 2 manifolds By the classification result in [FKMS97] there are six compact, simply connected homogeneous nearly G 2 manifolds: . We describe the homogeneous structure on each of these spaces.
-In the round S 7 the embedding of G 2 in Spin(7) is obtained by lifting the standard embedding of G 2 into SO(7).
The first four homogeneous spaces are normal, and for those the nearly G 2 metric g on G/H is related to the Killing form B of G by g = − 3 40 B. The choice of the scalar constant 3 40 is based on our convention τ 0 = 4. The general formula for the constant was derived in [AS12, Lemma 7.1]. In the remaining two homogeneous spaces the nearly G 2 metric is not a scalar multiple of the Killing form of G (see [Wil99]). Let m be the orthogonal complement of the Lie algebra h of H in g with respect to g. Then m is invariant under the adjoint action of h that is, [h, m] ⊂ m and thus all the six homogeneous nearly G 2 manifolds are naturally reductive. The reductive decomposition g = h ⊕ m equips the principal H-bundle G → G/H with a G-invariant connection whose horizontal spaces are the left translates of m. This connection is known as the characteristic homogeneous connection. On homogeneous nearly G 2 manifolds the characteristic homogeneous connection has holonomy contained in G 2 . If we denote by Z m the projection of Z ∈ g on m, the torsion tensor T for any X, Y ∈ m is given by and is totally skew-symmetric. Thus by the uniqueness result in [CS04] it is the canonical connection with respect to the nearly G 2 structure on G/H [HN12]. The canonical connection is a G 2 -instanton as proved in [HN12, Proposition 3.1].
The adjoint representation ad : H → GL(m) gives rise to the associated vector bundle G× ad m on G/H. Similarly since G/H has a nearly G 2 structure we have the adjoint action of G 2 on m which we again denote by ad and the isotropy homomorphism λ : H → G 2 which we can use to construct the associated vector bundle G × ad •λ m. The canonical connection is a connection on both G × ad m and G × ad •λ m with structure group H and G 2 respectively. Therefore it is natural to study the infinitesimal deformation space of the canonical connection in both these situations. Since H ⊂ G 2 , the deformation space as an H-connection is a subset of the deformation space as a G 2 -connection.
We can completely describe the deformation space when the structure group is H but for structure group G 2 we can only find the deformation space for the normal homogeneous nearly G 2 manifolds listed in Table 1 since our methods do not work for non-normal homogeneous metrics. However since H is abelian in both of the non-normal cases Theorem 3.7 tells us that the canonical connection is rigid as an H-connection. But we cannot say anything about the deformation space for the structure group G 2 in those two cases.
Thus the only cases left to consider are listed in Table 1. The remainder of this article is devoted to computing the infinitesimal deformation space of the canonical connection with the structure group H and G 2 for the homogeneous spaces listed in Table 1.

Infinitesimal deformations of the canonical connection
which is also known as the induced G-representation Ind G H V . For any connection A on G the covariant derivative associated to A on any bundle associated to A is denoted by ∇ A . Let s ∈ Γ(G × ρ V ) and f s : G → V be the G-equivariant function given by For the canonical connection on G → M , X h = X for every vector field. Thus the covariant derivative ∇ can is given by By the Peter-Weyl Theorem [Kna86, Theorem 1.12] the space of sections can also be formulated as follows. If we denote by G irr the set of equivalence classes of irreducible Hrepresentations then Claim: The left G-action is given by g.
The proof of the claim is now complete.
We can compute the covariant derivative on The above can be written as Thus we get that for the canonical connection the covariant derivative of a section s ∈ Γ(G × ρ V ) with respect to some X ∈ m translates into the derivative X(f s ), which is minus the differential of the left-regular representation (ρ L ) * (X)(f s ), see [MS10]. Note that Cas m is just used for notational convenience and as m may not be a Lie algebra apriori. Also in Cas h the trace is taken over H. To study the deformation space of the canonical connection ∇ can on these homogeneous spaces we rewrite the Schrödinger-Lichnerowicz formula (3.8) in terms of the Casimir operator of h and g and then use the Frobenius reciprocity formula to compute the deformation space of the canonical connection in each case. Let F be the curvature associated to ∇ can then the operator −2ǫ F can be reformulated in terms of Cas h by doing similar calculations as in [CH16,Lemma 4 Let (E, ρ E ) be an H-representation. We denote the tensor product of representations on m * and E by ρ m * ⊗E . For every t ∈ R, D t,A denotes the Dirac operator on G × ρ m * ⊗E (m * ⊗ E) ⊗ / S associated to the connection ∇ A and ∇ t on G × ρ m * ⊗E (m * ⊗ E) and / S respectively. From now on we use the same symbol to denote the Lie group representation and the associated Lie algebra representation wherever there is no confusion. On a naturally reductive space Kostant's formula for cubic Dirac operator relates the square of the Dirac operator to suitable Casimir operators and scalar terms (see [Kos99,Agr03,MS10]). We now use Proposition 3.4 to prove a similar result Proposition 4.2. Let ∇ can be the canonical connection on a homogeneous nearly G 2 manifold M = G/H. Let (E, ρ E ) be an H-representation and ǫ be a smooth section of G × ρ m * ⊗E (m * ⊗ E).
Proof. We begin by analyzing the rough Laplacian term in the Schrödinger-Lichnerowicz formula for (D −1/3,can ) 2 ǫ · η from (3.8) and then substitute the F -dependent term from (4.2) in the same. We denote by ρ L the left regular representation of G. From above calculations we know that at the center of a normal orthonormal frame acts as −Ric of the canonical connection on 1-forms which is equal to − 16 3 id from Proposition 2.3. Substituting all the terms in (3.8) for t = −1 we get which completes the proof.
Since all the homogeneous spaces considered in Table 1 are naturally reductive and H ⊂ G 2 , there is an adjoint action of H on m, h and g 2 and thus H-representations on m * ⊗ h and m * ⊗ g which we denote by ρ m * ⊗h , ρ m * ⊗g 2 . The corresponding Lie algebra representations are denoted similarly. The infinitesimal deformation space of the instanton ∇ can is a subspace of Γ(m * ⊗ E) where E can be either h or g 2 .
From Propositions 3.1 and 4.2 it is clear that if ǫ is an infinitesimal deformation of ∇ can on the bundle m * ⊗ E over G/H then where the trace in both the Casimirs is taken over G.
Using (4.4) we can reformulate the infinitesimal deformation space of the canonical connection. Since the Casimir operator acts as scalar multiple of the identity on irreducible representations we can solve (4.4) for irreducible subrepresentations of L. From Theorem 3.2 the deformations of the canonical connection are the −2 eigenfunctions ǫ · η of D −1,can . To explicitly compute the deformation space first we need to find the solutions for (4.4) which by above proposition is identical to the space of 49 9 eigenfunctions ǫ · η of (D −1/3,can ) 2 . For α ∈ Λ 1 Ad P by Lemma 2.2 D t,A α · η = D 0,A α · η + t 2 ϕ · α · η = D 0,A α · η − t 2 α · η. Therefore the ± 7 3 eigenfunctions ǫ ·η of D −1/3,can correspond to the −2 and 8 3 eigenfunction of D −1,A respectively. By Proposition 3.5 we have the following decomposition The first summand on the right hand side is isomorphic to the space of infinitesimal deformations of ∇ can by Theorem 3.2. So in the second step we check which of the subspaces in ker((D −1/3,can ) 2 − 49 9 id) ∩ (Γ(m * ⊗ E) · η) lie in the −2 eigenspace of D −1,can . The Killing spinor η is parallel with respect to ∇ −1 therefore by the definition of the Dirac operator and Proposition 3.6 we can restrict D −1,can and (D −1/3,can ) 2 to operators from Γ(m * ⊗ E) → Γ(m * ⊗ E). On a homogeneous space we can explicitly compute the canonical connection as we describe below.
Step 1: Calculating ker For each V i we find all the complex irreducible G-representations W i,j , j = 1 . . . n i , that satisfy the equation In order to see whether Because of Schur's Lemma this multiplicity is given by dim(Hom(W i,j , m * C ⊗V i ) H ). Repeating this process for all the i, j's and summing over all irreducible G-representations W i,j along with their multiplicity we get, (4.6) Step 2: Calculating ker(D −1,can + 2id) ∩ Γ(m * ⊗ E) : To figure out which of the W i,j 's found in Step 1 are in the ker(D −1,can + 2id) we need to calculate the covariant derivative ∇ can on Hom then Hom(W, m * ⊗ E) H is non-trivial. By Schur's Lemma the dimension of Hom(W, m * ⊗ E) H is the number of common irreducible H-subrepresentations in Res H G W and m * ⊗ E. Let W α be such a common irreducible H-representation. We denote by V | U the subspace of V isomoprohic to U then Hom(W | Wα , (m * ⊗ E)| Wα = Span{φ α }. Let τ * be the Lie algebra g representation associated to the G-representation Using this we can calculate the Dirac operator at eH by (4.7) The above method can be extended by linearity to compute the Dirac operator on Γ(m * ⊗ E). Note that we have omitted the Killing spinor η since it is parallel with respect to η so does not effect the eigenspace.
In the following sections we implement the above procedure on each of the four homogeneous spaces.
In the case of nearly G 2 manifolds / D and the 7-dimensional vol commute and thus we do not have such an isomorphism between the ±λ eigenspaces of the Dirac operator. In fact there is no such automatic relation between ker( / D 2 − λ 2 id) and ker( / D + λid) as §4.4 reveals.
Remark 4.4. The Dirac operator is always self-adjoint therefore the above method of finding a particular eigenspace of a Dirac operator D can be used more generally in any bundle associated to the spinor bundle over a homogeneous spin manifold. Often times it is easier to find the eigenspaces of the square of the Dirac operator D 2 similar to the case in hand. Once we know the λ 2 -eigenspace of D 2 we can apply D on them to see which of them lie in the λ or −λeigenspace of D.

Eigenspaces of the square of the Dirac operator
In this section we follow Step 1 of the above procedure. To see which of the irreducible representations of G satisfy (4.4), we need to compute the Casimir operator on complex irreducible representations. Given any irreducible representation ρ λ with highest weight λ we use the Freudenthal formula to compute ρ λ (Cas g ). We drop the constant 40 3 in our definition of Casimir operator for this section as it does not play any role in comparing the Casimir operators. Let µ = 1 2 (sum of the positive roots of g) then the Freudenthal formula states that ρ λ (Cas g ) = B(λ, λ) + 2B(µ, λ). The adjoint representation g 2 is the unique 14-dimensional irreducible representation of G 2 . The complex irreducible representations of G 2 are identified with respect to their highest weights of the form (p, q) ∈ Z 2 ≥0 and are denoted by V (p,q) . Here V (1,0) is the 7-dimensional standard G 2 -representation and V (0,1) is the 14-dimensional adjoint representation. The reductive splitting of the Lie algebra is given by spin(7) = g 2 ⊕ m.
But since there are no positive integral solutions of this equation there are no deformations of the canonical connection on Spin(7)/G 2 .
But since there are no integral solutions for the equation the deformation space is trivial in this case.
Case 2: E = g 2 The adjoint representation of (g 2 ) C splits as an so(3) representation into S 2 C 2 ⊕ S 10 C 2 . The first component in the splitting has already been studied in case 1 and hence has no contribution to the deformation space. For the second component ρ 10 (Cas so(3) ) = 1.
Under the embedding given above a Cartan subalgebra of sp(1) u , sp(1) d is given by Span{H 1 } and Span{(E 2 , E 3 )} respectively. Let P, Q be the standard 2-dimensional representation of sp(1) u , sp(1) d respectively. Then the unique (n + 1)−dimensional irreducible sp(1) u (respectively sp(1) d ) representation is given by S n P (respectively S n Q). From previous calculations we have B(H 1 , H 1 ) = −12 thus the eigenvalue of Cas sp(1)u on S n P is given by ρ n (Cas sp(1)u ) = 1 12 (n 2 + 2n).
Using the embeddings of su(2) and u(1) given above we see that Cartan subalgebras of su(2) and u(1) in su(3)⊕su(2) are given by span{(E 1 , E 3 )} and span{H 2 } respectively. By calculations completely analogous to the previous case we then get that if we represent the irreducible (n+1)dimensional su(2) d representations by S n W where W is the standard su(2) d representation and the 1-dimensional u(1) representation with highest weight k by F (k) we get by the Freudenthal formula (4.8) ρ n (Cas su(2) d ) = 1 20 (n 2 + 2n), ρ k (Cas u(1) ) = 1 36 k 2 .
As su(2) d ⊕ u(1) representations the 7-dimensional space m C decomposes as whereas the 3-dimensional adjoint representation of (su(2) d ) C is irreducible and hence is isomorphic to S 2 W .
The adjoint representation su(2) d ⊕ u(1) splits as irreducible su(2) d ⊕ u(1) representations as follows: Since U (1) is abelian we know by Theorem 3.7 that the component u(1) is abelian and thus gives rise to no deformations of the canonical connection. Therefore we only need to check for deformations corresponding to S 2 W . For that we need to look for representations V (m 1 ,m 2 ,l) such that 1 9 (m 2 1 + m 2 2 + m 1 m 2 + 3m 1 + 3m 2 ) + 1 8 (l 2 + 2l) = 8 20 , which as seen before has no integral solutions. Hence the canonical connection admits no deformations in this case.

Eigenspaces of the Dirac operator
All the G-representations listed in Table 2 lie in ker((D −1/3,can ) 2 − 49 9 id) ∩ Γ(m * ⊗ E) which by (4.5) is equal to (ker(D −1,can + 2id) ⊕ ker(D −1,can − 8 3 id)) ∩ ∩Γ(m * ⊗ E). Since the canonical connection is translation invariant it takes an irreducible G-representation to itself. Hence the irreducible subspaces found in Table 2 lie in either ker(D −1,can − 8 3 id) or ker(D −1,can + 2id) where the subspaces in the latter space constitute the infinitesimal deformations of the canonical Homogeneous space h g 2 Spin(7)/G 2 0 0  Table 2 lies in ker(D −1,can + 2id) for each of the homogeneous spaces. for all the homogeneous spaces G/H in Table 1 the metric corresponding to the nearly G 2 structure ϕ is given by − 3 40 B where B is the Killing form of G. For 1-forms X, Y the Clifford product between X and Y · η is given by (4.9) Thus we have all the ingredients in (4.7) to calculate the action of the Dirac operator D −1,can on each irreducible subspace in Table 2.
Any section of the bundle associated to m * ⊗ g 2 in ker((D −1/3.can ) 2 − 49 9 id) can be represented by (α, v) for some v ∈ V (0,2) | S 6 C 2 ∼ = m * C . The action of the canonical connection on such a section is then given by ∇ −1,can where the Lie bracket is in so(5). We can now calculate the action of the Dirac operator, D −1,can on (α, e 1 ) · η at the point eH as follows. We omit the ·η from the computations to reduce notational clutter and will continue to do so in every case.
Remark 4.5. We can immediately see from above that the only other possible eigenvalue for which sp(2) is an eigenspace of D −1,can is − 8 3 for c 2 = − 2 3 c 1 . This shows that not all spaces in ker((D −1/3,can ) 2 − 49 9 id) are in ker(D −1,can + 2id).
From our previous work we know that the canonical connection has no deformations as an SU(2) × U(1) connection so we only have to consider the case E = g 2 .
As an SU(2) × U(1) representation, . We have already seen that S 2 W gives rise to no deformations. From previous calculations we know that ker 1,0) respectively. Therefore there are 6 subspaces of Γ(m * ⊗ g 2 ) to consider here.
Summary of the results: For three out of the four considered normal homogeneous spaces the canonical connection is rigid as an H-connection. As a G 2 -connection the canonical connection has a non-trivial infinitesimal deformation space except for the round S 7 . Summing up all the results found above we get the following theorem.
Theorem 4.6. The infinitesimal deformation space for the canonical connection on the four normal homogeneous nearly G 2 spaces G/H when the structure group is H or G 2 is isomorphic to where V (0,1) is the unique 5-dimensional complex irreducible Sp(2)-representation.

Integrability of the deformation spaces
We now describe some of the deformation spaces obtained in Theorem 4.6. Let M be a nearly G 2 manifold. We first observe that for the structure group G 2 the space of non-trivial deformations in Theorem 4.6 are either isomorphic to or contains as a subrepresentation one or multiple copies of the Lie algebra g of the automorphism group G. A vector field X on M preserves the G 2 -structure ϕ if L X ϕ = 0. We denote by X the space of vector fields on M preserving the G 2 -structure. Since the G 2 -structure on G/H is G invariant, the space g is contained in X . Note that if X ∈ X then L X ψ = L X g = 0.
Given a parallel section in Γ(g 2 (T * M )⊗Ad P ) ⊂ Γ(Λ 2 T * M ⊗Ad P ), one can define an operator that associates to each vector field in X an infinitesimal deformation of a G 2 instanton on M . Such an operator was defined in [CH16] where a similar situation arises when one computes the deformation space of the canonical connection on the homogeneous 6-dimensional nearly Kähler manifolds.
The next proposition asserts that if we fix a section ξ ∈ Γ(g 2 (T * M ) ⊗ Ad P ) ⊂ Γ(Λ 2 T * M ⊗ Ad P ), then for any vector field X ∈ X on M the Ad P valued 1-form ǫ X = i X ξ ∈ Γ(T * M ⊗ Ad P ) defines an infinitesimal deformation of the nearly G 2 instanton A in the sense of (3.2). The proof of the proposition follows verbatim from the proof of [CH16, Proposition 9] and is hence omitted.
The above proposition implies that for each ξ ∈ Γ(g 2 (T * M ) ⊗ Ad P ) such that ∇ −1,A ξ = 0, there is a copy of g in the deformation space of A. Thus the multiplicity of g in the deformation space can be found by identifying the parallel sections of g 2 (T * M ) ⊗ Ad P . On G/H, when we see P as a G 2 -bundle, every parallel section of g 2 (T * M ) ⊗ Ad P corresponds to an H-invariant element of the H-representation g 2 ⊗ g 2 (since Ad P ∼ = g 2 ) and vice-versa. The number of linearly independent H-invariant elements of g 2 ⊗ g 2 is equal to the multiplicity of the trivial H-representation in g 2 ⊗ g 2 .
Observe that since A is a G 2 instanton, the curvature F A ∈ Γ(g 2 (T * M ) ⊗ Ad P ). When A = ∇ can is the canonical connection on G/H and F is the curvature, ∇ −1,can F = 0 since Hol(∇ can ) ⊆ G 2 . Hence by Proposition 4.7 for every X ∈ X , ǫ X = i X F defines an infinitesimal deformation of A = ∇ can . Using the Bianchi identity and the definition of ǫ X we have that Since under the action of a gauge transformation φ, the curvature F transforms by φF φ −1 , for all X ∈ X there exists an infinitesimal gauge transformation φ X such that Also i X A defines an infinitesimal gauge transformation, hence [φ X + i X A, F ] is an action of an infinitesimal gauge transformation on F . Thus for all X ∈ X the deformations i X F arise from gauge transformations and hence do not descend to the moduli space.
Thus for finding the multiplicity of g in the deformation space (modulo gauge transformations) of the canonical connection on G/H, we need to find the number of trivial sub-representations of H in g 2 ⊗ g 2 apart from the one that corresponds to F . In all the cases we consider, the trivial H-representation occurs with multiplicity one in the subrepresentation g 2 ⊗ h of g 2 ⊗ g 2 . The trivial representation coming from g 2 ⊗ h corresponds to the H-invariant element F . We deal with the four normal homogeneous spaces one by one. The notation for the irreducible H-representations in all the cases is the same as used in §4.3.
There are two trivial components occurring in the above decomposition from S 2 C 2 ⊗ S 2 C 2 and S 10 C 2 ⊗ S 10 C 2 respectively but since the component coming from S 2 C 2 ⊗ S 2 C 2 corresponds to F , up to gauge transformations the deformation space of the canonical connection on SO(5)/SO(3) contains only one copy of g = so(5) as shown in Theorem 4.6.
The trivial sp(1) ⊕ sp(1) components of g 2 ⊗ g 2 coming from S 2 P ⊗ S 2 P and S 2 Q ⊗ S 2 Q correspond to F and thus can be ignored. The only trivial component that corresponds to an infinitesimal deformation modulo gauge transformations comes from P S 3 Q ⊗ P S 3 Q, hence again g = sp(2) ⊕ sp(1) appears with multiplicity 1 in the deformation space which is consistent with our findings in Theorem 4.6.
The first two components in the above decomposition correspond to h hence the only trivial su(2) ⊕ u(1)-subrepresentations of g 2 ⊗ g 2 that correspond to non-trivial deformations come from the spaces S 2 W F (3) ⊗ S 2 W F (−3) and F (6) ⊗ F (−6). Hence as proved in Theorem 4.6 the space g = su(3) ⊕ su(2) occurs in the deformation space with multiplicity 2.
On the squashed 7-sphere the canonical connection splits into two connections with Hol = Sp(1) u and Sp(1) d respectively. From §4.3 the deformations only come from the Sp(1) u part which is the pullback of the standard instanton on S 4 . If we view S 4 as the symmetric homogeneous space Sp(2)×Sp(1) Sp(1)a×Sp(1) b ×Sp(1)c and denote by P, Q, R ∼ = C 2 the irreducible representation of the three Sp(1) factors respectively we have the orthogonal decomposition sp(2) ⊕ sp(1) = sp(1) a ⊕ sp(1) b ⊕ sp(1) c ⊕ n.
The squashed sphere becomes a bundle over S 4 by reducing to the subgroup Sp(1) 2 corresponding to the identification Q = R so the factor Sp(1) d acts diagonally. The complexified tangent space of Sp(2)×Sp(1) Sp(1)u×Sp(1) d is then m ∼ = S 2 Q + P Q.
The standard instanton on S 4 is the unique Sp(2)-invariant ASD connection on S 4 with charge 1. As a bundle over S 4 , the Levi-Civita connection induces the standard instanton on P . It is also the homogeneous connection on the Spin(4) = Sp(1) 2 bundle over S 4 obtained by left-translating the subspace n in sp(2) ⊕ sp(1) = 3sp(1) ⊕ n by Sp(2) × Sp(1). Thus the horizontal distribution corresponding to the standard instanton is n.
On the other hand the canonical connection on the squashed 7-sphere is the characteristic homogeneous connection defined by the horizontal distribution m in the decomposition sp(2) ⊕ sp(1) = 2sp(1)⊕m = 2sp(1)⊕(S 2 Q⊕n). The canonical connection on squashed 7-sphere reduces to Sp(1) 2 and preserves the horizontal distribution D defined by n which is stable under both Ad(Sp(1) 3 ) and Ad(Sp(1) 2 ).
Let M be the moduli space of charge-1 instantons on S 4 with structure group SU(2). Then, there is a diffeomorphism from M to B 5 ⊂ R 5 which to an instanton associates its center. The standard instanton on S 4 is the charge-1 instanton that corresponds to the center of the ball, that is to 0 ∈ B 5 , and is the unique homogeneous charge-1 instanton. As the name suggests, the homogeneous charge-1 instanton is invariant with respect to the Sp(2)-action. The pullback of the homogeneous charge-1 instanton to the squashed S 7 is a G 2 -instanton (see [BO19], [Cla14]). As shown in [AHS78] the moduli space of the standard instanton on S 4 can be identified as a topological space and as a differentiable manifold with R + × H (see [DK90, sec 4.1]). As shown above the Sp(1) part of the canonical connection on the squashed 7-sphere is the pullback of the standard instanton, hence the deformation space of the canonical connection on the squashed 7-sphere must contain the deformation space of the standard ASD instanton on S 4 and thus be at least 5-dimensional. From Table 4.6 we know that the moduli space of the deformations of the canonical connection on the squashed 7-sphere is exactly 5-dimensional and hence we get the following Corollary.
Theorem 4.8. The deformations of the canonical connection on the squashed 7-sphere are lifts of the deformations of the standard ASD connection on S 4 and are thus integrable.
As of the deformation subspace isomorphic to 2su(3) of the canonical connection on SU(3) × SU(2)/SU(2) × U(1) with structure group G 2 , the author is unaware of any such explicit description. It would be interesting to see whether these deformations are genuine.