Deformations of G2-Instantons on Nearly G2 Manifolds

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dϕ = τ ϕ, 0 ∗ϕ then the 3-form ϕ deﬁnes a nearly parallel G2 structure. The existence of such a G2 structure on a manifold was shown to be equivalent to the existence of a spin structure with a real Killing spinor in [BFGK91]. Nearly G2 manifolds are manifolds with a nearly parallel G2 structure. Friedrich– Kath–Moroianu–Semmelmann in [FKMS97] showed that excluding the case of the 7-dimensional sphere there are three types of nearly parallel G2 structures depending on the dimension of the space K / of all Killing spinors . The dimension of K / is bounded above by 3, giving rise to the S S three diﬀerent types,

1. dim(K /) = 1: type 1 or proper nearly G manifolds. S 2 2. dim(K /) = 2: type 2 or Sasaki-Einstein manifolds. S 3. dim(K /) = 3: type 3 or 3-Sasakian manifolds. S

Nearly G2 manifolds were introduced as manifolds with weak holonomy G2 by Gray in [Gra71]. These manifolds are positive Einstein. Some examples of such manifolds are

type 1: (S7, round), (S7, squashed), the Aloﬀ–Wallach spaces N , k = l k,l 6 type 2: N1,1, Stiefel manifold V5,2 SU(3)×SU(2) type 3: SU(2)×U(1)

The cones over these manifolds have holonomy contained in Spin(7), speciﬁcally Spin(7), SU(4), Sp(2) when the nearly parallel G2 structure on the link is of type 1, 2, 3 respectively. This property makes these spaces particularly important in the construction and understanding of manifolds with torsion free Spin(7)-structures. 7 Let M be a manifold with a G2 structure ϕ and let η be the Killing spinor associated to ϕ. A connection A on M is a G2-instanton if its curvature FA satisﬁes the algebraic condition

F ϕ = F . A ∧ ∗ϕ A The above condition is equivalent to F η = 0 as shown in §3. When the G structure is parallel A · 2 (the case when the constant τ0 = 0) these instantons clearly solve the Yang–Mills equation ∗ d∇F = 0. The analogous result was proved in the nearly G2 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 G2 manifolds important from the point of view of gauge theory in higher dimensions. However G2-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 G2-instantons on parallel G2 manifolds were constructed in [Cla14], [Wal16] and [SEW15]. In [BO19] the authors proved the existence of nearly G2-instantons on certain Aloff–Wallach spaces and classified invariant G2-instantons on these spaces with gauge group U(1) and SO(3). Recently, Waldron [Wal20] proved that the pullback of the standard instanton on S7 obtained from ASD instantons on the 4-sphere via the quaternionic Hopf fibration lies in a smooth, complete, 15-dimensional family of G2-instantons. In [CH16] Charbonneau–Harland studied the infinitesimal deformation space of irreducible instantons with semi-simple structure group on nearly Kähler 6-manifolds by identifying it with

2 Ragini Singhal January 7, 2021 the eigenspace of a Dirac operator. In this article, we investigate the infinitesimal deformation space of G2-instantons on nearly G2 manifolds by applying a similar approach. A significant difference between nearly G2 manifolds and the 6 dimensional nearly Kähler manifolds is that the Killing spinors η, vol η are linearly dependent in the former and independent in the latter · case. This prevents us from having a result like [CH16, Proposition 4(iii)] where the authors relate the λ2-eigenspace of the square of the Dirac operator to the λ-eigenspace of the Dirac which makes the computation of the infinitesimal deformation space much more convenient. In fact we show in §4 that such a relation does not exist in the nearly G2 case by explicitly computing the kernel of the elliptic operator for the homogeneous nearly G2 manifolds. In [Dri20] the author uses the spinorial approach to describe the deformation space of instantons on asymptotically conical G2 manifolds. In §2 we describe a 1-parameter family of connections on the spinor bundle / over nearly S G2 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 G2 manifolds and proved the obstructedness of infinitesimal deformations of the nearly G2 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. We prove the following main theorems for a nearly G -instanton A on a principal bundle 2 P with curvature F . Let EM be a vector bundle associated to and the Dirac operator D−1,A A P is as defined in (3.3).

Theorem 3.2. The space of inﬁnitesimal deformations of a G2-instanton A on a principal bundle over a nearly G manifold is isomorphic to the kernel of the elliptic operator P 2 D−1,A + 2 Id : Γ(Λ1(/) EM) Γ(Λ1(/) EM). S ⊗ → S ⊗

Theorem 3.7. Any G2-instanton A on a principal G-bundle over a compact nearly G2 manifold M is rigid if

(i) the structure group G is abelian, or

(ii) all the eigenvalues of the operator

1 1 L : Λ AdP Λ AdP A ⊗ → ⊗ w 2wyF 7→ − A are greater than 28 . − 5 A similar result as above has been proved in [BO19, Proposition 8] when the structure group is abelian or the eigenvalues are less than 6. For the proof of the upper bound the authors used the Weitzenböck formula on the connection associated to the Levi-Civita connection and A. In contrast, the proof of the lower bound on the eigenvalue uses the Schrödinger–Lichnerowicz formula for the family of Dirac operators constructed in §3. We describe the infinitesimal deformation space of the canonical connection on all the homogeneous nearly G2 manifolds whose nearly G2 metric is normal. By considering the actions of the Lie groups H and G2 on G/H we can view the canonical connection as an H-connection or a G2-connection. We compute its infinitesimal deformation spaces in both of these cases. The results are recorded in Theorem 4.6. It will be interesting to see if these infinitesimal deformations are genuine. As of now, the author is unaware of any known family of nearly G2-instantons for which the infinitesimal deformations are the ones found in Theorem 4.6.

3 Deformations of Nearly G2-instantons January 7, 2021

This article is organized as follows. In §2 we give some preliminaries on nearly parallel G2 structures to set up conventions and notations. In §3 we introduce the inﬁnitesimal deformation space of nearly parallel G2-instantons on nearly G2 manifolds and prove that it is isomorphic to the kernel of a Dirac operator. In §4 we prove Theorem 4.6.

Acknowledgements. The author would like to thank her supervisors Benoit Charbonneau and Spiro Karigiannis for the innumerable discussions, immense support and advice during the project. The author would also like to thank Gonçalo Oliveira for recommending Theorem 3.7(ii) and Uwe Semmelmann for his suggestions. The author is also thankful to Shubham Dwivedi for many useful interactions regarding the project.

2 Preliminaries

2.1 Nearly parallel G2 structures Let M be a 7-dimensional Riemannian manifold equipped with a positive 3-form ϕ Ω3 (M). ∈ + The 3-form ϕ induces an orientation and a metric on M and thus a Hodge star operator on ∗ϕ the space of differential forms (see [Bry87]). The G2 structure ϕ is called a nearly parallel G2 structure on M if it satisfies the following differential equation for some non-zero τ R, 0 ∈ dϕ = τ ϕ. (2.1) 0 ∗ϕ We denote the 4-form ϕ by ψ in the remainder of this article. The condition dϕ = τ ψ implies ∗ϕ 0 dψ = 0, thus the nearly parallel G2 structure ϕ is co-closed.

Every manifold with a G2 structure is orientable and spin, and thus admits a spinor bundle /. Let LC be the Levi-Civita connection of the induced metric on M. A spinor η Γ(/) is a S ∇ ∈ S real Killing spinor if for some non-zero δ R, ∈ LC η = δX η X Γ(T M). (2.2) ∇X · ∀ ∈ There is a one-to-one correspondence between nearly parallel G2 structures and real Killing spinors on M. Given a nearly parallel G2 structure ϕ that satisfies (2.1) there exists a real Killing 1 τ0 τ0 spinor η that satisfies (2.2) with δ = 8 τ0 and vice-versa. Switching 8 to 8 corresponds to − + − changing the orientation of the cone M 2 R . See [BFGK91] and [Bär93] for more details. ×r The constant τ0 can be altered by rescaling the metric and readjusting the orientation. In this article we use τ0 = 4. With this choice of τ0 our nearly G2 structure ϕ and Killing spinor η satisfies the following equations respectively

dϕ = 4ψ, 1 LC η = X η. (2.3) ∇X −2 ·

Manifolds with nearly parallel G2 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 3 2 21 2 ϕ, then the Ricci curvature Ricg = 8 τ0 g and the scalar curvature Scalg = 7Ricg = 8 τ0 . A G2 structure on M induces a splitting of the spaces of diﬀerential forms on M into irreducible G2 representations. The space of 2-forms Λ2(M) decomposes as

Λ2(M) = Λ2(M) Λ2 (M), 7 ⊕ 14

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2 where Λl has pointwise dimension l. More precisely, we have the following description of the space of forms :

Λ2(M)= Xyϕ X Γ(T M) = β Λ2(M) (ϕ β)= 2β , 7 { | ∈ } { ∈ | ∗ ∧ − } Λ2 (M)= β Λ2(M) β ψ = 0 = β Λ2(M) (ϕ β)= β . 14 { ∈ | ∧ } { ∈ | ∗ ∧ } Note that we are using the convention of [Kar09] which is opposite to that of [Joy00] and [Bry06]. 2 The space Λ14 is isomorphic to the Lie algebra of G2 denoted by g2. Since the group G2 2 preserves the G2 structure ϕ, it preserves the real Killing spinor η induced by ϕ. The space Λ14 can be equivalently deﬁned as

Λ2 = ω Λ2 ω η = 0 . (2.4) 14 { ∈ | · } We make use of this identiﬁcation when deﬁning the instanton condition on M in §3.

2.2 The spinor bundle

For a 7-dimensional Riemannian manifold M with a nearly parallel G2 structure ϕ, the spinor bundle / is a rank-8 real vector bundle over M and is isomorphic to the bundle R T M = Λ0 Λ1. S ⊕ ⊕ At each point p M, we can identify the ﬁber of / with R T M = R R7 = Re(O) Im(O)= O. ∈ S ⊕ p ∼ ⊕ ∼ ⊕ If η is the real Killing spinor on M induced by ϕ then we have the isomorphism

/ = (Λ0T M η) (Λ1T M η) = Λ0T M Λ1T M. S · ⊕ · ∼ ⊕ Under this isomorphism any spinor s = (f η, α η) / can be written as s = (f, α) Λ0 Λ1. · · ∈ S ∈ ⊕ The 3-form ϕ induces a cross product on vector ﬁelds X,Y Γ(T M). Throughout this ×ϕ ∈ article we use ei to denote both tangent vectors and 1-forms, identiﬁed using the metric. All the computations are done in a local orthonormal frame e ,...,e and any repeated indices { 1 7} are summed over all possible values. With respect to this local orthonormal frame, we have (X Y ) = X Y ϕ . The octonionic product of two octonions (f , X ) and (f , X ) is given ×ϕ l i j ijl 1 1 2 2 by,

(f , X ) (f , X ) = (f f X , X ,f X + f X + X X ). 1 1 · 2 2 1 2 − h 1 2i 1 2 2 1 1 × 2 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

Y (f, Z)= ( Y,Z ,fY + Y Z). (2.5) · − −h i × Note that the product defined above differs from [Kar10] by a negative sign due to our choice of the representation of Cl on / [LM89, Chapter 1.8]. We define the Clifford multiplication of 7 S any p-form β = β e e e with a spinor by, i1...ip i1 ∧ 2 ∧···∧ ip β (f, X)= β (e (e . . . (e (f, X)) . . .)). · i1...ip i1 · i2 · · ip · We record an identity for Clifford algebras for later use and refer the reader to [LM89, Proposition 3.8, Ch1] for the proof. Proposition 2.1. For α Λp(M) ∈ e α e = ( 1)p+1(n 2p) α. j · · j − − Xj

5 Deformations of Nearly G2-instantons January 7, 2021

The Cliﬀord multiplication between a p-form α and a 1-form v can be written as [LM89, Proposition 3.9]

v α = v α vyα. · ∧ − The vector bundle / is a G -representation and since G is the isotropy group of the 3-form ϕ S 2 2 the map µ ϕ µ from the bundle of spinors / to itself is an isomorphism. The same argument 7→ · S holds for the 4-form ψ. Lemma 2.2. The subbundles of / isomorphic to Λ0 and Λ1 are eigenspaces of the operations of S Cliﬀord multiplication by ϕ and ψ. The associated eigenvalues are

Λ0 Λ1 ϕ 7 1 − ψ 7 1. − Proof. The bundle / is a G -representation. The spaces Λ0, Λ1 are its irreducible subrepre- S 2 sentations and thus are eigenspaces of the operators deﬁned by the Cliﬀord multiplication by ϕ, ψ respectively. By Schur’s Lemma there exist real constants λ0, λ1,µ0,µ1 such that for all f Λ0, α Λ1 ∈ ∈ ϕ f = λ f, ϕ α = λ α, · 0 · 1 ψ f = µ f, ψ α = µ α. · 0 · 1 Proposition 2.1 then implies e ϕ e = ϕ and e ψ e = ψ thus i i · · i i i · · i P P 7 λ f = ϕ f = e ϕ e f, 0 · i · · i · Xi=1 7 µ f = ψ f = e ψ e f. 0 · i · · i · Xi=1 Using the fact that e f Λ1 and summing over i we get i · ∈ λ0 + 7λ1 = 0, (R1)

µ0 + 7µ1 = 0. (R2) We ﬁnd the eigenvalues corresponding to Λ0 by explicit calculations and use relations (R1) and (R2) to show the result for Λ1. Let (f, 0) Λ0 be a spinor. In the local orthonormal frame ∈ e ,...,e , we have ϕ = 1 ϕ e e e , where ϕ is skew-symmetric in each pair of indices. 1 7 6 ijk i ∧ j ∧ k ijk Using (2.5) we get that 1 1 ϕ (f, 0) = ϕ e (e (e (f, 0))) = ϕ e (e (0, fe )) · 6 ijk i · j · k · −6 ijk i · j · k 1 = ϕ e ( fδ , fϕ e ) 6 ijk i · − kj jkt t 1 = ϕ ( fϕ , fδ e + fϕ ϕ e ). −6 ijk − ijk − kj i jkt itp p

By using the skew-symmetry of ϕ and the contraction identities ϕijkϕijl = 6δkl, ϕijkϕijk = 42 (see [Kar09]), we get 1 ϕ (f, 0) = (42f, 6fδ ϕ e ) = (7f, 0). · 6 − it itp p

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Similarly, in above local orthonormal frame, ψ = 1 ψ e e e e and using (2.5) we get 24 ijkl i ∧ j ∧ k ∧ l 1 1 ψ (f, 0) = ψ e (e (e (e (f, 0)))) = ψ e (e (e (0, fe ))) · 24 ijkl i · j · k · l · −24 ijkl i · j · k · l 1 = ψ e (e ( fδ , fϕ e )) 24 ijkl i · j · − kl klp p 1 = ψ e ( fϕ δ , fδ e + fϕ ϕ e ) −24 ijkl i · − klp jp − kl j klp jpt t 1 = ψ (fδ δ fϕ ϕ δ , fϕ δ e fδ ϕ e + fϕ ϕ ϕ e ). 24 ijkl kl ij − klp jpt it klp jp i − kl ijs s klp jpt itr r Here we can use the skew-symmetry of ψ, the contraction identity ψ ϕ = 4ϕ along with ijkl klp − ijp the contraction identities of ϕ mentioned before to obtain 1 1 ψ (f, 0) = (24δ δ f, 0) = (24.7f, 0) = (7f, 0). · 24 il il 24

Substituting λ0 = 7 and µ0 = 7 in Relations (R1), (R2) respectively proves the desired result.

A common feature between nearly Kähler 6-manifolds and manifolds with nearly parallel G2 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 . It was proved by Cleyton–Swann in [CS04, Theorem 6.3] that ∇ ⊂ 2 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 G2 structure.

For the nearly G2 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 ∈ t g( t Y,Z)= g( LC Y,Z)+ ϕ(X,Y,Z). (2.6) ∇X ∇X 3

Let T t be the torsion (1, 2)-tensor of t. Since the connection LC is torsion free ∇ ∇ g(X, T t(Y,Z)) = g(X, t Z) g(X, t Y ) g(X, [Y,Z]) ∇Y − ∇Z − t t = g( LC Z, X)+ ϕ(Y,Z,X) g( LC Y, X) ϕ(Z,Y,X) ∇Y 3 − ∇Z − 3 g(X, [Y,Z]) − 2t = ϕ(X,Y,Z). 3 Therefore the torsion tensor T t is given by 2t T t(X,Y )= ϕ(X,Y, ) (2.7) 3 ·

7 Deformations of Nearly G2-instantons January 7, 2021 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 / as ∇ S t t µ = LCµ + (i ϕ) µ. (2.8) ∇X ∇X 6 X · The space of real Killing spinors is isomorphic to Λ0 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 ϕ, · · − X t t η = 0 η + (i ϕ) η ∇X ∇X 6 X · 1 t = X η (X ϕ + ϕ X) η −2 · − 12 · · · 1 t = X η (7X η X η) −2 · − 12 · − · t + 1 = X η. − 2 · Therefore η is parallel with respect to the connection −1. The connection −1 thus has holon- ∇ ∇ omy group contained in G2 with totally skew-symmetric torsion and is therefore the canonical connection on the nearly G2 manifold M described in [CS04].

Proposition 2.3. The Ricci tensor Rict of the connection t is given by ∇ 2t2 Rict = (6 )g. − 3 Proof. By using the expression of the Ricci tensor for a connection with a totally skew-symmetric torsion from [FI02], we have t 2t2 Rict(X,Y ) = Ric0(X,Y ) d∗ϕ(X,Y ) g(i ϕ, i ϕ) − 3 − 9 X Y The Ricci tensor for the Levi-Civita connection is given by Ric0 = 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 e1,...,e7 using the contraction identity ϕijkϕijl = 6δkl as follows 1 g(i ϕ, i ϕ)= X Y ϕ ϕ g(e e , e e ) X Y 4 k γ ijk αβγ i ∧ j α ∧ β i,j,k,α,β,γX 1 = X Y (ϕ ϕ ϕ ϕ ) 4 k γ ijk ijγ − ijk jiγ i,j,k,γX

= 3 XkYγδkγ = 3g(X,Y ). Xk,γ Summing up all the terms together give the desired identity for Rict.

3 Deformation theory of instantons

Let M be a principal K-bundle. We denote by AdP the adjoint bundle associated to . Let P → 2 ∗ P A be a connection 1-form on and F Γ(Λ T M AdP ) be the curvature 2-form associated P A ∈ ⊗ to A given by 1 F =dA + [A A]. A 2 ∧

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There are many ways to deﬁne the instanton condition on A. If (M, g) is equipped with a G- structure such that G O(n), there is a subbundle g(T ∗M) Λ2T ∗M whose ﬁbre is isomorphic ⊂ ⊂ ∗ to g = Lie(G). The connection A is an instanton if the 2-form part of FA belongs to g(T M). In global terms, A is an instanton if

∗ 2 ∗ F Γ(g(T M) AdP ) Γ(Λ T M AdP ). A ∈ ⊗ ⊂ ⊗

Note that in dimension 7 if M is equipped with a G2 structure then this condition implies that A is an instanton if the 2-form part of F g (T ∗M) = Γ(Λ2 ). A ∈ 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 Λ2 Λ2 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 Λ2 are eigenspaces of the operator. Then FA is an instanton if

( Q F )= νF . ∗ ∗ ∧ A A for some ν R. In dimension 7 it turns out that Q = ψ (see [HN12]) and the above condition ∈ is equivalent to F Γ(Λ2 ) when ν = 1. A ∈ 14 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

F η = 0. A ·

When M has a G2 structure and η is the corresponding spinor, (2.4) implies that a the above condition is satisﬁed if and only if A is a G2-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 G2 structure ϕ all the above 2 deﬁnitions are equivalent. They all imply that the curvature FA associated to A lies in Γ(Λ14) and thus satisﬁes all of these equivalent conditions:

F η = 0, A · FA ϕ = F, ∧ ∗ (3.1) F ψ = 0, A ∧ FA ⌟ ϕ = 0.

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 G2 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 G2-instanton we can prove this fact by direct computation. For an instanton A, (3.1) and the second Bianchi identity imply

(dA)∗F = dA F A ∗ ∗ A = dA(ϕ F ) ∗ ∧ A = 4 (ψ F ) = 0. ∗ ∧ A

9 Deformations of Nearly G2-instantons January 7, 2021

3.1 Inﬁnitesimal deformation of instantons

7 Let M be a nearly G2 manifold. We are interested in studying the infinitesimal deformations of nearly G2-instantons on M. An infinitesimal deformation of a connection A represents an ∗ ∗ infinitesimal change in A and thus, is a section of T M AdP . If ǫ Γ(T M AdP ) is an ⊗ ∈ ⊗ infinitesimal deformation of A, the corresponding change in the curvature FA up to first order is given by dAǫ. A standard gauge fixing condition on this perturbation is given by (dA)∗ǫ = 0. So in total the pair of equations whose solutions define an infinitesimal deformation of an instanton A is given by (dAǫ) η = 0, (dA)∗ǫ = 0. (3.2) · On a nearly G2 manifold we can define a 1-parameter family of Dirac operators t Dt,A = DA + ϕ 2 ·

The 1-parameter family of connections on the spinor bundle / deﬁned in (2.8) and the S connection A on can be used to construct a 1-parameter family of connections on the associated P t,A t vector bundle / AdP . We denote by , the connection induced by and A for all t R S⊗ ∇ ∇ ∈ respectively. We denote by Dt,A the Dirac operator associated to t,A. The following proposition ∇ associates the solutions to (3.2) to a particular eigenspace of Dt,A for each t. The proposition was proved in [Fri12] for t = 0. ∗ t,A Proposition 3.1. Let ǫ be a section of T M AdP , and let D be the Dirac operator constructed ⊗ from the connections t,A for t R. Then ǫ solves (3.2) if and only if ∇ ∈ t + 5 Dt,A(ǫ η)= ǫ η. (3.3) · − 2 · Proof. Let e , a = 1 . . . 7 be a local orthonormal frame for T ∗M. Then { a } D0,A(ǫ η)= e 0(ǫ η) · a ·∇a · = (e 0ǫ) η + e ǫ 0η a ·∇a · a · ·∇a = (dAǫ + (dA)∗ǫ) η + e ǫ 0η. · a · ·∇a Applying Proposition 2.1 to the 1-form part of ǫ we get e ǫ e η = 5ǫ η. So if η is a real a · · a · · Killing spinor then (2.3) together with the above identity imply 1 D0,A(ǫ η)=(dAǫ + (dA)∗ǫ) η e ǫ e η · · − 2 a · · a · 5 = (dAǫ + (dA)∗ǫ ǫ) η. − 2 · It follows from (2.8) and the identity e i ϕ = 3ϕ that a a · a P t Dt,A = D0,A + ϕ 2 · Since ǫ η Λ1 η, by Lemma 2.2 we have · ∈ · t 5 Dt,A(ǫ η)= dAǫ + (dA)∗ǫ + − − ǫ η. · 2 · The equation Dt,A(ǫ η) = t+5 ǫ η is thus equivalent to (dAǫ + (dA)∗ǫ) η = 0, which in · − 2 · · turn is equivalent to the pair of equations (dAǫ) η = 0, (dA)∗ǫ = 0 since these two components · live in complementary subspaces.

10 Ragini Singhal January 7, 2021

Since η is parallel with respect to −1 we can view D−1,A as an operator on Λ1 Ad deﬁned ∇ ⊗ P by D−1,A(ǫ η) = (D−1,Aǫ) η. The following theorem is an immediate consequence of the above · · proposition.

Theorem 3.2. The space of inﬁnitesimal deformations of a G2-instanton A on a principal bundle over a nearly G manifold M is isomorphic to the kernel of the operator P 2 −1,A 1 1 D +2 Id : Γ(Λ AdP ) Γ(Λ AdP ). (3.4) ⊗ → ⊗

t+5 t,A 1 Remark 3.3. By Proposition 3.1, the eigenspace of the operator D on Λ η AdP is − 2 · ⊗ 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 7 ker(D−1/3,A + id) = ker(D−1,A + 2id). (3.5) 3 ∼ We can obtain an expression for the square of the Dirac operators constructed above using the Schrödinger–Lichnerowicz formula in the case of skew-symmetric torsion obtained by Agricola– Friedrich in [AF04]. 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 and µ Γ(/ EM). Let A be P ∈ S ⊗ any connection on . Then for all t R, P ∈ 1 t t2 (Dt/3,A)2µ = ( t,A)∗ t,Aµ + Scal µ + dϕ µ ϕ 2µ + F µ. (3.6) ∇ ∇ 4 g 6 · − 18k k · Proof. Let e ,...,e be an orthonormal frame for the tangent bundle. As before we obtain { 1 7} t Dt,Aµ = (D0,A + ϕ )µ. 2 · Squaring both sides we obtain, t 2 (Dt/3,A)2µ = D0,A + ϕ µ 6 · t t2 = (D0,A)2µ + (D0,A(ϕ µ)+ ϕ D0,Aµ)+ ϕ ϕ µ. 6 · · 36 · · The ﬁrst term of the above expression is given by the Schrödinger–Lichnerowicz formula 1 (D0,A)2µ = ( 0,A)∗ 0,Aµ + Scal µ + F µ. (E1) ∇ ∇ 4 g · The anti-commutator in the second term is given by

D0,A(ϕ µ)+ ϕ D0,Aµ = e 0,A(ϕ µ)+ ϕ e 0,Aµ · · a ·∇a · · a ·∇a = (e 0,Aϕ) µ + (e ϕ + ϕ e ) 0,Aµ a ·∇a · a · · a ·∇a =dϕ µ +d∗ϕ µ 2(e ⌟ ϕ) 0,Aµ (E2) · · − a ·∇a but since M is nearly G2, ϕ is coclosed, therefore

D0,A(ϕ µ)+ ϕ D0,Aµ =dϕ µ 2(e ⌟ ϕ) 0,Aµ · · · − a ·∇a

11 Deformations of Nearly G2-instantons January 7, 2021

For the 3-form ϕ, ϕ ϕ = ϕ 2 (e ⌟ ϕ) (e ⌟ ϕ) and (e ⌟ ϕ) (e ⌟ ϕ)= 3 ϕ 2 + (e ⌟ · k k − a ∧ a a · a − k k a ϕ) (e ⌟ ϕ) which imply ∧ a ϕ ϕ µ = ϕ 2µ (e ⌟ ϕ) (e ⌟ ϕ) µ, · · k k − a ∧ a · = ϕ 2µ ((e ⌟ ϕ) (e ⌟ ϕ) + 3 ϕ 2) µ k k − a · a k k · = 2 ϕ 2µ (e ⌟ ϕ) (e ⌟ ϕ) µ. (E3) − k k − a · a · At the center of a normal frame, t t ( t,A)∗ t,Aµ = ( 0,A + (e ⌟ ϕ))( 0,A + (e ⌟ ϕ))µ ∇ ∇ − ∇a 6 a ∇a 6 a t t = 0,A 0,Aµ (e ⌟ ϕ) 0,Aµ 0,A((e ⌟ ϕ) µ) −∇a ∇a − 6 a ·∇a − 6∇a a · t2 (e ⌟ ϕ) (e ⌟ ϕ) µ − 36 a · a · t t = ( 0,A)∗ 0,Aµ (e ⌟ ϕ) 0,Aµ ( d∗ϕ µ + (e ⌟ ϕ) 0,Aµ) ∇a ∇a − 6 a ·∇a − 6 − · a ·∇a t2 ((e ⌟ ϕ) (e ⌟ ϕ)) µ. − 36 a · a · Again using the fact that d∗ϕ = 0 we get t t2 ( 0,A)∗ 0,Aµ = ( t,A)∗ t,Aµ + (e ⌟ ϕ) 0,Aµ + ((e ⌟ ϕ) (e ⌟ ϕ)) µ. (E4) ∇a ∇a ∇ ∇ 3 a ·∇a 36 a · a · Substituting the three terms in the expression of (Dt/3,A)2µ using (E1), (E2), (E3), (E4) we get the result.

t/3,A 2 When the connection A is an instanton on a nearly G2 manifold the expression for (D ) 2 can be simpliﬁed further. For the G2 structure ϕ, ϕ = 7 and under our choice of convention k k t/3,A 2 0 dϕ = 4ψ and Scalg = 42. Thus we can calculate the action of (D ) on spinors in Λ η and Λ1 η as follows. · Let η Γ(Λ0M EM) be a real Killing spinor then Lemma 2.2 implies ψ η = 7η and ∈ ⊗ · F η = 0 by (3.1). Thus by above proposition we obtain, A · 7 (Dt/3,A)2η = ( t,A)∗ t,Aη (t2 12t 27)η. (3.7) ∇ ∇ − 18 − − Now suppose ǫ is an inﬁnitesimal deformation of A. Then ǫ η Γ(Λ1M EM). From Lemma · ∈ ⊗ 2.2 we know that ψ ǫ η = ǫ η and since F η = 0, F ǫ η = (F ǫ + ǫ F ) η = 2(ǫyF ) η. · · − · · · · · · · − · Thus by above proposition 1 (Dt/3,A)2(ǫ η) = ( t,A)∗ t,A(ǫ η) (7t2 + 12t 189)ǫ η 2(ǫyF ) η. (3.8) · ∇ ∇ · − 18 − · − ·

In the special case when the bundle EM is equal to AdP , the holonomy group H G of the ⊂ connection A acts on the Lie algebra g of G. Let us denote by g g the subspace on which 0 ⊂ H acts trivially. Let g1 be the orthogonal subspace of g0 with respect to the Killing form of G. The corresponding splitting of the adjoint bundle is given by AdP = L L . By Proposition 0 ⊕ 1 3.4 (D−1/3,A)2 is self adjoint and hence respects the decomposition

1 1 0 0 / AdP = (Λ M L ) (Λ M L ) (Λ M L ) (Λ M L ). S⊗ ⊗ 0 ⊕ ⊗ 1 ⊕ ⊗ 0 ⊕ ⊗ 1 We use the shorthand ΛiL for ΛiM L where i, j = 0, 1. For compact M we have the following j ⊗ j proposition.

12 Ragini Singhal January 7, 2021

Proposition 3.5. Let A be a G -instanton on a principal G-bundle with holonomy group H 2 P and suppose AdP splits as above. Then

(i) ker((D−1/3,A)2 49 id) = ker((D−1/3,A)2 49 id) (Λ1L Λ0L ). − 9 − 9 ∩ 1 ⊕ 0 −1/3,A 2 49 1 −1/3,A 7 −1/3,A 7 1 (ii) ker((D ) 9 id) Λ L1 = ker(D + 3 id) ker(D 3 id) Λ L1. − ∩ ⊕ − ∩ Proof. To prove (i) we need to show that ker((D−1/3,A)2 ( 7 )2id) (Λ0L Λ1L ) is trivial. − 3 ∩ 1 ⊕ 0 1. Let µ ker((D−1/3,A)2 ( 7 )2id) Λ0L . Thus we have by (3.7) , ∈ − 3 ∩ 1 7 0= (µ, (D−1/3,A)2 ( )2)µ) ZM − 3 49 7 2 = (µ, ( −1,A)∗ −1,Aµ + ( )µ) ZM ∇ ∇ 9 − 3 = −1,Aµ 2. ZM k∇ k

But since the action of the holonomy group of A ﬁxes no non-trivial elements in g1 and the holonomy group of −1 acts trivially on Λ0 we get µ = 0. ∇ 2. Let ǫ η ker((D−1/3,A)2 ( 7 )2id) Λ1L . By the deﬁnition of L the curvature F acts · ∈ − 3 ∩ 0 0 A trivially on ǫ η in (3.8) and we get, · 7 0= (ǫ η, (D−1/3,A)2 ( )2)ǫ η) ZM · − 3 · 97 7 2 = (ǫ η, ( −1)∗ −1(ǫ η) + ( )ǫ η) ZM · ∇ ∇ · 9 − 3 · 48 = −1(ǫ η) 2 + ǫ η 2 ZM k∇ · k 9 ZM k · k hence ǫ η = 0. · For proving (ii) we already know that ker((D−1/3,A)+ 7 ker((D−1/3,A)+ 7 Λ1L 3 }⊕ 3 } ∩ 1 ⊂ ker((D−1/3,A)2 49 id) Λ1L . The reverse inclusion can be seen using the fact that since D−1/3,A − 9 ∩ 1 and (D−1/3,A)2 commute they have the same eigenvectors. Moreover since D−1/3,A is self adjoint, ǫ µ ker((D−1/3,A)2 49 id) Λ1L implies D−1/3,Aǫ µ = 7 ǫ µ thus the corresponding · ∈ − 9 ∩ 1 k · k 3 k · k eigenvalues of D−1/3,A can only be 7 . ± 3 Remark 3.6. Note that part (i) for the above proposition holds only for D−1/3,A and not for any other Dt,A where t = 1/3 since the proof explicitly uses the fact that η is parallel with 6 − respect to −1. But since Dt,A is self adjoint for all t R, for any λ R we have the following ∇ ∈ ∈ decomposition

t,A 2 2 1 t,A t,A 1 ker (D ) λ id Λ AdP = ker D λid ker D + λid Λ AdP . − ∩ − ⊕ ∩ 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 g1 = 0 and L1 is trivial. Thus by Remark 3.3 the space of inﬁnitesimal deformations of the G2-instanton A which is isomorphic to −1/3,A 7 1 −1/3,A 7 1 ker(D + ) Λ AdP = ker(D + ) Λ L is zero dimensional. 3 ∩ 3 ∩ 1

13 Deformations of Nearly G2-instantons January 7, 2021

In [BO19, Proposition 24] the authors prove that the G2-instanton A is rigid if all the eigenvalues of the operator

1 1 L : Λ AdP Λ AdP A ⊗ → ⊗ w 2wyF 7→ − A are smaller than 6. We prove the lower bound for the eigenvalue as follows. Let λ be the smallest ∗ eigenvalue of L . If ǫ Γ(T M AdP ) is an inﬁnitesimal deformation of A then from (3.8) and A ∈ ⊗ Theorem 3.2 we know that 2 t,A ∗ t,A 5t 3t 17 ( ) ǫ η = + ǫ η LA(ǫ) η. ∇ ∇ · 12 2 − 4 · − · 5t2+18t−51 R Taking the inner product with ǫ η on both sides we get that if λ> min 12 t = · n | ∈ o 28 then ǫ = 0 is the only solution. Thus we get the following result. − 5 Theorem 3.7. Any G2-instanton A on a principal G-bundle over a compact nearly G2 manifold M is rigid if

(i) the structure group G is abelian, or

(ii) the eigenvalues of the operator L are either all greater than 28 or all smaller than 6. A − 5 Some immediate consequences of Theorem 3.7 are that the ﬂat instantons are rigid. Also if all the eigenvalues of LA are equal then A has to be rigid.

4 Instantons on homogeneous nearly G2 manifolds

4.1 Classiﬁcation of homogeneous nearly G2 manifolds

In [FKMS97] the authors classify all the compact, simply connected homogeneous nearly G2 1 manifolds. To exhibit this classification a certain amount of notation must be set. Let Sk,l = diag(eikθ, eilθ, e−i(k+l)θ),θ R denote the embedding of U(1) into SU(3) needed to define the { ∈ } Aloff–Wallach spaces Nk,l. Any homogeneous nearly G2 manifold is one of the six manifolds listed in Table 1. We describe the homogeneous structure on each of these spaces.

7 7 Sp(2)×Sp(1) (S , ground) = Spin(7)/G2, (S , gsquashed)= Sp(1)×Sp(1) , SO(5)/SO(3),

M(3, 2) = SU(3)×SU(2) , N(k, l) = SU(3)/S1 k, l Z, Q(1, 1, 1) = SU(2)3/U(1)2. U(1)×SU(2) k,l ∈

Table 1: Homogeneous nearly G2 manifolds

7 - In the round S the embedding of G2 in Spin(7) is obtained by lifting the standard embedding of G2 into SO(7). - For the squashed metric on S7 the two copies of Sp(1) in Sp(2) Sp(1) denoted by Sp(1) × u and Sp(1)d [AS12] are

a 0 1 0 Sp(1) := , 1 : a Sp(1) , Sp(1) := , a : a Sp(1) . u 0 1 ∈ d 0 a ∈

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SO(5) - In the Steifel manifold V5,2 = SO(3) , the Lie group SO(3) is embedded into SO(5) via the 2 R3 5 dimensional irreducible representation of SO(3) on Sym0( ).

- In SU(3)×SU(2) the embedding of SU(2) (denoted by SU(2) ) and U(1) in SU(2) SU(2) is U(1)×SU(2) d × deﬁned as [AS12]

eiθ 0 0 a 0 SU(2) := , a : a SU(2) , U(1) :=  0 eiθ 0  , 1 : θ R d 0 1 ∈ ∈  0 0 e−2iθ       - In the Aloﬀ-Wallach spaces Nk,l where k, l are coprime positive integers the embedding of 1 Sk,l = U(1)k,l in SU(3) is described

eikθ 0 0 S1 =  0 eilθ 0  ,θ R k,l ∈  0 0 e−i(k+l)θ      3 - In Q(1, 1, 1) we denote the two copies of U(1) inside SU(2) as U(1)u, U(1)d where their respective embeddings are given by

eiθ 0 e−iθ 0 U(1) = Span , , I ,θ R , u 0 e−iθ 0 eiθ 2 ∈ eiθ 0 e−iθ 0 U(1) = Span I , , ,θ R . d 2 0 e−iθ 0 eiθ ∈

The ﬁrst four homogeneous spaces are normal, and for those the nearly G2 metric g on G/H is related to the Killing form B of G by g = 3 B. The choice of the scalar constant 3 is based on − 40 40 our convention τ0 = 4. The general formula for the constant was derived in [AS12, Lemma 7.1]. In the remaining two homogeneous spaces the nearly G2 metric is not a scalar multiple of the Killing form of G (see [Wil99]).

7 7 Sp(2)×Sp(1) (S , ground) ∼= Spin(7)/G2, (S , gsquashed) ∼= Sp(1)×Sp(1) ,

SO(5)/SO(3), M(3, 2) SU(3)×SU(2) . ∼= U(1)×SU(2)

Table 2: Normal homogeneous nearly G2 manifolds

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 homogeneous ⊂ spaces listed in Table 1 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 G2 manifolds the characteristic homogeneous connection has holonomy contained in G . If we denote by Zm the projection of Z g on m, the torsion tensor T for any 2 ∈ X,Y m is given by ∈

T (X,Y )= [X,Y ]m, −

15 Deformations of Nearly G2-instantons January 7, 2021 and is totally skew-symmetric. Thus by the uniqueness result in [CS04] it is the canonical connection with respect to the nearly G2 structure on G/H [HN12]. The canonical connection is a G2-instanton as proved in [HN12, Proposition 3.1]. The adjoint representation ad: H GL(m) gives rise to the associated vector bundle G m → ×ad on G/H. Similarly since G/H has a nearly G2 structure we have the adjoint action of G2 on m which we again denote by ad and the isotropy homomorphism λ: H G which we can use to → 2 construct the associated vector bundle G ◦ m. The canonical connection is a connection on ×ad λ both G m and G ◦ m with structure group H and G respectively. Therefore it is natural ×ad ×ad λ 2 to study the infinitesimal deformation space of the canonical connection in both these situations. Since H G , the deformation space as an H-connection is a subset of the deformation space ⊂ 2 as a G2-connection. We can completely describe the deformation space when the structure group is H but for structure group G2 we can only find the deformation space for the normal homogeneous nearly G2 manifolds listed in Table 2 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 G2 in those two cases. Thus the only cases left to consider are listed in Table 2. The remainder of this article is devoted to computing the infinitesimal deformation space of the canonical connection with the structure group H and G2 for the homogeneous spaces listed in Table 2.

4.2 Infinitesimal deformations of the canonical connection Let M = G/H be a homogeneous manifold. Consider the principal H-bundle G M. If (V, ρ) is → an H-representation then the space of smooth sections Γ(G ρ V ) of the associated vector bundle ∞ × G ρ V is isomorphic to the space C (G, V )H of H-equivariant smooth functions G V . The × ∞ → space C (G, V )H carries the left regular G-representation ρL defined by ρL(g)(f)= g.f = f lg−1 G ◦ which is also known as the induced G-representation IndH 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 V ) and f : G V be the G-equivariant function given by ∇ ∈ ×ρ s → s(gH) = [g,f (g)]. If we denote by X the horizontal lift of X Γ(T M) via A, then A acts s h ∈ ∇ on s as ( A s)(gH) = (g, X (f )(g)). ∇X h s For the canonical connection on G M, X = X for every vector field. Thus the covariant → h derivative can is given by ∇ ( cans)(gH) = (g, X(f )(g)). ∇X s By the Peter–Weyl Theorem [Kna86, Theorem 1.12] the space of sections can also be formulated as follows. If we denote by Girr the set of equivalence classes of irreducible H- representations then

Γ(G V )= Hom(W, V ) W . ×ρ H ⊗ WM∈Girr ∞ The embedding Hom(W, V )H W into C (G, V )H = Γ(G ρ V ) is given by sending (φ, w) to ⊗ −1 × the function f(φ,w) deﬁned by f(φ,w)(g)= φ(τ(g )w). Thus (φ, w) deﬁnes a section s(φ,w)(gH)= [g,f(φ,w)(g)] which we denote by (φ, w) as well.

Claim: The left G-action is given by g.f(φ,w) = f(φ,τ(g)w).

16 Ragini Singhal January 7, 2021

−1 Proof. Let k G. Then since g.f = f l −1 and f (g)= φ(τ(g )w) we have ∈ ◦ g (φ,w) −1 −1 −1 (g.f(φ,w))(k)= f(φ,w)(g k)= φ(τ((g k) )w) −1 = φ(τ(k )τ(g)w)= f(φ,τ(g)w)(k). The proof of the claim is now complete. We can compute the covariant derivative on s Hom(W, V ) W Γ(G V ) by (φ,w) ∈ H ⊗ ⊂ ×ρ can d tX X s(φ,w)(gH)= X(f(φ,w))(g)= f(e g) ∇ dt t=0

d d −tX = (f(φ,w) letX )(g)= (e .f)(g) dt t=0 ◦ dt t=0

d = f(φ,τ(e−tX )w) = f(φ,τ∗(X)w)(gH). dt t=0 −

The above can be written as can (φ, w)= (φ, τ∗(X)w). (4.1) ∇X − Thus we get that for the canonical connection the covariant derivative of a section s Γ(G V ) ∈ ×ρ with respect to some X m translates into the derivative X(f ), which is minus the differential ∈ s of the left-regular representation (ρL)∗(X)(fs), see [MS10]. 3 Let ai, i = 1 ...n be an orthonormal basis of g with respect to g = 40 B then the Casimir { 2} dim G − element Casg Sym (g) is defined by a a . On any g representation (V,µ) we can ∈ i=1 i ⊗ i define the Casimir invariant µ(Casg) gl(PV ) by ∈ n 2 µ(Casg)= µ(ai) . Xi=1

For the reductive homogeneous spaces G/H let ai, i = 1 . . . dim(H) and ai, i = dim(H) . . . dim(G) { }dim(H{) } be the basis of h and m respectively. If we deﬁne Cash = i=1 ai ai and Casm = dim(G) ⊗ a a we can decompose Casg as P dim(H) i ⊗ i P Casg = Cash + Casm.

Note that Casm is just used for notational convenience and as m may not be a Lie algebra apriori. Also in Cash the trace is taken over H. Remark 4.1. If one uses the metric cB instead of B then the Casimir operator is divided − − by the scalar c. 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 ∇ 2ǫyF can be reformulated in terms of Cash by doing similar calculations as in [CH16, Lemma 4] − which gives 2ǫyF = (ρm∗ (Cash) 1 + 1m∗ ρ (Cash) ρm∗⊗ (Cash))ǫ. (4.2) − ⊗ E ⊗ E − E ∗ Let (E, ρE ) be an H-representation. We denote the tensor product of reprentations on m ∗ R t,A ∗ and E by ρm ⊗E. For every t , D denotes the Dirac operator on G ρm∗⊗E (m E) / ∈A t ∗ × ⊗−1/3⊗,canS associated to the connection and on G ∗ (m E) and / respectively. For D ∇ ∇ ×ρm ⊗E ⊗ S we record the following proposition. 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.

17 Deformations of Nearly G2-instantons January 7, 2021

can Proposition 4.2. Let be the canonical connection on a homogeneous nearly G2 manifold ∇ ∗ M = G/H. Let (E, ρE ) be an H-representation and ǫ be a smooth section of G ρm∗⊗E (m E). Then × ⊗ −1/3,can 2 49 (D ) ǫ η = ( ρ (Casg)+ ρ (Cash))ǫ + ǫ) η. (4.3) · − L E 9 · Proof. We begin by analyzing the rough Laplacian term in the Schrödinger– Lichnerowicz formula for (D−1/3,can)2ǫ η from (3.8) and then substitute the F -dependent term from (4.2) in the same. · We denote by ρL the left regular representation of G. From above calculations we know that at the center of a normal orthonormal frame e , i = 1 . . . 7 of m with respect to g = 3 B, { i } − 40 −1,can ∗ −1,can −1,can −1,can 2 ( ) = = ρ (e ) = ρ (Casm). ∇ ∇ −∇ei ∇ei − L i − L H H G ∗ ∗ Since ResG ρL = ResG IndH (m E) ∼= m E we have that ρm∗⊗E(Cash) = ρL(Cash). Also 2 ⊗ ⊗ ρm∗ (e ) = ρm∗ (Cash) acts as Ric of the canonical connection on 1-forms which is equal to i − 16 id from Proposition 2.3. Substituting all the terms in (3.8) for t = 1 we get − 3 −

−1/3,can 2 97 (D ) ǫ η = ( ρ (Casm)ǫ + ǫ + (ρm∗ (Cash) 1 + 1m∗ ρ (Cash) ρm∗⊗ (Cash))ǫ) η · − L 9 ⊗ E ⊗ E − E · 97 16 = ( (ρ (Casm)+ ρ (Cash))ǫ + ( )ǫ + ρ (Cash)ǫ) η − L L 9 − 3 E · 49 = (( ρ (Casg)ǫ + ρ (Cash)ǫ + ǫ) η − L E 9 · which completes the proof.

Since all the homogeneous spaces considered in Table 2 are naturally reductive and H G , ⊂ 2 there is an adjoint action of H on m, h and g and thus H-representations on m∗ h and m∗ g 2 ⊗ ⊗ which we denote by ρm∗⊗h, ρm∗⊗g2 . 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 g2. 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 ⊗

ρE(Cash)ǫ = ρL(Casg)ǫ (4.4) 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 explic- − · itly compute the deformation space first we need to find the solutions for (4.4) which by above 49 −1/3,can 2 1 proposition is identical to the space of eigenfunctions ǫ η of (D ) . For α Λ AdP by 9 · ∈ Lemma 2.2 Dt,Aα η = D0,Aα η + t ϕ α η = D0,Aα η t α η. Therefore the 7 eigenfunctions · · 2 · · · − 2 · ± 3 ǫ η of D−1/3,can correspond to the 2 and 8 eigenfunction of D−1,A respectively. By Proposition · − 3 3.5 we have the following decomposition

49 ker (D−1/3,can)2 id Γ(m∗ E) = ker(D−1,can + 2id) Γ(m∗ E) − 9 ∩ ⊗ ∩ ⊗ (4.5) 8 ker(D−1,can id) Γ(m∗ E) − 3 ∩ ⊗

18 Ragini Singhal January 7, 2021

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 id) (Γ(m∗ E) η) lie in the 2 eigenspace of D−1,can. − 9 ∩ ⊗ · − 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((D−1/3,can)2 49 id) Γ(m∗ E) : − 9 ∩ ⊗ Let EC = n V be the decomposition of EC into complex irreducible H-representations. ⊕i=1 i For each Vi we find all the complex irreducible G-representations Wi,j, j = 1 ...ni, that satisfy the equation

ρVi (Cash)= ρWi,j (Casg).

G ∗ G ∗ In order to see whether Wi,j IndH (m E)C we ﬁnd the multiplicity mi,j of Wi,j in IndH (mC ⊂ ⊗ ∗ ⊗ V ). Because of Schur’s Lemma this multiplicity is given by dim(Hom(W , mC V ) ). Repeating i i,j ⊗ i H this process for all the i, j’s and summing over all irreducible G-representations Wi,j along with their multiplicity we get,

n ni −1/3,can 2 49 ∗ ker((D ) id) Γ(m E)C = m W . (4.6) − 9 ∩ ⊗ ∼ i,j i,j Mi=1 Mj=1

Step 2: Calculating ker(D−1,can + 2id) Γ(m∗ E) : ∩ ⊗ −1,can To ﬁgure out which of the Wi,j’s found in Step 1 are in the ker(D + 2id) we need to can ∗ ∗ calculate the covariant derivative on Hom(W , mC V ) W Γ(m E)C. ∇ i,j ⊗ i H ⊗ i,j ⊆ ⊗ If (W, τ) is an irreducible G-subrepresentation of IndG (m∗ E) then Hom(W, m∗ E) is H ⊗ ⊗ H non-trivial. By Schur’s Lemma the dimension of Hom(W, m∗ E) is the number of common ⊗ H irreducible H-subrepresentations in ResH W and m∗ E. Let W be such a common irreducible G ⊗ α H-representation. We denote by V the subspace of V isomoprohic to U then Hom(W , (m∗ |U |Wα ⊗ E) = Span φ . Let τ∗ be the Lie algebra g representation associated to the G-representation |Wα { α} (W, τ) then for X Γ(T M) and (φ = c φ , w) Hom(W, m∗ E) W , (4.1) ∈ α α ∈ ⊗ H ⊗ can P ∗ (φ, w)(eH) = φ(τ∗(X)w) m E. ∇X − ∈ ⊗ Using this we can calculate the Dirac operator at eH by

7 7 −1,can −1,can D (φ , w)(eH)= e (φ , w)(eH)= e φ (τ∗(e )w). (4.7) α − i ·∇ei α − i · α i Xi=1 Xi=1 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.

Remark 4.3. In a nearly Kähler 6-manifold whose structure is deﬁned by a real Killing spinor η, the spinor vol η is another independent real Killing spinor. Any Dirac operator D/ anti- · commutes with the Cliﬀord multiplication by vol that is D/vol = vol D/, hence for all λ R we 2 − · ∈ have ker(D/ λid) = ker(D/ +λid). Therefore ker(D/ λ2id) = 2 ker(D/ λid) and one can compute − ∼ − ∼ ± the λ eigenspace of D/ by computing the λ2 eigenspace of D/ 2 as done in [CH16, Proposition 4].

19 Deformations of Nearly G2-instantons January 7, 2021

In the case of nearly G2 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 2± such automatic relation between ker(D/ λ2id) 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 D2 similar to the case in hand. Once we know the λ2-eigenspace of D2 we can apply D on them to see which of them lie in the λ or λ- − eigenspace of D.

4.3 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 ρλ with highest weight λ we use the 40 Freudenthal formula to compute ρλ(Casg). We drop the constant 3 in our deﬁnition of Casimir operator for this section as it does not play any role in comparing the Casimir operators. Let 1 µ = 2 (sum of the positive roots of g) then the Freudenthal formula states that

ρλ(Casg)= B(λ, λ) + 2B(µ, λ). (4.8)

We compute the deformation space of the canonical connection for E = h and E = g2 as described earlier. In all the examples listed below, Case 1 is for E = h and Case 2 is for E = g2.

4.3.1 Spin(7)/G2

For this space, H = G2 so there is only one case to consider. The adjoint representation g2 is the unique 14-dimensional irreducible representation of G2. The complex irreducible representations of G2 are identiﬁed with respect to their highest weights of the form (p,q) Z2 and are denoted by V . Here V is the 7-dimensional standard ∈ ≥0 (p,q) (1,0) G2-representation and V(0,1) is the 14-dimensional adjoint representation. The reductive splitting of the Lie algebra is given by spin(7) = g m. 2 ⊕ We have the following isomorphisms of G2 representations,

hC = (g2)C ∼= V(0,1) mC ∼= V(1,0).

The isomorphism spin(7) ∼= so(7) implies that the eigenvalues of their Casimir operators on irreducible representations are equal. For so(7), let E be the 7 7 skew-symmetric matrix ij × with 1 at the (i, j)th entry and 0 elsewhere. We deﬁne H = E E ,H = E E and 1 45 − 23 2 67 − 45 H = E . A Cartan subalgebra for so(7) is given by Span H , i = 1, 2, 3 . A set of simple roots 3 45 { i } α , i = 1, 2, 3 is given by { i } i 0 0 α1 =  2i , α2 =  i  , α3 = i . − i i 0   −   

20 Ragini Singhal January 7, 2021

The Cartan matrix C of so(7) which is given by

2 1 1 − − C =  1 2 0  . − 2 0 2 − 

Then one can compute the simple co-roots Fis by αi(Fj) = Cij which give F1 = iH2, F2 = iH + 2iH and F = 2iH 2iH . The set of fundamental weights is dual to the set of the − 1 3 3 − 2 − 3 simple co-roots. We denote the fundamental weights in decreasing order by λ1, λ2 and λ3 which are dual to F3, F1, F2 respectively. We can compute easily that

B(H ,H ) B(H ,H ) B(H ,H ) 20 10 10 1 1 1 2 1 2 − − B(H2,H1) B(H2,H2) B(H2,H3) =  10 20 10  − B(H ,H ) B(H ,H ) B(H ,H ) 10 10 10  3 1 3 2 3 3  − −  which implies,

B(λ1, λ1) B(λ1, λ2) B(λ1, λ2) 3/40 1/10 1/20 B(λ2, λ1) B(λ2, λ2) B(λ2, λ3) = 1/10 1/5 1/10 . B(λ , λ ) B(λ , λ ) B(λ , λ ) 1/20 1/10 1/10  3 1 3 2 3 3   

Since half the sum of positive roots is given by λ1 + λ2 + λ3 in [Hum78, Section 13.3] therefore by (4.8) on an irreducible SO(7)-representation V(m1,m2,m3) with highest weight m1λ1 + m2λ2 + m λ , m ,m ,m 0 we have 3 3 1 2 3 ≥ 1 ρ (Cas )= (3m2 + 8m2 + 4m2 + 8m m + 4m m + 8m m + 18m + 32m + 20m ). λ so(7) 40 1 2 3 1 2 1 3 2 3 1 2 3 Now we compute the eigenvalues of the Casimir operator for the irreducible representations of g so(7). A Cartan subalgebra of g is given by Span H ,H . Here a pair of simple roots 2 ⊂ 2 { 1 2} β1, β2 is given by

i 0 β = , β = 1 2i 2 i − and the Cartan matrix C˜ for g2 is given by

2 1 C˜ = − 3 2 −

. Let µ1,µ2 be the fundamental weights in decreasing order then their duals with respect to B are iH 2iH , iH respectively and one can compute − 1 − 2 2 B(µ ,µ ) B(µ ,µ ) 1/15 1/10 1 1 1 2 = . B(µ2,µ1) B(µ2,µ2) 1/10 1/5

Again half the sum of the positive roots is given by µ1 +µ2. Using these values in the Freudenthal formula for an irreducible G2-representation V(p,q) with highest weight pµ1 + qµ2 we have 1 ρ (Cas )= (p2 + 3q2 + 3pq + 5p + 9q). (p,q) g2 15

Case 1: E = g2

21 Deformations of Nearly G2-instantons January 7, 2021

The adjoint representation (g2)C ∼= V(0,1). From above 4 ρ (Cas )= . (0,1) g2 5

Substituting the above found values into (4.4) we get that V(m1,m2,m3) can be an inﬁnitesimal deformation space for the canonical connection if 1 4 (3m2 + 8m2 + 4m2 + 8m m + 4m m + 8m m + 18m + 32m + 20m )= . 40 1 2 3 1 2 1 3 2 3 1 2 3 5 But since there are no positive integral solutions of this equation there are no deformations of the canonical connection on Spin(7)/G2.

4.3.2 SO(5)/SO(3)

The complex irreducible SO(5)-representations are characterized by highest weights (m1,m2) Z kC2 2+k−1 ∈ ≥0. The complex irreducible representations of SO(3) are given by S which is a k = k + 1 dimensional space. The 3-dimensional adjoint representation so(3)C and the 7-dimensional representation mC are irreducible SO(3)-representations therefore

6C2 mC ∼= S , 2C2 so(3)C ∼= S . A Cartan subalgebra of so(5) is given by Span H ,H where H = E ,H = E where { 1 2} 1 12 2 34 E is the 5 5 skew-symmetric matrix with 1 at the (i, j)th position and 0 elsewhere. With ij × respect to the Killing form B on so(5), H is orthogonal to H with B(H ,H )= 6 for i = 1, 2. 1 2 i i − Let λ , λ be the fundamental weights whose duals are i(H H ), 2iH respectively then half 1 2 1 − 2 2 the sum of positive roots is given by λ1 + λ2. Doing similar computations as above we get

B(λ , λ ) B(λ , λ ) 1/6 1/12 1 1 1 2 = . B(λ2, λ1) B(λ2, λ2) 1/12 1/12

Using (4.8) for the eigenvalues of the Casimir operator for irreducible representation V(m1,m2) of SO(5) with highest weight m λ + m λ for m ,m 0 we get, 1 1 2 2 1 2 ≥ 1 ρ (Cas )= (2m2 + m2 + 2m m + 6m + 4m ). (m1,m2) so(5) 12 1 2 1 2 1 2 Under the embedding of so(3) in so(5) the Cartan subalgebra of so(3) is given by Span 2H + { 1 H . Here the Cartan subalgebra is 1-dimensional and the fundamental weight µ is dual to 2} 1 4iH + 2iH . Using B(H ,H )= 6 one can compute that B(4H + 2H , 4H + 2H )= 120 1 2 i i − 1 2 1 2 − the eigenvalue of the Casimir operator on the irreducible representation SqC2 of so(3) is given by 1 ρ (Cas )= (q2 + 2q). q so(3) 120 Case 1: E = so(3) The adjoint representation of so(3)C is an irreducible so(3) representation with highest weight 2. Thus 1 ρ (Cas )= ρ (Cas )= . E so(3) 2 so(3) 15

22 Ragini Singhal January 7, 2021

We need to ﬁnd irreducible representations V(m1,m2) of so(5) that satisfy (4.4) which requires 1 1 (2m2 + m2 + 2m m + 6m + 4m )= . 12 1 2 1 2 1 2 15 But since there are no integral solutions for the equation the deformation space is trivial in this case.

Case 2: E = g2 The adjoint representation of (g )C splits as an so(3) representation into S2C2 S10C2. The 2 ⊕ 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

ρ10(Casso(3)) = 1.

Thus we need to ﬁnd so(5) representations V(m1,m2) such that 1 (2m2 + m2 + 2m m + 6m + 4m ) = 1, 12 1 2 1 2 1 2 which has one integral solution namely m1 = 0,m2 = 2. Thus V(0,2) ∼= so(5)C is the only SO(5)-representation for which Casg has eigenvalue 1. As so(3) representations

V = S2C2 S6C2, (0,2) ∼ ⊕ 8 ∗ 10 2 2k 2 mC S C = S C . ⊗ ∼ Mk=2

∗ 10C2 6C2 Thus V(0,2) and mC S have 1 common irreducible so(3) representation namely S . Thus G⊗ ∗ 10C2 −1/3,can 2 V(0,2) occurs in IndH (mC S ) with multiplicity 1. Therefore in this case (ker((D ) 49 ∗ ⊗ − id) Γ(m g ))C = V . 9 ∩ ⊗ 2 ∼ (0,2)

Sp(2)×Sp(1) 4.3.3 Sp(1)×Sp(1) The Lie algebra sp(2) sp(1) decomposes as ⊕ sp(2) sp(1) = sp(1) sp(1) m ⊕ u ⊕ d ⊕ and the embeddings sp(1)u, sp(1)d are given by

a 0 0 0 sp(1)u = , 0 : a sp(1) , sp(1)d = , a : a sp(1) n 0 0 ∈ o n 0 a ∈ o where we follow the notations used in [AS12]. Let H1 = (E1, 0),H2 = (E2, 0) and H3 = (0, E3) then a Cartan subalgebra of sp(2) sp(1) is given by Span H ,H ,H where ⊕ { 1 2 3} i 0 0 0 000 0 00 0 0 0 i 0 0 i 0 E =   , E =   , E = . 1 0 0 i 0 2 000 0 3 0 i  −    − 00 0 0 0 0 0 i    −  If B denote the Killing form of Sp(2) Sp(1) we can compute that H s are orthogonal with × i respect to B and B(H ,H )= 12 for i = 1, 2 and B(H ,H )= 8. The fundamental weights i i − 3 3 −

23 Deformations of Nearly G2-instantons January 7, 2021

λ , λ , λ are dual to i(H H ), iH , iH respectively and half the sum of positive roots is given 1 2 3 1 − 2 1 3 by λ1 + λ2 + λ3. By identical calculations as in other cases we get

B(λ1, λ1) B(λ1, λ2) B(λ1, λ3) 1/12 1/12 0 B(λ2, λ1) B(λ2, λ2) B(λ2, λ3) = 1/12 1/6 0  . B(λ , λ ) B(λ , λ ) B(λ , λ ) 0 0 1/8  3 1 3 2 3 3    Applying the Freudenthal formula (4.8) we get that the Casimir operator of sp(2) sp(1) acts on ⊕ the irreducible representations V with highest weight m λ + m λ + lλ ,m ,m , l 0 (m1,m2,l) 1 1 2 2 3 1 2 ≥ with the eigenvalue 1 1 ρ (Cas )= (m2 + 2m2 + 2m m + 4m + 6m )+ (l2 + 2l). (m1,m2,l) sp(2)⊕sp(1) 12 1 2 1 2 1 2 8 Under the embedding given above a Cartan subalgebra of sp(1) , sp(1) is given by Span H u d { 1} and Span (E , E ) respectively. Let P,Q be the standard 2-dimensional representation of { 2 3 } sp(1)u, sp(1)d respectively. Then the unique (n + 1) dimensional irreducible sp(1)u (respec- n − n tively sp(1)d) representation is given by S P (respectively S Q). From previous calculations we have B(H ,H )= 12 thus the eigenvalue of Cas on SnP is given by 1 1 − sp(1)u 1 ρ (Cas )= (n2 + 2n). n sp(1)u 12

Similarly with the help of previous work one can calculate B((E2, E3), (E2, E3)) = 20. Thus p − Cassp(1)d acts on S Q as the scalar multiple of 1 ρ (Cas )= (n2 + 2n). n sp(1)d 20 The adjoint representation sp(1) is an irreducible 3-dimensional sp(1) representation and hence we have the following decompositions into Sp(1) Sp(1) representations u × d 2 2 2 (sp(1) )C = S P, (sp(1) )C = S Q, mC = S Q PQ u ∼ d ∼ ∼ ⊕ where PQ denotes the tensor product of P and Q and we omitted the tensor product sign for clarity and will continue to do so. Case 1: E = sp(1) sp(1) u ⊕ d We need to ﬁnd the irreducible sp(2) sp(1) representations V that satisfy (4.4) for each ⊕ (m1,m2,l) irreducible component of hC that is (sp(1)u)C and (sp(1)d)C . For sp(1)u this equation takes the form 1 1 8 (m2 + 2m2 + 2m m + 4m + 6m )+ (l2 + 2l)= . 12 1 2 1 2 1 2 8 12

The integral solution (m1,m2, l) for this equation is (0, 1, 0). Thus the only irreducible sp(2) 2 ⊕ sp(1) representations for which Casg has eigenvalue is V . As sp(1) sp(1) -representations 3 (0,1,0) u⊕ d we have the following decomposition

V = PQ C, (0,1,0) ∼ ⊕ 2 2 3 (sp(1) m)C = S PS Q S PQ P Q. u ⊗ ∼ ⊕ ⊕

The irreducible Sp(1) Sp(1) representation in (sp(1)u m)C common with V(0,1,0) is PQ with × G ∗ ⊗ multiplicity 1. Thus V occurs in Ind (m sp(1) )C with multiplicity 1. Therefore the (0,1,0) H ⊗ u

24 Ragini Singhal January 7, 2021

∗ solutions to (4.4) in Γ(m sp(1) )C is the 5-dimensional complex Sp(2) Sp(1) representation ⊗ u × V(0,1,0).

For the next irreducible hC component (sp(1)d)C (4.4) for V(m1,m2,l) becomes 1 1 8 (m2 + 2m2 + 2m m + 4m + 6m )+ (l2 + 2l)= , 12 1 2 1 2 1 2 8 20 which has no integral solutions and thus it has no contribution to the deformation space. Thus from Proposition 4.2 we conclude that (ker((D−1/3.can)2 49 id) Γ(m∗ sp(1) − 9 ∩ ⊗ u ⊕ sp(1) )C = (V ) when the structure group is Sp(1) Sp(1) . d ∼ (0,1,0) u × d Case 2: E = (g2)C The adjoint representation of g decomposes into irreducible sp(1) sp(1) as follows: 2 u ⊕ d 2 2 3 (g )C = S P S Q PS Q. 2 ⊕ ⊕ We have already seen the contribution of the ﬁrst two irreducible components in the summation. For the third component

ρ1,3(Cassp(1)u⊕sp(1)d ) = 1, so here we need to ﬁnd the sp(2) sp(1) representations V such that ⊕ (m1,m2,l) 1 1 (m2 + 2m2 + 2m m + 4m + 6m )+ (l2 + 2l) = 1. 12 1 2 1 2 1 2 8 The sp(2) sp(1)-representations that satisfy (4.4) are V and V , which decompose into ⊕ (2,0,0) (0,0,2) sp(1) sp(1) representations as u ⊕ d 2 2 2 V = sp(2)C = S P S Q P Q, V = (sp(1) )C = S Q. (2,0,0) ∼ ∼ ⊕ ⊕ (0,0,2) ∼ d ∼ Moreover

3 ∗ 2 4 2 2 5 3 4 2 PS Q mC = S PS Q S PS Q P (S Q S Q Q) S Q S Q. ⊗ ∼ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 3 ∗ 2 Thus V(2,0,0) and PS Q mC have two common irreducible representations PQ,S Q and V(0,0,2) 3 ∗ ⊗ 2 and PS Q mC have one common irreducible representation S Q. So by Frobenius reciprocity ⊗ G ∗ 3 V(2,0,0) and V(0,0,2) lie in IndH (mC PS Q) with multiplicity 2, 1 respectively. Thus the solution of ∗ ⊗ (4.4) in Γ(m g2)C is the 28 dimensional Sp(2) Sp(1) complex representation 2V(2,0,0) V(0,1,0) ⊗ × − 49 ∗⊕ ⊕ V . So again by Proposition 4.2 we conclude that ker((D 1/3.can)2 id) Γ(m g )C = (0,0,2) − 9 ∩ ⊗ 2 ∼ 2V V V when the structure group is G . (2,0,0) ⊕ (0,1,0) ⊕ (0,0,2) 2

SU(3)×SU(2) 4.3.4 SU(2)×U(1) The embeddings of su(2) and u(1) in su(3) su(2) which we denote by su(2) and u(1) following × d [AS12] in su(3) su(2) are given by ⊕ i 0 0 a 0 su(2) = , a : a su(2) , u(1) = span 0 i 0  , 0 . d 0 0 ∈ n o n 0 0 2i o  −  A Cartan subalgebra of su(3) su(2) is given by span H = (E , 0),H = (E , 0),H = (0, E ) ⊕ { 1 1 2 2 3 3 } where 0 1 0 i 0 0 0 1 E1 =  1 0 0 , E2 = 0 i 0  , E3 = . − 1 0 0 0 0 0 0 2i −    −  25 Deformations of Nearly G2-instantons January 7, 2021

We can check that the H s are orthogonal with respect to the Killing form B on SU(3) SU(2). i × As earlier we denote by λ , λ , λ the fundamental weights which are dual to i (H H ), i (H + 1 2 3 2 1 − 2 2 1 H2), iH3 respectively. By direct computations we get

B(H ,H ) B(H ,H ) B(H ,H ) 12 0 0 1 1 1 2 1 3 − B(H2,H1) B(H2,H2) B(H2,H3) =  0 36 0  , − B(H ,H ) B(H ,H ) B(H ,H ) 0 0 8  3 1 3 2 3 3   −  therefore

B(λ1, λ1) B(λ1, λ2) B(λ1, λ3) 1/9 1/18 0 B(λ2, λ1) B(λ2, λ2) B(λ2, λ3) = 1/18 1/9 0  . B(λ , λ ) B(λ , λ ) B(λ , λ ) 0 0 1/8  3 1 3 2 3 3   

Half the sum of the positive roots is λ1 + λ2 + λ3 and thus by Freudenthal formula (4.8) for a su(3) su(2) representation V with highest weight m λ +m λ +lλ where m ,m , l 0 ⊕ (m1,m2,l) 1 1 2 2 3 1 2 ≥ 1 1 ρ (Cas )= (m2 + m2 + m m + 3m + 3m )+ (l2 + 2l). m1,m2,l su(3)⊕su(2) 9 1 2 1 2 1 2 8 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 , E ) and span H respectively. By calculations ⊕ { 1 3 } { 2} completely analogous to the previous case we then get that if we represent the irreducible (n+1)- n dimensional su(2)d representations by S 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) 1 ρ (Cas )= (n2 + 2n), n su(2)d 20 1 ρ (Cas )= k2. k u(1) 36

As su(2) u(1) representations the 7-dimensional space mC decomposes as d ⊕ 2 mC = S W W F (3) W F ( 3), ∼ ⊕ ⊕ − whereas the 3-dimensional adjoint representation of (su(2)d)C is irreducible and hence is isomorphic to S2W . Case 1: E = su(2) u(1) d ⊕ The adjoint representation su(2) u(1) splits as irreducible su(2) u(1) representations as d ⊕ d ⊕ follows:

2 (su(2) u(1))C = S W C. d ⊕ ∼ ⊕ 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 2 deformations corresponding to S W . For that we need to look for representations V(m1,m2,l) such that 1 1 8 (m2 + m2 + m m + 3m + 3m )+ (l2 + 2l)= , 9 1 2 1 2 1 2 8 20 which as seen before has no integral solutions. Hence the canonical connection admits no deformations in this case.

26 Ragini Singhal January 7, 2021

Case 2: E = g2 The adjoint representation (g )C splits as su(2) u(1) representation as follows: 2 d ⊕ 3 3 2 (g )C = S W F (3) S W F ( 3) S W F (6) F ( 6) C. 2 ⊕ − ⊕ ⊕ ⊕ − ⊕ We need to follow the same procedure as above for each of the components. For each component we need to ﬁnd the su(3) su(2) representation V that satisﬁes (4.4). We have already ⊕ (m1,m2,l) solved this for S2W C so we just need to compute it for the rest. ⊕ 3 From above calculations ρS WF (3)(Cash) = 1 therefore V(m1,m2,l) should satisfy 1 1 (m2 + m2 + m m + 3m + 3m )+ (l2 + 2l) = 1. 9 1 2 1 2 1 2 8 The only possible solutions are V ,V . As su(2) u(1) representations V = S2W (0,0,2) (1,1,0) ⊗ (0,0,2) ∼ and V(1,1,0) ∼= su(3)C. Further one can compute 2 V(0,0,2) ∼= su(2)C ∼= S W, 2 V = su(3)C = S W W F (3) W F ( 3) C, (1,1,0) ∼ ∼ ⊕ ⊕ − ⊕ 3 ∗ 5 3 4 2 4 2 S W F (3) mC = (S W S W W )F (3) (S W S W )F (6) S W S W. ⊗ ∼ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 3 ∗ 2 Thus V(0,0,2) and S W F (3) mC has one common component S W with multiplicity 1 and 3 ∗ ⊗ 2 V(1,1,0) and S W F (3) mC has two common components S W, W F (3) both with multiplicity 1 ⊗ G ∗ 3 each. So by Frobenius reciprocity Ind (mC S W F (3)) contains a copy of V 2V . H ⊗ (0,0,2) ⊕ (1,1,0) The representation S3W F ( 3) is the dual of the representation S3W F (3) and since SU(2) − ⊗ U(1) representations are isomorphic to their duals the result for this case is same as the above G ∗ 3 and Ind (mC S W F ( 3)) also contains a copy of V 2V . H ⊗ − (0,0,2) ⊕ (1,1,0) For the u(1) representation F (6), ρ6(Casu(1)) = 1. Thus again the only solutions are ∗ V ,V by the previous case. The su(2) u(1) representation F (6) mC has the fol- (0,0,2) (1,1,0) ⊕ ⊗ lowing decompostion

∗ 2 F (6) mC = S W F (6) W F (9) W F (3), ⊗ ∼ ⊕ ⊕ G ∗ thus V is not contained in Ind (mC F (6)) but V is with multiplicity 1. Since F ( 6) = (0,0,2) H ⊗ (1,1,0) − ∼ F (6)∗ this case is similar to the above case. − 49 ∗ Summing up all the parts together we get that ker((D 1/3,can)2 Γ(m g )C = 2(V − 9 }∩ ⊗ 2 ∼ (0,0,2) ⊕ 3V(1,1,0)) when the structure group is G2. Table 3 lists the ker((D−1/3,can)2 49 id) Γ(m∗ E) when E = h and E = g for all the − 9 ∩ ⊗ 2 homogeneous spaces listed in Table 2. Note that for the remaining two homogeneous spaces 3 2 N , k = l and SU(2) /U(1) our methods does not apply when E = g2 although since H is k,l 6 abelian for both of them there are no deformations for the E = h case. The space V (0,1) listed in Table 3 denotes the unique irreducible 5-dimensional complex representation of sp(2).

4.4 Eigenspaces of the Dirac operator

All the G-representations listed in Table 3 lie in ker((D−1/3,can)2 49 id) Γ(m∗ E) which by − 9 ∩ ⊗ (4.5) is equal to (ker(D−1,can + 2id) ker(D−1,can 8 id)) Γ(m∗ E). Since the canonical ⊕ − 3 ∩∩ ⊗ connection is translation invariant it takes an irreducible G-representation to itself. Hence the irreducible subspaces found in Table 3 lie in either ker(D−1,can 8 id) or ker(D−1,can + 2id) − 3 where the subspaces in the latter space constitute the inﬁnitesimal deformations of the canonical

27 Deformations of Nearly G2-instantons January 7, 2021

Homogeneous space h g2 Spin(7)/G2 0 0

SO(5)/SO(3) 0 so(5)

Sp(2)×Sp(1) V (0,1) 2sp(2) sp(1) V (0,1) Sp(1)×Sp(1) R ⊕ ⊕ R

SU(3)×SU(2) 0 2su(2) 6su(3) SU(2)× U(1) ⊕

Nk,l 0 unknown

SU(2)3/U(1)2 0 unknown

Table 3: ker((D−1/3,can)2 49 id) Γ(m∗ E) − 9 ∩ ⊗ connection by Theorem 3.2. Thus now it remains to identify which of the subspaces in Table 3 lies in ker(D−1,can + 2id) for each of the homogeneous spaces. for all the homogeneous spaces G/H in Table 2 the metric corresponding to the nearly G structure ϕ is given by 3 B where 2 − 40 B is the Killing form of G. For 1-forms X,Y the Cliﬀord product between X and Y η is given · by X Y η = X,Y η ϕ(X,Y,.) η. (4.9) · · h i − · 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 3.

4.4.1 SO(5)/SO(3) From the previous section we know that there are no deformation of the canonical connection when the structure group is SO(3). For the structure group G2 we calculated that the smooth ∗ −1/3.can 2 49 sections of G ∗ (m g ) in ker((D ) id) = V = so(5)C. If we denote by ×ρm ⊗g2 ⊗ 2 − 9 ∼ (0,2) ∼ E the skew-symmetric matrix with 1 at (i, j), 1 at (j, i) and 0 elsewhere and deﬁne ij − 2 2 √3 e1 := (E12 2E34), e2 := (√2E45 (E23 E14)), 3 − 3 − √2 − 2√5 2 √3 e3 := E25, e4 := (√2E35 (E13 + E24)), 3 3 − √2 √10 √10 2√5 e := (E E ), e := (E + E ), e := E , 5 3 24 − 13 6 − 3 23 14 7 3 15 then e , i = 1 . . . 7 deﬁnes a basis of m∗ which is orthonormal with respect to the metric 3 B. { i } − 40 With respect to this basis the nearly G2 structure ϕ is given by

ϕ = e124 + e137 + e156 + e235 + e267 + e346 + e457. We have seen that for SO(5)/SO(3) the canonical connection has no deformation as an SO(3) con- −1,can nection. Now we need to check whether the SO(5)-representation V(0,2) lies in the ker(D 8 ∗ − ∗ − C 1,can C 3 id) Γ(m g2) or ker(D + 2id) Γ(m g2) . As seen before the common irre- ∩ ⊗ ∩ ∗ ⊗ 6C2 ∗ ducible so(3) representation in V(0,2) so(3) and (m g2)C is S = mC. We denote the 1- ∗ | ⊗ ∼ dimensional space Hom(V(0,2), (m g2)C) = Span(α). Let µi, i = 1 . . . 11 be a basis of the 11- ⊗ 10 2 dimensional subspace of (g2)C isomorphic to the so(3) representation S C . Then the subspace

28 Ragini Singhal January 7, 2021

∗ 10 2 ∗ 6 2 of mC S C (m g )C isomorphic to S C is given by Span v , i = 1 . . . 7 where ⊗ ⊂ ⊗ 2 { i } e e v = 2 (5(µ µ ) + 3√15µ )+ e (µ + µ ) 4 (5µ + 3√15(µ + µ )) 1 − 14 ⊗ 1 − 7 9 3 ⊗ 5 11 − 14 ⊗ 2 3 4 + e5 (µ3 µ4)+ e6 µ9 + e7 (µ6 µ10), ⊗ − ⊗ e ⊗ − v =e µ + e ( 2µ + µ ) 3 (47µ + 37µ + 3√5µ ) e (µ + 2µ ) 2 1 ⊗ 9 2 ⊗ − 5 4 − 28 ⊗ 1 7 9 − 4 ⊗ 6 10 e e 5 µ + 7 ( 37µ + 3√15(µ + µ )), − 14 ⊗ 8 28 ⊗ − 2 3 4 e e e e v = 1 (µ µ )+ 2 (2µ + µ )+ 3 (47µ + 3√5(µ + µ )) 4 (µ 2µ ) 3 − 2 ⊗ 3 − 4 2 ⊗ 6 10 56 ⊗ 2 3 4 − 2 ⊗ 5 − 11 e e 6 µ + 7 ( 37µ + 6√15µ ), − 28 ⊗ 8 56 − 1 9 e 5e e v = 1 (5µ + 3√15(µ + µ )) + 2 µ 3 (3√15µ + 41µ + 13µ ) 4 − 28 ⊗ 2 3 4 28 ⊗ 8 − 56 ⊗ 2 3 4 e e e 5 (µ 2µ )+ 6 (µ + 2µ )+ 7 (3√15(µ µ ) + 41µ ), − 2 ⊗ 5 − 11 2 ⊗ 6 10 56 ⊗ 1 − 7 9 e e v =e (µ + µ ) 2 (3√15(µ µ ) + 13µ )+ 4 (3√15µ + 41µ + 13µ ) 5 1 ⊗ 5 11 − 28 ⊗ 1 − 7 9 28 ⊗ 2 3 4 e e 2e + 5 (47µ + 3√15(µ + µ )) + 6 (47µ + 37µ + 3√15µ )+ 7 µ , 28 ⊗ 2 3 4 28 ⊗ 1 7 9 28 ⊗ 8 e 2e e v =e ( µ + µ )+ 2 (3√15µ + 13µ + 41µ )+ 3 µ + 4 (3√15(µ µ ) + 41µ ) 6 1 ⊗ − 6 10 28 ⊗ 2 3 4 7 ⊗ 8 28 ⊗ 1 − 7 9 e e + 5 (37µ + 47µ 3√15µ )+ 6 ( 37µ + 3√15(µ + µ )), 28 ⊗ 1 7 − 9 28 ⊗ − 2 3 4 e e 5e v = 1 (5(µ µ ) + 3√15µ ) 3 (3√15(µ µ ) + 13µ )+ 4 µ 2e (µ + µ ) 7 14 ⊗ 1 − 7 9 − 28 ⊗ 1 − 7 9 14 ⊗ 8 − 5 ⊗ 6 10 e + e ( 2µ + µ ) 7 (3√15µ + 13µ + 41µ ). 6 ⊗ − 5 11 − 28 ⊗ 2 3 4 6 2 The subspace of V(0,2) isomorphic to S C is SpanC ei, i = 1 . . . 7 and the SO(3) equivariant ∗ { } homomorphism α between V and (m g )C is given by (0,2) ⊗ 2 α(e )= v , α(e )= v , α(e )= v , 1 1 2 7 3 − 5 α(e )= 2v , α(e ) = 2v , α(e )= v , α(e )= v . 4 − 4 5 3 6 − 2 7 6 ∗ −1/3.can 2 49 Any section of the bundle associated to m g2 in ker((D ) 9 id) can be represented ∗ ⊗ − by (α, v) for some v V 6C2 = mC. The action of the canonical connection on such a section ∈ (0,2)|S ∼ is then given by −1,can(α, v)(eH) = α([X, v]) where the Lie bracket is in so(5). We can now ∇X − calculate the action of the Dirac operator, D−1,can on (α, e ) η at the point eH as follows. We 1 · omit the η from the computations to reduce notational clutter and will continue to do so in · every case. 7 D−1,can(α, e )(eH)= e −1,can(α, e )(eH) 1 i ·∇ei 1 Xi=1 2 = − (e α(e )+ e α(e )+ e α( e )+ e α(e )+ e α( e )+ e α( e )) 3 2 · 4 3 · 7 4 · − 2 5 · 6 6 · − 5 7 · − 3 2 = (2e v e v + e v + e v + 2e v e v ) 3 2 · 4 − 3 · 6 4 · 7 5 · 2 6 · 3 − 7 · 5 2 = ( 3v ) η = 2α(e ). 3 − 1 · − 1 −1,can Thus by the translation invariance of the canonical connection V(0,2) ker(D + 2id) ∗ ⊆ ∩ Γ(m g )C. ⊗ 2

29 Deformations of Nearly G2-instantons January 7, 2021

Sp(2)×Sp(1) 4.4.2 Sp(1)×Sp(1)

−1/3.can 2 49 From the previous section we know that for E = sp(1) sp(1) the ker((D ) 9 id) ∗ ⊕ ∗ − ∩ Γ(m E)C = V . Let e , i = 1 . . . 7 be an orthonormal basis of m with respect to the ⊗ ∼ (0,1,0) { i } metric 3 B given by − 40 1 0 0 1 0 0 1 0 0 e := , 3i , e := , 3j , e := , 3k , 1 3 0 2i − 2 3 0 2j − 3 3 0 2k − √5 0 1 √5 0 i √5 0 j √5 0 k e := , 0 , e := , 0 , e := , 0 , e := , 0 . 4 3 1 0 5 3 i 0 6 3 j 0 7 3 k 0 − With respect to this basis the nearly G2 form is given by

ϕ = e e e e + e e e , 123 − 145 − 167 − 246 257 − 347 − 356 From Table 3 we know that as an Sp(1) Sp(1) connection the deformation space of the × canonical connection is an irreducible subrepresentation of V(0,1,0) and is thus trivial or (V(0,1,0))R . We need to check whether this space lies in the 2 eigenspace of D−1,A − The Sp(2) Sp(1)-representation V is 5 dimensional. We need to ﬁnd the space Hom(V , (m∗ × (0,1,0) (0,1,0) ⊗ (sp(1)u sp(1)d))C)Sp(1)×Sp(1). The common irreducible Sp(1) Sp(1) representations in V(0,1,0) ∗⊕ × ∗ and (m sp(1) )C is PQ. Let S2P = Span I, J, K then the subspace of (m sp(1) )C ⊗ u { } ⊗ u isomorphic to the space PQ is given by SpanC v , v , v , v where { 1 2 3 4} v = e I + e J + e K, v = e I + e J e K, 1 5 ⊗ 6 ⊗ 7 ⊗ 2 − 4 ⊗ 7 ⊗ − 6 ⊗ v = e I e J + e K, v = e I e J e K. 3 − 7 ⊗ − 4 ⊗ 5 ⊗ 4 6 ⊗ − 5 ⊗ − 4 ⊗

Let the subspace of V(0,1,0) isomorphic to PQ be given by Span w1, w2, w3, w4 and the homo- ∗ { } morphism space Hom(V , (m sp(1) )C)= Span(β) where β is deﬁned by (0,1,0) ⊗ u w v + iv , w v iv , 1 7→ 3 4 2 7→ 1 − 2 w v + iv , w v iv . 3 7→ 1 2 4 7→ 3 − 4 Using this isomprhism one can compute that the only non-trivial gl(V ) elements with (0,1,0)|PQ respect to the basis w , w , w , w are { 1 2 3 4} i 0 0 0 0 1 0 0 0 i 0 0 − 2 0 i 0 0  2 10 00 2 i 0 0 0  τ∗(e )= − , τ∗(e )= , τ∗(e )= . 1 3 0 0 i 0 2 3 00 01 3 3 0 0 0 i      −  0 0 0 i 0 0 1 0 0 0 i 0   −   −   −  −1,can Also by the deﬁnition of the canonical connection, (β, w)(eH) = β(τ∗(X)w). Thus we ∇X − can calculate

7 7 −1,can −1,can (D (β, w ))(eH)= e (β, w )(eH)= e β((τ∗(e )w ) ) 1 i ·∇ei 1 − i · i 1 |PQ Xi=1 Xi=1 2 2 2 = (e β( iw )+ e β( w )+ e β( iw )) − 1 · 3 1 2 · 3 2 3 · 3 2 2 = (ie (v + iv )+ e (v iv )+ ie (v iv )) −3 1 · 3 4 2 · 1 − 2 3 · 1 − 2 2 = (3(v + iv )) = 2β(w ). −3 3 4 − 1

30 Ragini Singhal January 7, 2021

−1,can Thus we have shown that V(0,1,0) lies in the ker(D + 2id). For E = g the subspace of Γ(m∗ g ) in ker((D−1/3.can)2 49 id) is isomorphic to the 2 ⊗ 2 − 9 Sp(1) Sp(1) representation 2V V V . We have already dealt with the space × (2,0,0) ⊕ (0,1,0) ⊕ (0,0,2) V(0,1,0). The remaining spaces are 2V(2,0,0) ∼= 2sp(2) and V(0,0,2) ∼= sp(1). The two copies of ∗ 3 V(2,0,0) arise from Hom(V(2,0,0), mC PS Q)Sp(1)×Sp(1) and the one copy of V(0,0,2) arises from ∗ 3 ⊗ Hom(V , mC PS Q) × . Thus we have two cases: (0,0,2) ⊗ Sp(1) Sp(1) ∗ 3 Case: 1-Hom(V , mC PS Q) × V (0,0,2) ⊗ Sp(1) Sp(1) ⊗ (0,0,2) Let w , w , w be the standard basis of V = sp(1)C then the non-trivial actions of m on { 1 2 3} (0,0,2) ∼ sp(1)C are given by

0 0 0 0 0 2 0 2 0 [e1, .]= 0 0 2 , [e2, .]=  0 0 0 , [e3, .]= 2 0 0 . − 0 2 0 2 0 0 0 0 0   −   

Let µi, i = 1 . . . 8 be a basis of the Sp(1)u Sp(1)d subrepresentation of (g2)C isomorphic to { } × ∗ PS3Q. The 1-dimensional space Hom(V , (m g )C)= Span φ where φ maps (0,0,2) ⊗ 2 { } w e (µ µ )+ e (µ + µ )+ e (µ µ ) e (µ + µ ), 1 7→ 4 ⊗ 5 − 2 5 ⊗ 1 6 6 ⊗ 4 − 7 − 7 ⊗ 3 8 w e (µ 2µ ) e (µ + 2µ )+ e (µ 2µ ) e (µ + 2µ ), 2 7→ 4 ⊗ 3 − 8 − 5 ⊗ 4 7 6 ⊗ 1 − 6 − 7 ⊗ 2 5 w e (2µ + µ )+ e (µ 2µ ) e (2µ + µ )+ e (µ 2µ ). 3 7→ − 4 ⊗ 4 7 5 ⊗ 8 − 3 − 6 ⊗ 2 5 7 ⊗ 6 − 1 The connection −1,can(φ, w) = φ([X, w]) for w sp(1) where the Lie bracket is in the Lie ∇X − ∈ algebra sp(2) sp(1). Thus we can calculate ⊕ 7 7 D−1,can(φ, w )(eH)= e −1,can(φ, w )(eH)= e φ([e , w ]) 1 i ·∇ei 1 − i · i 1 Xi=1 Xi=1 = (e φ( 2w )+ e φ(2w )) − 2 · − 3 3 · 2 = 2(e (µ µ )+ e (µ + µ )+ e (µ µ ) e (µ + µ )) − 4 ⊗ 5 − 2 5 ⊗ 1 6 6 ⊗ 4 − 7 − 7 ⊗ 3 8 = 2φ(w ). − 1 − − ∗ Hence again by translation invariance of 1,can, V ker(D 1,can + 2id) Γ(m g )C. ∇ (0,0,2) ⊆ ∩ ⊗ 2 ∗ 3 Case: 2-Hom(V , mC PS Q) V (2,0,0) ⊗ Sp(1)×Sp(1) ⊗ (2,0,0) 2 2 The Sp(2) Sp(1)-representation V(2,0,0) ∼= sp(2)C ∼= S P S Q PQ. The subspace of × 2 ⊕ ⊕ (sp(2))C isomorphic to S Q,PQ is given by SpanC e1, e2, e3 , SpanC e4, e5, e6, e7 respectively. 3 { } { } ∗ As before the basis of PS Q (g2)C is denoted by µ1,µ2,...,µ8 and the subspace of (m 2 ⊂ { } ⊗ g2)C isomorphic to S Q is given by Span φ(w1), φ(w2), φ(w3) deﬁned above. The subspace of ∗ { } (m g )C isomorphic to PQ is given by Span v , v , v , v where ⊗ 2 { 1 2 3 4} v = e (µ + µ ) e (µ + 2µ ) e (2µ µ ), 1 1 ⊗ 1 6 − 2 ⊗ 4 7 − 3 ⊗ 3 − 8 v = e (µ µ ) e (µ 2µ )+ e (2µ + µ ), 2 1 ⊗ 2 − 5 − 2 ⊗ 3 − 8 3 ⊗ 4 7 v = e (µ + µ ) e (µ + 2µ ) e (2µ µ ), 3 − 1 ⊗ 3 8 − 2 ⊗ 2 5 − 3 ⊗ 1 − 6 v = e (µ µ ) e (µ 2µ )+ e (2µ + µ ). 4 − 1 ⊗ 4 − 7 − 2 ⊗ 1 − 6 3 ⊗ 2 5 ∗ Let A , A be a basis of the 2-dimensional space Hom(V , (m g )C) and let { 1 2} (2,0,0) ⊗ 2 Sp(1)u×Sp(1)d A = c1A1 + c2A2 for some real constants c1, c2 then we have that

A(e1)= c1w1, A(e2)= c1w2, A(e3)= c1w3

31 Deformations of Nearly G2-instantons January 7, 2021

A(e )= c v , A(e )= c v , A(e )= c v , A(e )= c v 4 − 2 2 5 2 1 6 − 2 4 7 2 3 2 and A1, A2 acts trivially on S P . ∗ ∗ Let s(A,w) Γ(m g2)C be the section corresponding to (A, w) Hom(V(2,0,0), (m ∈ ⊗ −1,can ∈ ⊗ g )C) × sp(2) then (A, w) = A(ad(X)w) = A([X, w] ) where the Lie bracket 2 Sp(1) Sp(1) ⊗ ∇X − | is in the Lie algebra sp(2). Using this action of −1,can we can calculate ∇ 7 7 (D−1,can(A, e ))(eH)= e −1,can(A, e )(eH)= e A([e , e ] ) 1 i ·∇ei 1 − i · i 1 | Xi=1 Xi=1 2 = ( e A(e )+ e A(e )+ e A(e ) e A(e )+ e A(e ) e A(6)) −3 − 2 · 3 3 · 2 4 · 5 − 5 · 4 6 · 7 − 7 · 2 = (c ( e w + e w )+ c (e v e ( v )+ e v e ( v ))) −3 1 − 2 · 3 3 · 2 2 4 · 1 − 5 · − 2 6 · 3 − 7 · − 4 4c 6c 4c 6c = 1 − 2 w = 1 − 2 A (e ). 3 1 3 1 1 By doing similar computations we get that

−1,can (D (A, fi))(eH) = 0, i = 1, 2, 3, 4c 6c (D−1,can(A, e ))(eH)= 1 − 2 A (e ), i = 1, 2, 3, i 3 1 i 20c + 6c (D−1,can(A, e ))(eH)= 1 2 A (e ), i = 4, 5, 6, 7. i − 9 2 i ∗ −1,can Therefore the subspace of Hom(V(2,0,0), (m g2)C)Sp(1)×Sp(1) in the ker(D + 2id) is 5 ⊗ given by the condition c2 = 3 c1 and is thus 1-dimensional. Therefore V(2,0,0) occurs in the − ∗ ker(D 1,can + 2id) Γ(m g )C with multiplicity 1. ∩ ⊗ 2 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 for c = 2 c . This shows that not all spaces in − 3 2 − 3 1 ker((D−1/3,can)2 49 id) are in ker(D−1,can + 2id). − 9

SU(3)×SU(2) 4.4.3 SU(2)×U(1) As before let e , i = 1 . . . 7 be an orthonormal basis of m∗ with respect to g. If we deﬁne { i } i 0 0 1 0 i I = , J = − , K = we have 0 i 1 0 i 0 − 1 2I 0 1 2J 0 1 2K 0 e := , 3I , e := , 3J , e := , 3K , 1 3 0 0 − 2 3 0 0 − 3 3 0 0 −

0 0 √2 0 0 √2i √5 √5 e4 :=  0 0 0  , 0 , e5 :=  0 0 0  , 0 , 3 3 √2 0 0 √2i 0 0 −      0 0 0 0 0 0 √5 √5 e6 := 0 0 √2 , 0 , e7 := 0 0 √2i , 0 . 3 3 0 √2 0 0 √2i 0  −      With respect to this basis the nearly G2 structure ϕ is given by ϕ = e + e e + e + e + e e . 123 145 − 167 246 257 347 − 356

32 Ragini Singhal January 7, 2021

∗ 2 As an SU(2) U(1) representation, mC = S W W F (3) W F ( 3) where × ∼ ⊕ ⊕ − S2W = Span e , e , e , W F (3) = Span e ie , e ie , W F ( 3) = Span e + ie , e + ie . { 1 2 3} { 4 − 5 6 − 7} − { 4 5 6 7} 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, (g )C = S3W (F (3) F ( 3)) S2W F (6) F ( 6). We × 2 ∼ ⊕ − ⊕ ⊕ ⊕ − have already seen that S2W gives rise to no deformations. From previous calculations we know − 49 ∗ 1/3,can 2 m 3 su C su C that ker((D ) 9 id) Γ( C S W F ( 3)) ∼= V(0,0,2) 2V(1,1,0) ∼= ( (2)) 2( (3)) ∗ − ∩−1/3,can⊗2 49 ± ⊕ ⊕ and Γ(mC F ( 6)) ker((D ) id) = V respectively. Therefore there are 6 ⊗ ± ∩ − 9 ∼ (1,1,0) subspaces of Γ(m∗ g ) to consider here. ⊗ 2 ∗ 3 Case: 1-Hom(V , mC S W F (3)) × V (0,0,2) ⊗ SU(2) U(1) ⊗ (0,0,2) We denote by µ , i = 1 . . . 4 a basis of S3W F (3). Let f , i = 1 . . . 3 be the standard basis of { i } i su(2) such that [f1,f2]= 2f3, [f1,f3] = 2f2, [f2,f3]= 2f1. Then the subspace of W F ( 3) ∗ − − − ⊗ S3W F (3) (m g )C isomorphic to (su(2))C is given by Span v , v , v where ⊂ ⊗ 2 { 1 2 3} 3i 5i v = (e + ie ) µ + (e + ie ) ( µ + µ ), 1 4 4 5 ⊗ 1 6 7 ⊗ 4 2 4 v = (e + ie ) ( iµ + µ ) + (e + ie ) ( iµ µ ), 2 4 5 ⊗ − 2 4 6 7 ⊗ − 1 − 3 5i 3i v = (e + ie ) ( µ + µ ) (e + ie ) µ 3 4 5 ⊗ − 4 1 3 − 4 6 7 ⊗ 2 ∗ and the space Hom(V , (m g )C)= Span γA where γA is deﬁned by (0,0,2) ⊗ 2 { } γA(f )= v , γA(f )= i(v v ), γA(f )= 2(v + v ). 1 2 2 1 − 3 3 − 1 3 For i = 1, 2, 3, since e = ( 2 f , f ) we have [e , v] = [f , v] for all v su(2). The action is i 3 i − i i − i ∈ trivial for i = 4 . . . 7 since [e ,f ] / Span f ,f ,f . We can thus calculate i j ∈ { 1 2 3} 7 D−1,can(γA,f )(eH)= e −1,can(γA,f )(eH) 1 i ·∇ei 1 Xi=1 = e γA(2f ) e γA(2f ) 2 · 3 − 3 · 2 = (4e (v + v ) + 2ie (v v )) − 2 · 1 3 3 · 1 − 3 = 2v = 2 γA(f ). − 2 − 1 ∗ 3 −1,can Hence Hom(V , mC S W F (3)) V ker(D + 2id). (0,0,2) ⊗ |Sp(1)×Sp(1) ⊗ (0,0,2) ⊆ ∗ 3 Case: 2-Hom(V , mC S W F (3)) V (1,1,0) ⊗ SU(2)×U(1) ⊗ (1,1,0) 2 Let a basis of the subspace of V(1,1,0) ∼= (su(3))C isomorphic to S W ∼= (su(2))C be given by I 0 J 0 K 0 p := , p := , p := . 1 0 0 2 0 0 3 0 0 where I, J, K are deﬁned previously. Then [p1,p2] = 2p3, [p1,p3] = 2p2, [p2,p3] = 2p1. The ∗ 3 ∗ 2 − − basis of mC S W F (3) mC g isomorphic to S W is given by Span w , w , w where ⊗ ⊂ ⊗ 2 { 1 2 3} µ + iµ µ iµ w = (e + ie ) 2 3 + (e + ie ) 1 − 4 , 1 4 5 ⊗ 2 6 7 ⊗ 2 µ 2iµ µ 2iµ w = (e + ie ) 4 − 1 + (e + ie ) 3 − 2 , 2 4 5 ⊗ 2 6 7 ⊗ 2

33 Deformations of Nearly G2-instantons January 7, 2021

µ + 2iµ µ 2iµ w = (e + ie ) 1 4 + (e + ie ) 2 − 3 . 3 − 4 5 ⊗ 2 6 7 ⊗ 2

Since (su(3))C = mC C, the subspace of (su(3))C isomorphic to W F (3) is given by ⊕ 2 3 ∗ SpanC e ie , e ie . The subspace of S W S W F (3) (m g )C isomorphic to { 4 − 5 6 − 7} ⊗ ⊂ ⊗ 2 W F (3) is given by Span u , u where { 1 2} µ + iµ 2µ + iµ µ + 2iµ u = ie 2 3 + e 1 4 ie 1 4 , 1 1 ⊗ 2 2 ⊗ 2 − 3 ⊗ 2 µ iµ 2µ iµ µ 2iµ u = ie 1 − 4 + e 2 − 3 + ie 2 − 3 2 1 ⊗ 2 2 ⊗ 2 3 ⊗ 2

(1,1,0) ∗ 3 (1,1,0) ∗ 3 If we denote the space Hom(V , mC S W F (3)) and Hom(V , mC S W F (3)) by ⊗ ⊗ Span A , Span A respectively then { 1} { 2}

A1(pi)= wi, i = 1, 2, 3, A (e ie )= u , A (e ie )= u . 2 4 − 5 1 2 6 − 7 2

Define A = c1A1 + c2A2 for some constants c1, c2. We need to find the conditions on c1, c2 such that (A, w) Γ(m∗ S3W F (3)) ker(D−1,can + 2id) for all w su(3). ∈ ⊗ ∩ ∈ −1,can Let s be the section corresponding to (A, w). Then for any vector field X, (A, w)= (A,w) ∇X A(ad(X)w) = A([X, w] ) where the Lie bracket is in the Lie algebra su(3). Using this action − | of −1,can we can calculate ∇ 7 D−1,can(A, p )(eH)= e −1,can(A, p )(eH) 1 i ·∇ei 1 Xi=1 2 = ( ( e A(2p )+ e A(2p ))e A( e )+ e A(e )+ e A(e )+ e A(e )) − 3 − 2 · 3 3 · 2 4 · − 5 5 · 4 6 · 7 7 · 6 2c u u u u = 1 ( e w + e w ) c ( e i 1 + e 1 + e i 2 e 2 ) − 3 − 2 · 1 3 · 2 − 2 − 4 · 2 5 · 2 6 · 2 − 7 · 2 4c + 3ic 4c + 3ic = 1 2 w = 1 2 A (e ). 3 1 3 1 1 − The operator D 1,can acts trivially on the subspaces of (su(3))C isomorphic to C and W F ( 3). − On the remaining subspaces we can compute the action of the Dirac operator as 4c + 3ic D−1,can(A, p )(eH)= 1 2 A (e ), i = 1, 2, 3, 1 3 1 i 20c 3ic D−1,can(A, e ie )(eH)= 1 − 2 A (e ie ), 4 − 5 9 2 4 − 5 20c 3ic D−1,can(A, e ie )(eH)= 1 − 2 A (e ie ). 6 − 7 9 2 6 − 7 − 10i Thus for any w (su(3))C, (A, w) ker(D 1,can + 2id) if and only if c = c . Thus only one ∈ ∈ 2 3 1 copy of su(3) lies in ker(D−1,can + 2id). Note that similarly to Remark 4.5 here also for c = 4i c , (A, w) ker(D−1,can 8 id). 2 − 3 1 ∈ − 3 ∗ 3 Case: 3-Hom(V , mC S W F ( 3)) × V (0,0,2) ⊗ − Sp(1) Sp(1) ⊗ (0,0,2) Let f , i = 1 . . . 3 be as before and denote by ν , i = 1 . . . 4 a basis of S3W F ( 3). Then the i { i } − subspace of W F (3) S3W F ( 3) isomorphic to S2W is given by Span w , w , w where ⊗ − { 1 2 3} 3i 5i w = (e ie ) (− ν ) + (e ie ) (− ν + ν ), 1 4 − 5 ⊗ 4 1 6 − 7 ⊗ 4 2 4

34 Ragini Singhal January 7, 2021

w = (e ie ) (iν + ν ) + (e ie ) (iν ν ), 2 4 − 5 ⊗ 2 4 6 − 7 ⊗ 1 − 3 5i 3i w = (e ie ) ( ν + ν ) + (e ie ) ( ν ) 3 4 − 5 ⊗ 4 1 3 6 − 7 ⊗ 4 2 and the space Hom(V , (m∗ S3W F ( 3))) = Span γB where γB is deﬁned by (0,0,2) C ⊗ − { } i 1 γB(f )= w , γB(f )= (w w ), γB(f )= i(w + w ). 1 2 2 2 2 1 − 3 3 − 1 3

The action of ei, i = 1 . . . 7 on fj, j = 1 . . . 3 is the same as Case 1 and thus we can calculate −1,can B D (γ ,f1) as

7 D−1,can(γB,f )(eH)= e −1,can(γB,f )(eH) 1 i ·∇ei 1 Xi=1 = e γB(2f ) e γB(2f ) 2 · 3 − 3 · 2 = 2ie (w + w ) e (w w ) − 2 · 1 3 − 3 · 1 − 3 = iw = 2 γB(f ). − 2 − 1 − ∗ This implies V ker(D 1,can + 2id) Γ(m g )C. (0,0,2) ⊆ ∩ ⊗ 2 ∗ 3 Case: 4-Hom(V , mC S W F ( 3)) V (1,1,0) ⊗ − SU(2)×U(1) ⊗ (1,1,0) 2 As above in Case 2, let a basis of the subspace of (su(3))C isomorphic to S W ∼= su(2) be given ∗ 3 ∗ 2 by Span p ,p ,p . The basis of mC S W F ( 3) (m g )C isomorphic to S W is given by { 1 2 3} ⊗ − ⊂ ⊗ 2 Span w , w , w where { 1 2 3} ν iν ν + iν w = (e ie ) 2 − 3 + (e ie ) 1 4 , 1 4 − 5 ⊗ 2 6 − 7 ⊗ 2 ν + 2iν ν + 2iν w = (e ie ) 4 1 + (e ie ) 3 2 , 2 4 − 5 ⊗ 2 6 − 7 ⊗ 2 ν 2iν ν + 2iν w = (e ie ) 1 − 4 + (e ie ) 2 3 . 3 − 4 − 5 ⊗ 2 6 − 7 ⊗ 2

The subspace of (su(3))C isomorphic to W F ( 3) is given by Span e4 + ie5, e6 + ie7 . The 2 3 ∗ − { } subspace of S W S W F ( 3) (m g )C isomorphic to W F ( 3) is given by SpanC u , u ⊗ − ⊂ ⊗ 2 − { 1 2} where

ν iν 2ν iν ν 2iν u = ie 2 − 3 + e 1 − 4 + ie 1 − 4 , 1 − 1 ⊗ 2 2 ⊗ 2 3 ⊗ 2 ν + iν 2ν + iν ν + 2iν u = ie 1 4 + e 2 3 ie 2 3 . 2 − 1 ⊗ 2 2 ⊗ 2 − 3 ⊗ 2 ∗ 3 (1,1,0) ∗ 3 Again if we denote the spaces Hom(V (1, 1, 0), mC S W F ( 3)) and Hom(V , mC S W F ( 3)) ⊗ − ⊗ − by Span B , Span B respectively then { 1} { 2} B1(pi)= wi, i = 1, 2, 3,

B2(e4 + ie5)= u1, B2(e6 + ie7)= u2.

Again as before we need to ﬁnd the conditions on c1, c2 such that (B = c1B1 + c2B2, w) − ∈ ker(D 1,can + 2id) for all w (su(3))C. By similar computations as Case 2, we can calculate, ∈ 7 D−1,can(B,p )(eH)= e −1,can(B,p )(eH) 1 i ·∇ei 1 Xi=1

35 Deformations of Nearly G2-instantons January 7, 2021

2 = ( ( e B(2p )+ e B(2p )) + e B( e )+ e B(e )+ e B(e )+ e B(e )) − 3 − 2 · 3 3 · 2 4 · − 5 5 · 4 6 · 7 7 · 6 2c u u u u = 1 ( e w + e w ) c ( e i 1 + e 1 + e i 2 e 2 ) − 3 − 2 · 1 3 · 2 − 2 − 4 · 2 5 · 2 6 · 2 − 7 · 2 4c 3ic 4c 3ic = 1 − 2 w = 1 − 2 B (e ). 3 1 3 1 1 − Once can check that D 1,can acts trivially on the subspaces of (su(3))C isomorphic to C, W F (3) and 4c 3ic D−1,can(A, p )(eH)= 1 − 2 B (e ), i = 1, 2, 3, 1 3 1 i 20c + 3ic D−1,can(A, e + ie )(eH)= 1 2 B (e + ie ), 4 5 9 2 4 5 20c + 3ic D−1,can(A, e + ie )(eH)= 1 2 B (e + ie ). 6 7 9 2 6 7

− 10i Thus for all w (su(3))C, (B, w) ker(D 1,can + 2id) if and only if c = c which proves ∈ ∈ 2 − 3 1 that only one copy of su(3) lies in ker(D−1,can + 2id) in this case as well. It immediately follows from the given action that for c = 4i c , (B, w) ker(D−1,can 8 id). 2 3 1 ∈ − 3 ∗ Case: 5-Hom(V , mC F (6)) × V (1,1,0) ⊗ SU(2) U(1) ⊗ (1,1,0) From before we know that the subspace of (su(3))C isomorphic to W F (3) is given by Span e { 4 − ie5, e6 ie7 . if we denote by µ a basis vector for the 1-dimensional representation F (6), the − } ∗ subspace of mC F (6) isomorphic to W F (3) is given by SpanC (e4 + ie5) µ, (e6 + ie7) µ . ⊗ ∗ { ⊗ ⊗ } Let Hom(V , mC F (6)) = Span α . We can deﬁne α as follows, (1,1,0) ⊗ { } α(e ie ) = (e + ie ) µ, α(e ie )= (e + ie ) µ. 4 − 5 6 7 ⊗ 6 − 7 − 4 5 ⊗ −1,can Since V is isomorphic to the adjoint representation (su(3))C, (α, v)(eH) = α([X, v]) (1,1,0) ∇X − where X m, v W F (3) su(3). Thus we can compute ∈ ∈ ⊂ 7 D−1,can(α, e ie )(eH)= e −1,can(α, e ie )(eH) 4 − 5 i ·∇ei 4 − 5 Xi=1 2i 2 2i = (e α( (e ie )) + e α( (e ie )) + e α( (e ie ))) − 1 · 3 4 − 5 2 · 3 6 − 7 3 · 3 6 − 7 2 = (ie (e + ie ) µ e (e + ie ) µ ie (e + ie ) µ) −3 1 · 6 7 ⊗ − 2 · 4 5 ⊗ − 3 · 4 5 ⊗ = 2(e + ie ) µ = 2α(e ie ). − 6 7 ⊗ − 4 − 5 ∗ −1,can Therefore Hom(V , mC F (6)) V ker(D + 2id) and thus lies in the (1,1,0) ⊗ SU(2)×U(1) ⊗ (1,1,0) ⊂ deformation space.

∗ Case: 6-Hom(V , mC F ( 6)) × V (1,1,0) ⊗ − SU(2) U(1) ⊗ (1,1,0) The subspace of (su(3))C isomorphic to W F ( 3) is given by SpanC e4 +ie5, e6 +ie7 . We denote ∗ − { } F ( 6) = Span ν . Then mC F ( 6) isomorphic to W F ( 3) is given by Span (e4 ie5) − { } ⊗ −∗ − { − ⊗ ν, (e ie ) ν . Let Hom(V , mC F ( 6)) = Span β then 6 − 7 ⊗ } (1,1,0) ⊗ − { } β(e + ie )= (e ie ) ν, β(e + ie ) = (e ie ) ν. 4 5 − 6 − 7 ⊗ 6 7 4 − 5 ⊗

36 Ragini Singhal January 7, 2021

−1,can Since V = (su(3))C, (β, v)(eH) = β([X, v]) where X m, v W F ( 3) (1,1,0) ∼ ∇X − ∈ ∈ − ⊂ (su(3))C. Thus we can compute

7 D−1,can(β, e + ie )(eH)= e −1,can(β, e + ie )(eH) 4 5 i ·∇ei 4 5 Xi=1 2i 2 2i = (e β(− (e + ie )) + e β( (e + ie )) + e αA(− (e + ie ))) − 1 · 3 4 5 2 · 3 6 7 3 · 3 6 7 2 = (ie (e ie ) ν + e (e ie ) ν ie (e ie ) ν) −3 1 · 6 − 7 ⊗ 2 · 4 − 5 ⊗ − 3 · 4 − 5 ⊗ = 2(e ie ) ν = 2β(e + ie ), 6 − 7 ⊗ − 4 5 −1,can ∗ which by translation invariance of D shows that Hom(V , mC F ( 6)) (1,1,0) ⊗ − SU(2)×U(1) ⊗ V ker(D−1,can + 2id). (1,1,0) ⊂

4.5 Summary For three out of the four considered normal homogeneous spaces the canonical connection is rigid as an H-connection. As a G2-connection the canonical connection has a non-trivial inﬁnitesimal deformation space except for the round S7. 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 G2 spaces G/H when the structure group is H or G2 is isomorphic to

G/H Structure group H G2 Spin(7)/G2 0 0

SO(5)/SO(3) 0 so(5)

Sp(2) Sp(1) × V (0,1) sp(2) sp(1) V (0,1) Sp(1) Sp(1) R ⊕ ⊕ R × SU(3) SU(2) × 0 2su(2) 4su(3) SU(2) U(1) ⊕ × where V (0,1) is the unique 5-dimensional complex irreducible Sp(2)-representation.

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