Deformations of nearly K\"ahler instantons

We formulate the deformation theory for instantons on nearly K\"ahler six-manifolds using spinors and Dirac operators. Using this framework we identify the space of deformations of an irreducible instanton with semisimple structure group with the kernel of an elliptic operator, and prove that abelian instantons are rigid. As an application, we show that the canonical connection on three of the four homogeneous nearly K\"ahler six-manifolds G/H is a rigid instanton with structure group H. In contrast, these connections admit large spaces of deformations when regarded as instantons on the tangent bundle with structure group SU(3).

Lemma 1. Let α ∈ Λ 1 V and β ∈ Λ p V . Then in Cl(V, g), Recall that the space S of spinors for (V, g) is a real eight-dimensional vector space equipped with a positive-definite symmetric bilinear form (·, ·) which carries a representation of the Clifford algebra Cl(V, g). Let ψ ∈ S be any spinor of unit length, and let ψ T ∈ S * be the conjugate of ψ with respect to the symmetric bilinear form. Then ψ ⊗ ψ T is a self-adjoint element of S ⊗ S * ∼ = Cl(V, g).
Let us note here for future reference that the stabiliser subgroup in Spin(V, g) of ψ is isomorphic to SU (3), and that the corresponding subgroup of SO(V, g) is again isomorphic to SU (3). The spinor ψ defines a linear map φ → φ · ψ from Λ 0 ⊕ Λ 1 ⊕ Λ 6 to S. This map is easily seen to be an isometry with respect to the metrics g and (·, ·) so must be injective. Since its domain and target have equal dimension it is an isomorphism: Lemma 2. The subspaces of S isomorphic to Λ 0 , Λ 6 and Λ 1 are eigenspaces of the operations of Clifford multiplication by P and Q. The associated eigenvalues are Proof. The operations of multiplication by P and Q are SU(3)-equivariant, since P and Q are constructed from ψ and ψ is SU(3)-invariant. Therefore the subspaces of S isomorphic to Λ 0 ⊕ Λ 6 and Λ 1 are fixed by P and Q. The subspace Λ 1 forms an irreducible representation of SU(3), so P and Q act by scalar multiplication on this subspace. Since the action of Q is self-adjoint and commutes with the action of Vol g on Λ 0 ⊕ Λ 6 , Q must act as multiplication by some real constant q 1 on this space. Since the action of P is self-adjoint and anti-commutes with the action of Vol g , there must exist constants p 1 and p 2 such that P · ψ = p 1 ψ + p 2 Vol g · ψ and P · Vol g · ψ = p 2 ψ − p 1 Vol g · ψ. Given additionally that the actions of P and Q are both traceless, they must take the following form with respect to the decomposition S ∼ = Λ 0 ⊕ Λ 6 ⊕ Λ 1 : Equation (4) then implies that 1 + p 1 − q 1 = 8, 1 − p 1 − q 1 = 0, p 2 = 0 and 1 + q 1 /3 = 0. The unique solution of this system of equations is p 1 = 4, p 2 = 0 and q 1 = −3, giving the advertised result.
Having identified a complex structure one may define a Kähler form ω in the usual way and a unique (up to normalisation) complex 3-form Ω of type (3,0). Although these are not needed in what follows we pause to explain how these are related to the forms P and Q. With suitable normalisation of Ω, it holds that ω = * Q and that Ω = P + i * P.
Proofs of these equations are supplied in Appendix A. Let us remark here that Lemma (2) implies that P 2 = 1 8 Tr S (P 2 ) = 4. This relation implies that Ω is given a standard normalisation in which Ω = (e 1 + ie 2 ) ∧ (e 3 + ie 4 ) ∧ (e 5 + ie 6 ) in some orthonormal basis e 1 , . . . , e 6 . If ψ solves the real Killing spinor Equation (1) then its length is constant. Therefore any six-dimensional nearly Kähler manifold admits non-vanishing forms P and Q defined as above. Since ψ is non-vanishing it defines an SU(3)-structure on M , such that ψ and the forms P and Q are parallel with respect to any connection with holonomy group contained in SU (3).
Lemma 3. The differential forms P and Q satisfy the differential identities, dP = 4Q, d * Q = 3 * P.
Proof. The exterior derivative of any form φ may be calculated using the Levi-Civita connection via the identity dφ = ∇ LC ∧ φ.
Let e a be a local orthonormal frame for the cotangent bundle, and write ∇ LC = e a ⊗ ∇ LC a . Then, from Equation (4)  Therefore dP = 4Q and dQ = 0. Similarly, since P · Vol g = * P and Q · Vol g = * Q, ∇ ∧ (Vol g + * P − * Q) = Therefore d * Q = 3 * P and d * P = 0.
It should also be noted that the real Killing spinor equation is equivalent to the differential equations (∇ X J)X = 0 ∀X ∈ Γ(T M ), for J [29]. Thus Kähler six-manifolds may equivalently be defined to be almost Hermitian manifolds whose non-integrable almost complex structure satisfies this identity.
Vector fields X that preserve the metric g and the Killing spinor ψ are called automorphic. If X is automorphic, then X preserves also P, Q, ω, Ω and J.
A key feature of nearly Kähler geometry is the presence of a distinguished connection on the tangent bundle with holonomy SU(3) and skew parallel torsion. Let t ∈ R be a parameter, and let ∇ t be the connection constructed from the Levi-Civita connection ∇ LC as follows: The torsion tensor T t of the connection ∇ t is proportional to P : The connection ∇ t acts on sections η of the spin bundle as follows: It follows from Equation (1) and Lemma 2 that, for any vector field X, Therefore ψ is parallel with respect to the connection ∇ 1 , and ∇ 1 has holonomy group contained in SU(3). The connection ∇ 1 is known as the "canonical" or "characteristic" connection. For later use, we note here the following formula for the Ricci curvature tensor of the connection ∇ t : Proposition 1. The Ricci tensor Ric t of the connection ∇ t is equal to (5 − t 2 ) times the metric g.
Proof. Friedrich and Ivanov derive [24] an expression for the Ricci tensor of a connection with totally skewsymmetric torsion in terms of the Ricci tensor of the Levi-Civita connection and the torsion 3-form. In the case where the torsion three-form is tP their formula reads By Lemma 3, d * P = 0. As has already been noted, the Ricci tensor Ric 0 of the Levi-Civita connection equals 5 times the metric g.
The remaining term may be evaluated by a direct calculation, using the identity With the aid of Lemma 2 one finds that and similarly that Finally, {X, P } · Z · ψ can be evaluated by taking inner products with elements of SM . It holds that since the element [X, Z] in the Clifford algebra is an antisymmetric endomorphism of SM . Similarly, (Vol g · ψ, {X, P } · Z · ψ) = −4g(JX, Z)ψ and (W · ψ, {X, P } · Z · ψ) = 0 for all W ∈ T M . Therefore These formulae together allow evaluation of the trace: The result follows.

Instantons and deformations
Let A be a connection on a principal K-bundle P over a nearly Kähler six-manifold (M, g, ψ) and let F be its curvature. Then A is called an instanton if its curvature F satisfies Note that in this equation only the two-form part of F is acting on ψ. Thus if the adjoint bundle associated to the principal bundle P is denoted Ad P , the left hand side is a section of Ad P ⊗ SM . The instanton Equation (9) can be reformulated in a number of ways. Firstly, the SU(3)-structure defines a subbundle su(3)M of End(T M ) and also, via the metric-induced isomorphism End(T M ) ∼ = Λ 2 M , of Λ 2 M . The instanton Equation (9) is equivalent to the statement, Secondly, as on Kähler manifolds, the instanton equation given by Equation (10) is equivalent to the Hermitian-Yang-Mills equation Thirdly, the instanton equation is equivalent to It is straightforward to see that Equation (9) implies Equation (12): Equation (9) implies that the two-form part of F acts trivially on Λ 0 ⊕ Λ 6 and sends Λ 1 to itself with respect to the decomposition S ∼ = Λ 0 ⊕ Λ 1 ⊕ Λ 6 ; Lemma 2 then implies that F Q = − 1 2 {F, Q} = − 1 2 {F, 1} = −F . It can further be proved that (12) implies (11) by classifying the eigenvalues and eigenspaces of the operator on two-forms given by contraction with Q [31].
Instantons on nearly Kähler manifolds have the desirable property of solving the Yang-Mills equationsee [32] for a proof based on Equation (9) and the Killing spinor equation, or [63, Prop 2.10] for a proof based on Equation (12) and the differential identities given in Lemma 3. The canonical connection ∇ 1 on the tangent bundle of a nearly Kähler manifold is always an instanton [32].
The purpose of this note is to study perturbations of instantons. A perturbation of a connection A is a section ǫ of Ad P ⊗ T * M , and to leading order the corresponding perturbation of the curvature F is d A ǫ.
The gauge freedom in the perturbation can be fixed by imposing the standard condition d A * ǫ = 0; thus an infinitesimal perturbation of an instanton A is given by a solution ǫ to the equations, The purpose of the next proposition is to identify solutions of the above system with eigenmodes of a Dirac operator.
Proposition 2. Let ǫ be a section of Ad P ⊗ T * M , let t ∈ R, and let D t,A be the Dirac operator constructed from the connections ∇ t and A. Then ǫ solves equations (13) if and only if Proof. Let e a be a local orthonormal frame for T * M . The identity a ψ is easily verified. The Killing spinor Equation (1) and the identity e a · ǫ · e a = 4ǫ then imply that It follows from Equation (8) that the Dirac operator D t,A is given by Then by Lemma 2, we have From this identity one obtains that the equation The latter is equivalent to the pair (13) of equations, because the two components d A ǫ · ψ and (d A ) * ǫ · ψ belong to the complementary subspaces (Λ 1 M ⊕ Λ 6 M ) ⊗ Ad P and Λ 0 M ⊗ Ad P of SM ⊗ Ad P .
It is worth noting that Equation (14) is independent of t. The linearised instanton equations can also be formulated as part of an elliptic complex [48]. However, we have found the spinorial formulation more practical to work with, not least because there already exists a large body of literature containing useful identities for Dirac operators with torsionful connections.
By the previous proposition, to prove that an instanton is rigid it suffices to prove that 2 does not belong to the spectrum of the restriction of the Dirac operator D t,A to Λ 1 M ⊗ Ad P ⊂ SM ⊗ Ad P . To this end, we need a Schrödinger-Lichnerowicz formula for the square of the Dirac operator. Such a formula has been obtained in the case A = 0 by Agricola and Friedrich [3]; the following proposition provides a gauged version of their formula. A complete proof is presented below in order to keep this discussion self-contained.
Proposition 3. Let EM be the vector bundle obtained from P through some representation E of G and let η ∈ Γ(EM ⊗ SM ). Let A be any connection on P and let t ∈ R. Then (Note that in this formula the two-form part of F acts by Clifford multiplication on SM and the Ad P part of F acts on EM in the usual way) Proof. Let e 1 , . . . , e 6 be a local orthonormal frame for the tangent bundle. The square of the Dirac operator expands as follows: By the usual Schrödinger-Lichnerowicz formula, the first of the three terms on the right of this expression is The second term is One simplifies the third term using the identity α · α = α 2 − (e a α ∧ e a α) valid for any three-form α. Note that the expression for α · α has no components in Λ 2 M or Λ 6 M , because α · α is self-adjoint while two-forms and six-forms are skew-adjoint. Thus (e a P ∧ e a P ) · η.
Now, for any two-form β, we have β · β = − β 2 + β ∧ β. Given that a e a P 2 = 3 P 2 , we have (e a P ) · (e a P ) = −3 P 2 + (e a P ) ∧ (e a P ). Hence at the center of a normal frame (where the Christoffel symbols vanish), we have Combining the above equations yields the desired result.
The proof of the preceding proposition is very general and makes no assumptions about the dimension of M , the existence of Killing spinors, or whether A is an instanton. In the case of interest, the scalar curvature is equal to 30 (when λ = 1/2), P 2 = 4, and dP = 4Q (see Lemma 3), hence This formula should be compared with [4, Equation (2)] in the case A = 0, t = 1. From Lemma 2 one learns that Q acts as multiplication by 4 on Λ 1 M ⊂ SM and as multiplication by −3 on Λ 0 M ⊕ Λ 6 M ⊂ SM . By virtue of the instanton equation, the curvature term acts trivially on where it should be noted that F acts on ǫ by contraction of forms and via the action of the Lie algebra of the gauge group on E. Hence we obtain for The most useful case of the identity (18) is when t = 1, for the following two reasons. Firstly, t = 1 is the value that maximises the right hand side of the identity, and hence yields the strongest lower bound on the square of the Dirac operator. Secondly, when t = 1 the Laplace operator on the right hand side of the identity is the one for the canonical connection, which (as demonstrated in the next section) has useful representation-theoretical properties on homogeneous spaces. Since ψ is parallel with respect to ∇ 1 , the t = 1 case of the identity is equivalent to From the point of view of analysing instantons, the most useful case of Proposition 3 is when the vector bundle EM equals Ad P . Let H ⊂ K denote the holonomy group of the connection A. The group H acts on the Lie algebra k of K. Let k 1 ⊂ k be the subspace on which H acts trivially, and suppose that there is a complementary subspace k 0 ⊂ k such that k ∼ = k 0 ⊕ k 1 (when k admits an H-invariant non-degenerate bilinear form, such a complementary subspace exists). There is a corresponding splitting of the adjoint bundle: Proposition 4. Let A be an instanton on P with holonomy group H and suppose that Ad P splits as above. Then Moreover, the volume form induces on both ker((D 1/3,A ) 2 − 4) ∩ (Ω 0 L 0 ⊕ Ω 6 L 0 ) and ker((D 1/3,A ) 2 − 4) ∩ Ω 1 L 0 almost-complex structures that swap the two copies of the vector space in the decompositions given above.
Proof. Note that ∇ 1,A respects the decomposition as does (D 1/3,A ) 2 (by Proposition 3). Therefore to establish the first identity we must show that ker((D 1/3,A ) 2 − 4)∩Ω 1 L 1 and ker(( Therefore ∇ 1,A η = 0. However, Λ 0 L 0 ⊕ Λ 6 L 0 has no non-zero parallel sections, because the fibre k 0 of L 0 has no non-zero elements fixed by the action of the holonomy group H of A (and the holonomy group of ∇ 1 acting on Λ 0 ⊕ Λ 6 is trivial).
To establish the second identity we argue as above that any element of ker((D 1/3,A ) 2 − 4) ∩ (Ω 0 L 1 ⊕ Ω 6 L 1 ) is parallel. By the general holonomy principle the space of parallel sections of Λ 0 L 1 is isomorphic to the fixed set of H in k 1 , which is the whole of k 1 by definition. Similarly, the space of parallel sections of Λ 6 L 1 is isomorphic to k 1 . The sum of these two spaces is naturally isomorphic to k ⊗ C, with the almost complex structure given by multiplication with Vol g .
To establish the third identity we note that the connection A fixes the subbundle L 0 ⊂ Ad P and, by the first part of this proposition, ker(( Multiplication by the volume form defines a linear map from this vector space to itself. This map squares to −1 so is an almost complex structure. It also swaps the two summands on the right hand side because it anti-commutes with D 1/3,A . Therefore the total space is isomorphic to the complexification of one of the two factors. The preceding proposition has important consequences in two particular cases. Firstly, if the structure group K is abelian then k 0 = 0 and ker((D 1/3,A ) 2 − 4) ∩ Ω 1 Ad P is zero-dimensional, so the space of deformations of the instanton is a subspace of a zero-dimensional space and hence is zero-dimensional. Thus: Theorem 1. Any instanton on a principal bundle with abelian structure group is rigid.
Secondly, we recall that a connection on a principal bundle is called irreducible if its holonomy group equals the structure group of the principal bundle. If A is an irreducible connection and the structure group of P is semisimple then k 1 = 0. In this case the previous proposition implies that: Theorem 2. The space of deformations of an irreducible instanton on a principal bundle with semisimple structure group is isomorphic to the kernel of the elliptic operator We end this section with some comments on a geometrical interpretation of the eigenspace of (D 1/3,A ) 2 acting on Ω 1 Ad P with eigenvalue 4. Given any orthogonal connection ∇ on the tangent bundle and any connection A on a principal bundle with curvature F , the ∇-Yang-Mills equation for A is When ∇ is the Levi-Civita connection, this equation is just the usual Yang-Mills equation. The instanton equation on a nearly Kähler six-manifold implies the ∇ t -Yang-Mills equation for any t ∈ R. Indeed, the term involving the torsion is proportional to F P , and vanishes as F is a (1,1)-form while P is the real part of a (3,0)-form.
The Yang-Mills equation for the canonical connection ∇ 1 , with ǫ a section of Ad P ⊗ T * M . The next proposition proves an identity which relates this equation to the operator (D 1/3,A ) 2 .
Proposition 5. Let EM be the vector bundle obtained from a K-principal bundle P over a nearly Kähler six-manifold through some representation E of K and let ǫ ∈ Γ(EM ⊗ T * M ). Let A be any connection on P. Then Proof. The Weitzenböck identity states that With our normalisation conventions, Ric 0 is equal to 5 times the identity. The Laplacian (∇ 0,A ) * ∇ 0,A is related to the Laplacian (∇ 1,A ) * ∇ 1,A that appears in Equation (19) as follows: Combining the above two identities with Equation (19) yields the desired result.
Suppose now that A is an instanton and ǫ is a section of Λ 1 M ⊗ Ad P which solves the linearised torsionful Yang-Mills Equation (20) and the gauge-fixing condition (d A ) * ǫ = 0. Then by the previous proposition One can easily check that both ǫ + and ǫ − satisfy (d A ) * ǫ = 0, and hence by the previous proposition both solve Equation (20). Thus we have proved: Proposition 6. Let A be an instanton on a principal bundle P over a nearly Kähler six-manifold and let ǫ ∈ Γ(Ad P ⊗ T * M ). Then ǫ satisfies the linearised torsionful Yang-Mills Equation (20) and the gauge-fixing

Instantons on homogeneous nearly Kähler manifolds
There are precisely four homogeneous nearly Kähler six-manifolds: (2), In all four cases, the nearly Kähler metric on G/H is induced from a multiple of the Cartan-Killing form on the Lie algebra g of G. In particular, the metric normalised as in Section 2 is induced from the positive symmetric bilinear form [42], Let m denote the orthogonal complement with respect to B of the Lie algebra h of H. The subspace m is invariant under the adjoint action of H, making the homogeneous space reductive. The tangent and cotangent bundles of G/H may be identified with the bundles associated to the H-principal bundle G → G/H via the natural representations The canonical connection on the H-principal bundle G → G/H is by definition the h-valued part of the left-invariant Maurer-Cartan form on G. The curvature of this connection is G-invariant, and may be identified with the H-invariant element of Λ 2 m * ⊗ h given by (here π h denotes projection onto h). The canonical connection on the principal bundle induces a connection on the tangent bundle, and it is well-known that this connection coincides with the canonical connection associated with the nearly Kähler structure. To verify this fact, it suffices to verify that the holonomy is contained in SU (3) and that the torsion is skew-symmetric, and then appeal to [24,Thm 10.1]. Alternatively, one could further verify that the torsion is parallel and appeal to [2,Thm 4.2]. In particular, this connection is an instanton.
The four nearly Kähler coset spaces therefore provide an ideal testing ground for the study of nearly Kähler instanton deformations. There are in fact two natural deformation problems to consider, as the canonical connection provides a connection on both the H-principal bundle G → G/H and the SU(3)-principal bundle associated with the SU(3)-structure. The remainder of this article is devoted to answering the following two questions on the four nearly Kähler coset spaces G/H: 1. Does the canonical connection admit any deformations as an instanton with gauge group contained in H?
2. Does the canonical connection admit any deformations as an instanton with gauge group contained in SU (3)?
To answer these questions, we solve the equation D 1/3 (ǫ · ψ) = 2ǫ · ψ for a section ǫ of the bundle associated to the H-representation h ⊗ m * in the first case and su(3) ⊗ m * in the second case. Note that a positive answer to the first question implies a positive answer to the second. For the case of SU(3)/U(1) 2 , the first question has already been answered in the negative in Theorem 1 using a simple positivity argument. In the remainder of this section we show that the same argument is inapplicable for the remaining three coset spaces, and in the following section complete answers are derived using group-theoretical analysis.
To this end, we first present a formula for the F -dependent term in Equation (18) in terms of a Casimir operator for h. We define Cas h ∈ Sym 2 (h) to be the inverse of the metric on h obtained by restriction of B. If I 1 , . . . , I dim(H) is an orthonormal basis for h then . (Here and throughout we denote by the same symbol representations of a Lie group and its Lie algebra).

Lemma 4.
Let (E, ρ E ) be any representation of H. Let F ∈ Λ 2 m * ⊗ h be as in Equation (21) and let ǫ ∈ m * ⊗ E. Then Proof. Pick orthonormal bases I i with i ∈ {1, . . . , dim(H)} for h and I a with a ∈ {dim(H)+1, . . . , dim(H)+6} for m. To simplify notation, we use indices i, j, k ∈ {1, . . . , dim(H)} and a, b, c ∈ {dim(H)+1, . . . , dim(H)+6}. Let e a be the basis for m * dual to I a . The structure constants f k ij , f b ia , f i ab , f c ab are defined by the formulae The result follows.
For the positivity argument used in Theorem 1 to be applied to any other instanton, it is necessary that the curvature term in Equation (19) is greater than −4. The following proposition shows that this condition does not hold for any of the homogeneous nearly Kähler manifolds other than SU(3)/U(1) 2 .
Proposition 7. Let M = G/H be a homogeneous nearly Kähler manifold and let A be the canonical connection on T M . Then the operator ǫ → −2F ǫ on m * ⊗ h has the following eigenvalues and eigenspace dimensions: • case SU(2) 3 /SU (2): Proof. In all cases the operator is evaluated using the Casimir expression given in Lemma 4, with E = h or an irreducible subspace thereof. We use the Freudenthal formula for Cas h , which states that (for any Lie algebra h) in the irreducible representation with highest weight λ, with δ equal to half of the sum of the positive roots of h (see for instance [35, p. 122]).

Case: G 2 /SU(3)
We first choose the Cartan subalgebra of su(3) consisting of diagonal matrices and call a weight positive if it evaluates to a positive quantity on diag(1, 0, −1). The matrices H 1 = diag(1, −1, 0) and H 2 = diag(0, 1, −1) are dual to the fundamental weights λ 1 and λ 2 . A word of caution is advisable here and as one computes the similar quantity in the other cases: h is not a Cartan subalgebra of g, it is the Lie algebra of H and B is not (− 1 12 ) th of the Cartan-Killing form of h: the trace is taken over all of g. One may verify by direct calculation that Hints are provided in Appendix B. It follows that One finds that δ = λ 1 + λ 2 . Therefore, in the complex representation (V (m1,m2) , ρ (m1,m2) ) with highest weight m 1 λ 1 + m 2 λ 2 one finds that The representations that appear in Equation (22) break up into irreducibles as follows: 2) ).

As vector spaces, this decomposition reads
and with respect to this decomposition The representations that appear in Equation (22) break up into irreducibles as follows:

As vector spaces, this decomposition reads
and with respect to this decomposition By direct calculation one finds that B(I i , I j ) = δ ij . Since su(2) ∼ = sp(1), we can reuse some of the previous computation. Let's use notation with prime (I ′ 1 , I ′ 2 , I ′ 3 , B ′ , Cas ′ ) to denote the previous case. Since [i, j] = 2k, we have [I i , I j ] = 2ǫ i,j,k I k , hence One finds that The Casimir has a single eigenvalue −4 on m * C . On the subspace V (2,0) of h C it has eigenvalue −8, and on the tensor product In [63] the eigenvalues of an operator proportional to ǫ → −2F ǫ are calculated for the canonical connection on S 6 . It was claimed that all eigenvalues were non-negative, leading to the conclusion that this instanton is rigid. The previous proposition shows that, on the contrary, this operator has both negative and positive eigenvalues. Support for the accuracy of our calculation is provided by the following observation. Since both the action of two-forms by contraction on 1-forms and the adjoint action of su(3) are traceless, the operator ǫ → −2F ǫ must be traceless. The result presented in Proposition 7 is consistent with the tracelessness of this operator. In contrast, the calculation leading to [63,Lemma 3.4] implies that this operator is not traceless, so cannot be correct. In fact, any traceless non-negative operator is necessarily zero, so this curvature operator is non-negative only in trivial cases.
Thus the proof of rigidity of the instanton on S 6 given by [63,Thm 3.3] is invalid. For reasons explained above we have not been able to prove the rigidity of the instanton on S 6 using a positivity argument along the lines of [63]. In the next section we prove the rigidity of this instanton using more powerful methods.

The spectrum of the Laplacian
In this section the space of deformations of the canonical connection on each of the homogeneous nearly Kähler manifolds is determined. To compute this space, we first derive a representation-theoretic expression for the whole of the operator on the right hand side of the Schrödinger-Lichnerowicz formula given by Equation (19), and then determine its spectrum using Frobenius reciprocity and standard formulae for the eigenvalues of the quadratic Casimir in particular representations.
Let E be any representation of H ⊂ G, and let L 2 (G; m * ⊗ E) denote the L 2 completion of the space of m * ⊗ E-valued functions on G. This linear space carries the following left representations of G: • The representation induced by the representation ρ m * on m * (denoted by the same symbol ρ m * ); • The representation induced by the representation ρ E on E (denoted by the same symbol ρ E ).
The L 2 completion of the space of sections of the vector bundle T * (G/H) ⊗ P × H E can be identified with the fixed set L 2 (G; m * ⊗ E) H ⊂ L 2 (G; m * ⊗ E) of the combined actions of ρ R , ρ E and ρ m * . On the level of Lie algebras, elements of L 2 (G; m * ⊗ E) H are functions f satisfying Proposition 8. Let A be the canonical connection on a nearly Kähler coset space G/H, let F be its curvature, let ψ be the Killing spinor, let (ρ E , E) be a representation of H, and let ǫ be a smooth section of T * (G/H) ⊗ E(G/H). Then Proof. By the Schrödinger-Lichnerowicz formula (19) the square of the Dirac operator can be expressed as a sum of three terms involving a Laplacian, a curvature operator, and scalar multiplication. The covariant derivative ∇ 1,A from which the Laplacian is built is equal to the covariant derivative on m * ⊗ E induced by the canonical connection of the homogeneous space. It is a standard result (see [42]) that the Laplacian can be identified with the action of a Casimir on C ∞ (G; m * ⊗ E) H : (Note that the right action of Cas m on C ∞ (G; m * ⊗ E) descends to an action on C ∞ (G; m * ⊗ E) H because Cas m is H-invariant).
The curvature term may be expressed as a sum of Casimirs by virtue of Lemma 4. Inserting these expressions into the Schrödinger-Lichnerowicz formula (19) yields because the left and right actions of Cas g = Cas h + Cas m on C ∞ (G; m * ⊗ E) agree. Finally, the operator ρ m * (Cas h ) is equal to minus the action of the Ricci curvature of the canonical connection on the cotangent bundle. Proposition 1 tells us the Ricci curvature is equal to 4 times the identity. Combining the above observations yields the advertised result.
We then have that and Frobenius reciprocity tells us that the multiplicity of (W αγ , σ αγ ) in L 2 (G; U αβ ) H as representations of G is equal to the multiplicity n(α, β, γ) of U αβ in W αγ as representations of H. Since L 2 (G; E ⊗ m * C ) H = α,β n α L 2 (G; U αβ ) H and since by construction, the space of solutions to Equation (31) in L 2 (G; U αβ ) H isomorphic to γ n(α, β, γ)W αγ , the space of solutions to Equation (31) This vector space is the complexification of ker (D 1/3,A ) 2 −4 ∩Ω 1 L 0 , so ker (D 1/3,A ) 2 −4 ∩Ω 1 L 0 is a real representation of G whose complexification is isomorphic to the above sum of irreducibles. By Proposition 4, ker (D 1/3,A ) 2 − 4 ∩ Ω 1 L 0 is isomorphic to two copies of ker D 1/3,A − 2 ∩ Ω 1 L 0 . Therefore we have proved the following result.
Lemma 5. The space of instanton perturbations is a real representation of G whose complexification is isomorphic to In the notation to be introduced below, W R (1,0) is the real representation of Sp(2) whose complexification is irreducible with highest weight (1, 0); it is the unique five-dimensional real irreducible representation.
The spaces of deformations of A within the SU(3)-principal bundle associated with the SU(3)-structure on G/H are isomorphic to the following representations of G: In this table g denotes the appropriate adjoint representation in each column.
Proof. We proceed case by case to follow the algorithm explained above.
Case: G 2 /SU(3) with structure group H The adjoint representation E = su(3) of the gauge group SU(3) is irreducible. As noted in the proof of Proposition 7, the unique eigenvalue of ρ E (Cas h ) on this space is −9.
In order to calculate eigenvalues of Cas g2 we appeal once again to the Freudenthal formula (23). The Cartan subalgebra for the complexified (g 2 ) C is in fact the Cartan subalgebra of sl 3 C ⊂ (g 2 ) C . It is spanned by the matrices H 1 = diag(1, −2, 1) and H 2 = diag(0, 1, −1, ) dual to the fundamental weights λ 1 and λ 2 of g 2 . By direct calculation one finds that Therefore One finds that δ, defined to be half of the sum of the positive roots, is equal to λ 1 + λ 2 ; see indeed Equation (38). Therefore the Freudenthal formula of Equation (23) yields for the Casimir in the representation ν (m1,m2) of (g 2 ) C with highest weight m 1 λ 1 + m 2 λ 2 . The smallest eigenvalues for the Casimir operator are 0, −6 and −12. In particular −9 is not an eigenvalue of Cas g2 in any representation. Therefore Equation (31) admits no solutions ǫ ∈ C ∞ (G 2 ; m * ⊗ su (3)  Since −4 does not occur as an eigenvalue of the Cas g , Equation (31) admits no solutions on L 2 (G; su(2)⊗m * ) H and the instanton is stable.
Case: SU(2) 3 /SU(2) with structure group SU (3) Next we consider the case of gauge group SU(3). We now give a different basis of g = su(2)⊕su(2)⊕su (2). As on page 14, we let I i = I (2). The orthogonal complement m can either be seen as a six-dimensional real vector space with basis The action of su(2) on m defines a homomorphism su(2) → su(3) where ad(I i )(X j ) = ǫ ijk X k . Under the adjoint action of su(2), E = su(3) breaks up into two irreducible pieces: The component V 2 is just the embedded su(2) ⊂ su (3). We have already shown that Equation ( The Casimir Cas g attains eigenvalue −12 precisely in the irreducible representations V (2,0,0) , V (0,2,0) and V (0,0,2) . It remains to determine whether these occur as subrepresentations of L 2 (G; V 4 ⊗ m * ) H .
By Lemma 5, the space of instanton perturbations is isomorphic to a real subspace of This representation is isomorphic to the adjoint representation of G = SU(2) 3 .
The adjoint representation E = h C splits into irreducible pieces as The second component V (0,0) does not give rise to any instanton perturbations since instanton perturbations from this component correspond to perturbations of the part of the instanton with gauge group U(1), and we have already shown that abelian instantons admit no perturbations. Therefore we only consider the first component V (2,0) , for which ρ (2,0) (Cas h ) = −8.
Now we calculate the eigenvalues of Cas g in irreducible representations of G = Sp(2). We choose the Cartan subalgebra of sp(2) C to be the space of diagonal 2 × 2 skew-adjoint quaternionic matrices of the form diag(ai, bi), with a, b ∈ C and i, j, k denoting the imaginary quaternions. A weight is called positive if it evaluates to a positive real number on the matrix i diag(2i, i). The matrices H 1 = i diag(0, i) and H 2 = i diag(i, −i) are dual to the fundamental weights λ 1 , λ 2 . By direct calculation one finds that The weight δ, defined to be half of the sum of the positive roots, is equal to λ 1 + λ 2 . Therefore, by the Freudenthal formula of Equation 23, in the representation (W (m1,m2) , σ (m1,m2) ) with highest weight m 1 λ 1 + m 2 λ 2 . The following table lists the smallest eigenvalues of Cas g : (m 1 , m 2 ) (0,0) (0,1) (1,0) (0,2) (1,1) The only irreducible representation in which Cas g has eigenvalue −8 is W (1,0) . Next we identify the irreducible subrepresentations of E ⊗ m * C . Recall now from Equations (27) and (29) that as representations H, A direct computation shows that As representations of H, we therefore have We note that W (1,0) has two components in common with V (2,0) ⊗ m * C , namely V (1,1) and V (1,−1) . Therefore the space of solutions to Equation (31) in L 2 (G; V (2,0) ⊗ m * C ) is isomorphic to two copies of W (1,0) . By Lemma 5, the space of instanton perturbations is isomorphic to the real subspace of W (1,0) , which has real dimension 5.
Case: Sp(2)/Sp(1) × U(1) with gauge group SU (3) It is important here to keep in mind that (A, a) ∈ su(2) ⊕ u(1) ∼ = sp(1) ⊕ u(1) sits in su(3) as diag (A + a, −2a). Hence it acts on su(3) as ad ((A, a)) Hence the representation E := su(3) C of H splits into irreducible subrepresentations as Perturbations coming from the first two components have already been analysed, and it remains to consider the final two components V (1,3) and V (1,−3) . The eigenvalues of Cas h on these spaces are both −12. From the analysis of representations of sp(2) detailed above, we learn that the unique representation of g = sp(2) in which Cas g has eigenvalue −12 is W (0,2) , the adjoint representation.
As representations of H, we have Since the representation W (0,2) of G is the adjoint representation, we have that W (0,2) ∼ = h C ⊕ m C as representations of H. In view of Equations (27) and (28), we have that as representations of H. This decomposition has two components in common with V (1,3) ⊗ m * C (namely V (0,2) and V (1,1) ) and two components in common with V (1,−3) ⊗ m * C (namely V (0,−2) and V (1,−1) ). Therefore W (0,2) occurs in each of L 2 (G; V (1,3) ⊗ m * C ) and L 2 (G; V (1,−3) ⊗ m * C ) with multiplicity 2. Taking account of the previous calculation for gauge group H, the space of solutions to (31) in L 2 (G; E ⊗ m * C ) is isomorphic to By Proposition 4, the space of instanton perturbations is a real dimensional representation of G whose complexification is isomorphic to W (1,0) ⊕ 2W (0,2) . The subalgebra h ⊂ g is generated by

Case
The complexified orthogonal complement m C of h splits into two pieces of types (1,0) and (0,1) with respect to the almost complex structure. This complex structure can be given as a function of the 3-symmetry, as in [12,Eqn. (5)]. Given the third root of unity ζ = e We denote by (V (m1,m2) , γ (m1,m2) ) the complex one-dimensional representation of U(1) × U(1) such that γ (m1,m2) (H i ) = im i . From the considerations above, the representation E := su(3) C of U(1) × U(1) breaks up into irreducibles as The components V (0,0) correspond to perturbations with gauge group U(1) 2 , and these have already been analysed. Therefore we need only consider the remaining six components.
We have yet to compute Cas g . We use H 1 = diag(1, −1, 0) and H 2 = diag(0, 1, −1) as on page 13, but this time the trace involved in the definition of B is over sl 3 C only, not over g 2 . We therefore get This B is 3/4 times the B obtained in Equation (24), hence the Casimir is 4/3 times the Casimir of Equation (25)). Thus We can see that −12 is an eigenvalue of Cas g only in the adjoint representation of g. Therefore the tensor products of each of the irreducible components of E with m * C are (1,4) , The adjoint representation of G breaks up into irreducible representations of H as Thus g C has precisely two components in common with each of the six tensor products with m * C listed above. Therefore the space of solutions to Equation (31) in L 2 (G; E ⊗ m * C ) is isomorphic to 12g C . By Lemma 5, the space of instanton perturbations is isomorphic to 6g.
The spaces of solutions to the linearised instanton equations described in the previous theorem are at first sight surprisingly large, given that the expected dimension of the instanton moduli space is zero. In fact, we can account for all of the perturbations described in this theorem by just two simple observations. We deal first with the five-dimensional piece W R (1,0) in the case of Sp(2)/Sp(1)× U(1). This manifold is the twistor space for S 4 and its nearly Kähler structure is the canonical nearly Kähler structure on the twistor space. The pull-back of any instanton on a self-dual four-manifold to its twistor space solves the nearly Kähler instanton equation on the twistor space (see [45]). The canonical connection on Sp(2)/Sp(1) × U(1) splits into two connections with holonomy groups Sp(1) and U(1); the Sp(1)-part is the pull back of the unique Sp(2)-invariant instanton on S 4 with first Pontryagin number 1. This instanton belongs to a moduli space of dimension five [5]. Thus the space of deformations of the canonical connection on Sp(2)/Sp(1)× U(1) with gauge group contained in Sp(1) × U(1) is guaranteed to be at least five-dimensional, and the previous theorem states that this moduli space is in fact exactly five-dimensional. The remaining deformations identified in Theorem 3 are all isomorphic as representations of the automorphism group G to multiple copies of the Lie algebra g of automorphic vector fields on the nearly Kähler manifold. This suggests the existence of an operation that converts automorphic vector fields into instanton perturbations. Such an operation is identified in the following proposition.
Proposition 9. Let A be an instanton on a principal bundle P over a nearly Kähler six-manifold M . Let X be an automorphic vector field for the SU(3) structure and let χ be a section of su(3)M ⊗ Ad P ⊂ Λ 2 M ⊗ Ad P such that ∇ 1,A χ = 0. Let ǫ X = ι X χ ∈ Γ(T * M ⊗ Ad P ). Then ǫ X solves the infinitesimal instanton equation Proof. First we explore the consequences of X being an automorphic vector field. By definition, the Lie derivatives with respect to X of g, ω and Ω are zero. For any section u of (T * ) ⊗p M and any connection ∇ on T M with torsion T , Suppose that ∇ is a connection with holonomy contained in SU(3) and let u = g. Then the right hand side describes the natural action of the section ∇X + T (X, ·) of End(T M ) on g, while the left hand side of the identity vanishes. Therefore ∇X + T (X, ·) takes values in the sub-bundle of End(T M ) that fixes g and whose fibre is isomorphic to so (6). Similarly, the cases u = ω, Ω of the identity tell us that ∇X + T (X, ·) fixes ω and Ω. The conclusion then is that if ∇ is any connection with holonomy contained in SU(3) and X is an automorphic vector field for the SU(3)-structure then ∇X + T (X, ·) ∈ Γ(su(3)M ) ⊂ Γ(End(T M )). Now we show that ǫ X = ι X χ solves the infinitesimal instanton equation. Given any connection A on P and vector fields Y, Z, . We choose to use the canonical connection ∇ = ∇ 1 . This connection has holonomy contained in SU (3), so ∇ 1,A χ = 0 and therefore . We rewrite the right hand side of this equation as follows: The terms on the first line describe the linear action of ∇X + T (X, ·) on the 2-form part of χ. Since χ is a section of su(3)M ⊗ Ad P ⊂ Λ 2 M ⊗ Ad P and (by the above argument) the endomorphism ∇X + T (X, ·) fixes this subbundle, the terms in the first line also describe a section of this subbundle.
The terms in the second line describe the natural action of the two-form part of χ on the three-form P . More concretely, if we identify χ with a sectionχ of End(T M ) ⊗ Ad P such that then the second line is equal to P (χ(X), Y, Z) + P (X,χ(Y ), Z) + P (X, Y,χ(Z)) Since the two-form part of χ belongs to the subspace identified with su(3) and su(3) fixes P , these terms vanish. Therefore d A ǫ(Y, Z) is a section of Ad P ⊗ su(3) ⊂ Ad P ⊗ Λ 2 M , so solves the infinitesimal instanton equation.
Proposition 9 accounts for all of the remaining deformations identified in Theorem 3, as we now briefly explain.
Note first that the curvature F of the canonical connection on any coset space is a parallel section of su(3)M ⊗ Ad P , so the previous proposition may be applied to χ = F . The resulting deformations ǫ X = ι X F have d A ǫ X = L X F + [ι X A, F ]. Since F is invariant under automorphisms up to gauge, L X F = [λ X , F ] for some infinitesimal gauge transformation λ X . Since [ι X A, F ] also corresponds to the action of an infinitesimal gauge transformation, we conclude that the deformations in Proposition 9 obtained from F are in the direction of the gauge orbit.
In order to apply the proposition we must identify all parallel sections of su (3) • In the case of G 2 /SU(3), the representation su(3) C of SU(3) has one irreducible component; it corresponds to the curvature of the canonical connection so Proposition 9 yields no non-trivial instanton deformations.
• In the case of SU(2) 3 /SU (2), the representation su(3) C ∼ = V 2 ⊕ V 4 of SU(2) has two irreducible components and su(3) ⊗ su(3) contains two copies of the trivial representation, one in V 2 ⊗ V 2 and one in V 4 ⊗ V 4 ; the invariant element of V 2 ⊗ V 2 corresponds to the curvature of the canonical connection so the proposition yields for each element of g a one-dimensional space of instanton perturbations.
• In the case of Sp (2) In this appendix the identities (6) are proven. The proof of the first of these requires careful track to be kept of minus signs, so let us first point out that, by analysis of eigenvalues (using Lemma 2), * Q · * Q = −3 + 2Q.
To prove the second equality, it suffices to prove that (v − iJv) (P + i * P ) = 0 for all cotangent vectors v. Using Lemma 1, this statement is equivalent to proving that {v − iJv, P + i * P } = 0 for all vectors v. Having moved the statement to the Clifford algebra, we can now use the definition of J, namely Jv · ψ = Vol g · v · ψ. Since Vol g · v = * v, Jv acts on ψ ⊗ ψ T just as * v does, and from this it can be shown that {Jv, P } = { * v, P } and {Jv, * P } = { * v, * P }.
Moreover, for any three-form α, we have v · α = −( * v) · ( * α) and α · v = −( * α) · ( * v). Thus we have This appendix describes very succinctly but explicitly the embedding sl 3 C ⊂ (g 2 ) C and exhibits the results needed in the main part of the paper.
The term by term identification of the basis element is an isomorphism of Lie algebras. Let W be the standard representation of sl 3 C. Then we have the orthogonal decomposition (g 2 ) C = h ⊕ W ⊕ W * as representation of sl 3 C. So Tr (g2) C (ad(H i ) • ad(H j )) = Tr h (ad(H i ) • ad(H j )) + 2Tr(H i H j ). One can thus compute B(H i , H j ) = − 1 12 Tr (g2) C (ad(H j ) • ad(H j )) with i, j ∈ {5, 2} to obtain (once relabelling 5 into 1) Equation (24).
Note also as claimed on page 18.
Given that a vector (X, Y ) ∈ C 6 represents the element [12, p.12] gives the expression for the almost complex structure. In view of this definition, we have Given the complex ordered basis B = ( This action provides the homomorphism su(2) ⊂ su(3) necessary to understand Equation (33).