Instantons on hyperkähler manifolds

An instanton (E, D) on a (pseudo-)hyperkähler manifold M is a vector bundle E associated with a principal G-bundle with a connection D whose curvature is pointwise invariant under the quaternionic structures of TxM,x∈M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_x M,~x\in M$$\end{document}, and thus satisfies the Yang–Mills equations. Revisiting a construction of solutions, we prove a local bijection between gauge equivalence classes of instantons on M and equivalence classes of certain holomorphic functions taking values in the Lie algebra of GC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G^{\mathbb {C}}$$\end{document} defined on an appropriate SL2(C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {SL}_2({\mathbb {C}})$$\end{document}-bundle over M. Our reformulation affords a streamlined proof of Uhlenbeck’s compactness theorem for instantons on (pseudo-)hyperkähler manifolds.


Introduction
Beginning in the mid-1970s, the self-duality equations for Yang-Mills fields successfully captured the imagination of theoretical physicists and mathematicians, epitomised by Don-aldson's flight into previously unforeseen realms of four-manifold differential topology (reviewed for instance in [21,29]). A Yang-Mills field is a pair (E, D) on a Riemannian manifold (M, g), where E is a vector bundle associated with a principle G-bundle with a connection D whose curvature satisfies the Yang-Mills equation D * F = 0. The (anti-)self-duality equations, requiring that the curvature F of a connection D over a Riemannian four-manifold (M, g) takes values in the (anti-)self-dual eigenspace of the Hodge star operator, implies the Yang-Mills equation in virtue of the Bianchi identity D F = 0. Connections satisfying the (anti-)self-dual Yang-Mills (SDYM) equations are called (anti-) instantons. They are global minimisers of the Yang-Mills energy functional, S(A) = ||F|| 2 = M F ∧ * F vol g .
The quest for explicit instanton solutions [6] was initially physically motivated, for instance by the mystery of the phenomenon of quark confinement [33], but the remarkable properties of instantons soon attracted powerful mathematical treatment. First, Ward showed that solutions of the self-duality equations on R 4 are encoded in certain holomorphic data on twistor space [40], effectively converting the problem to an algebro-geometric one. Then, Atiyah et al. [4] obtained a correspondence between solutions of the SDYM equations on S 4 and certain real algebraic bundles on the complex projective 3-space CP 3 . They thus established the relation between self-duality and holomorphic structures, yielding in particular the dimension of the moduli space of solutions for any compact gauge group. This led to a sequence of ansätze yielding SDYM solutions in terms of arbitrary solutions of linear equations [5,10]. Subsequently, powerful algebro-geometric results were used to obtain a complete construction of all SDYM fields on S 4 [2,3,20].
These developments were followed by fundamental analytical results on variational methods for Yang-Mills theory. The moduli space of instantons is a subset of the quotient A/G of the space of all connections A with the group of all gauge transformations G. Locally representing the connections in Coulomb gauges Uhlenbeck [37,38] developed analytical tools to study the singularities of the compact moduli space of instantons. Uhlenbeck's work, together with the novel variational methods introduced by Taubes to study gauge invariant theories, prepared the path for Donaldson's seminal work.
These analytical results depended crucially on the fact that the Yang-Mills functional and therefore also the Yang-Mills equations are conformally invariant in four dimensions. Further, the above-mentioned constructions of SDYM solutions crucially used the fact that R 4 conformally compactifies to S 4 . All this would seem to impede any generalisation to Yang-Mills fields in higher dimensions. Indeed, it is known that a connection over the sphere S d , d ≥ 5, with sufficiently small L 2 -norm is necessarily flat [8]. However, the solvability of four-dimensional SDYM equations relies in particular on the fact that, being linear algebraic constraints on the curvature, they are first-order equations for the vector potential. This first-order property was partly responsible for the good analytical properties of the SDYM equations. Indeed, an insistence upon this familiar sight of partial-flatness conditions, requiring the vanishing of certain linear combinations of the curvature components, which automatically imply the second-order Yang-Mills equations, yields the required instanton equations in dimensions greater than four. This idea was originally pursued in [9], where it was shown that the required equations are restrictions of the curvature F to an eigenspace of an endomorphism on the space of 2-forms defined by an appropriate co-closed 4-form , * ( * ∧ F) = λF, ∈ 4 T * M, λ ∈ R * . (1.1) The co-closedness of suffices to show that a Yang-Mills curvature field satisfying (1.1) implies the second-order Yang-Mills equations. For d > 4, the 4-form is pointwise theory, lead to a new direct proof of Uhlenbeck's strong compactness theorem for instantons on hk manifolds [18,30,[36][37][38]42,43]. The paper is structured as follows. After the preliminary Sect. 2, we introduce the notion of harmonic space and discuss its relation with the twistor bundle of a (pseudo-)hyperkähler manifold M in Sect. 3. In Sect. 4, we discuss the analytic gauge condition of the pull-back of the instanton over H(M) and the construction of the instanton field over M from the corresponding holomorphic prepotential on H(M). Our main new contributions appear in Sects. 4 and 5, where we introduce a convenient normalisation for equivalent prepotentials, prove a new existence result for an essentially unique instanton corresponding to a given prepotential, obtain curvature estimates and present our new brief proof of the strong compactness theorem for instantons on hk manifolds.

Basics of hyperkähler manifolds
Given a 4n-dimensional real vector space W , we recall that a hypercomplex structure on W is a triple (I 1 , I 2 , I 3 ) of endomorphisms satisfying the multiplicative relations of the imaginary quaternions, I 2 α = − Id W and I α I β = I γ for all cyclic permutations (α, β, γ ) of (1, 2, 3). Similarly, a hypercomplex structure on a 4n-dimensional real manifold M is a triple (J 1 , J 2 , J 3 ) of integrable complex structures on M, with the property that each triple (I α := J α | x ), x ∈ M, is a hypercomplex structure on T x M.
These two notions are generalised as follows. Consider the three-dimensional subspace Q W of End (W ), which is the span Q W = span R (I 1 , I 2 , I 3 ) of the endomorphisms of a hypercomplex structure (I α ). A (pseudo-)quaternionic Kähler structure on W is an inner product ·, · on W which is Hermitian with respect to Q W , that is, every element J ∈ Q W is skew-symmetric with respect to ·, · . For what concerns manifolds, we have instead the following Definition 2.1 A 4n-dimensional (pseudo-)Riemannian manifold (M, g) of signature (4 p, 4q), p +q = n, is a (pseudo-)quaternionic Kähler (qk) manifold if it admits a subbundle Q ⊂ End (T M) of quaternionic structures on the tangent spaces satisfying the following two conditions: (i) each inner product g x is Hermitian for the quaternionic structure Q x ; (ii) the parallel transport of the Levi-Civita connection ∇ of g preserves Q.
Further, (M, g) is a (pseudo-)hyperkähler (hk) manifold if there exist three global ∇-parallel sections J 1 , J 2 , J 3 of Q (which are thus integrable complex structures) determining a hypercomplex structure on each tangent space of M.
It is well known that a qk manifold is Einstein. Moreover, it is hk if and only if its scalar curvature is zero. In this paper we focus on hk manifolds, denoting them by a 5-tuple (M, g, J α , α = 1, 2, 3), where the J α are the ∇-parallel sections generating the bundle Q.
An adapted frame at the point x ∈ M of an hk manifold (M, g, J α ) is a g-orthonormal frame u = (e A ) : R 4n → T x M mapping the standard triple of complex structures (i, j, k) of H n R 4n into the triple of complex structures (J α | x ) α=1,2,3 . Note that any adapted frame u = (e A ) allows the identification of the standard Sp 1 -and Sp p,q -actions on H n with uniquely associated actions of Sp 1 -and Sp p,q -actions on T x M. The family O g (M, J α ) of all adapted frames is a principle bundle over M with structure group Sp 1 ·Sp p,q and is invariant under the parallel transport of the Levi-Civita connection. Hence, if the bundle O g (M, J α ) admits a global section, the above actions of Sp 1 ·Sp p,q on the tangent spaces T x M combine to yield a global action of Sp 1 ·Sp p,q onto T M.
We now fix a few technical details, which we shall need.
As the standard representation of Sp 1 ·Sp p,q on H n = C 2n , we choose the one for which the element ( i 0 0 −i ) ∈ sp 1 acts on (H n ) C C 2 ⊗ C 2n as the left multiplication by the complex structure i. Hence, for any given adapted frame u ∈ O g (M, J α )| x , the corresponding action of ( i 0 0 −i ) on T x M coincides with the action of the complex structure I = u * (i). This action of Sp 1 ·Sp p,q extends by C-linearity to the standard action of SL 2 These isomorphisms are necessarily related to each other by the action of some element of Sp 1 ·Sp p,q ⊂ SL 2 (C)·Sp n (C).
Furthermore, as standard bases for C 2 and C 2n , we shall use For any (C-linearly extended) adapted frame u : the corresponding complex basis (e ja ) of T C x M is changed into , the basis (e ±a ) is then transformed into e ±a = A b a e ±b . Note that, since the vectors h 1 ⊗ e o a and h 2 ⊗ e o a are ±i-eigenvectors of the element ( i 0 0 −i ) i ∈ Sp 1 , the corresponding elements e +a , e −a ∈ T C x M are vectors of type (1, 0) and (0, 1) with respect to the complex structure I = u(i) of T x M.

Connections, gauges, potentials and Yang-Mills fields
In this section we briefly review certain basic facts about gauge theories which we shall need in what follows.
Given a manifold M and a (principal or vector) bundle π : E → M, we denote by X(M) the space of all vector fields of M and by (E) the set of all sections σ : M → E. Given X ∈ X(M) and a smooth (real or complex) function f , we write X · f to denote the directional derivative of f along X .
Let p : P → M be a principal G-bundle and p E :E = P× G,ρ V →M an associated vector bundle with fibre V R N , determined by a faithful linear representation ρ : G → GL(V ). We shall refer to an open subset U ⊂ M on which there exists a choice of gauge (local trivialisation) ϕ : P| U → U × G of the bundle P over U as the domain of the gauge ϕ (a minor abuse of the language). Given two gauges ϕ, ϕ with overlapping domains U, U , the transition function ϕ • ϕ −1 : The family of the automorphisms of G defined by h → g x ·h is what is usually called gauge transformation between the gauges ϕ and ϕ . Finally, we recall that if a principal G-bundle p : P → M admits a collection of gauges, whose domains form an open cover of M and whose associated gauge transformations h → g x ·h are determined by maps g : U → G taking values in a fixed subgroup G o ⊂ G, then such gauges determine a G o -subbundle A connection 1-form ω on P induces a unique covariant derivative on the associated bundle E. We recall that a covariant derivative on E is an operator D : In this gauge, any vector in T P| U is naturally identified with a sum X (x,g) + B (x,g) ∈ T (x,g) (U × G) T x U + T g G and the 1-form ω on P| U U × G can be pointwise expressed as a sum of the form where ω G is the Maurer-Cartan form of G and, for each g ∈ G, the map A (·,g) : U → T * U ⊗ g is a g-valued 1-form, which changes G-equivariantly with respect to g. The 1-form A := A (·,e) : U → T * M ⊗ g is called the potential of ω in the considered gauge. If two gauges P| U U × G, P| U U × G have overlapping domains U, U , the corresponding potentials A, A are related through the gauge transformation h → g x ·h by means of We now recall that a section σ : U → E| U U × V of the associated bundle has the form σ (x) = (x, s i (x)) for a smooth map (s i ) : U → V = R N . Hence, for each vector field X ∈ X(U), it is possible to consider the section of where A i j := ρ * •A with ρ * : g → gl(V ) the Lie algebra representation determined by the linear representation ρ : G → GL(V ) that gives the associated vector bundle E = P × G,ρ V . The gl(V )-valued 1-form A i j is called the potential of D induced by the connection ω. The operator (2.3) is gauge independent due to (2.2), thus globally well defined as a covariant derivative on the sections of E.
We recall that the curvature 2-form of ω is the g-valued 2-form on P defined by . Given a gauge ϕ : P| U → U × g and the associated gauge ϕ for E ϕ : the curvature tensors of ω and of D are the (g-or gl(V )-valued) 2-forms F ϕ and F x on U, defined by (2.4) Note that F ϕ can be recovered from the potential A of ω by the formula . (2.5) and that, if h → g x ·h is the gauge transformation between ϕ, ϕ , one has that This shows that the curvature tensor of ω does depend on the gauge, while the curvature tensor of D does not. Due to this, the curvatures F ϕ x combine and determine a globally defined gl(V )-valued 2-form F on M. It can be also checked that it satisfies the identity a property often used as an alternative definition of the curvature of D. All of the above notions and properties have analogues in the case of holomorphic bundles, which we now briefly recall.
If (M, J ) is a complex manifold, G is a complex Lie group and V is a complex vector space, a principal G-bundle p : P → M (resp. a complex vector bundle p : E → M with fibre V ) is called holomorphic if it is equipped with a complex structure J , such that the right action of G on P (resp. the vector bundle structure on E) is J -holomorphic and the projection p is a ( J , J )-holomorphic mapping. In this case, a trivialisation is called holomorphic if it is a local holomorphic map from (P, J ) (resp. (E, J )) to the Cartesian product M × G (resp. M × V ), equipped with the product complex structure.
A connection form ω on a holomorphic G-bundle (P, J ) is called J -invariant if the corresponding horizontal spaces H (x,g) = ker ω (x,g) are invariant under the complex structure J . This is equivalent to say that, in any holomorphic trivialisation ϕ : P| U → U × G, the corresponding potential A : U → T * U ⊗ g takes actually values in T 10 * M ⊗ g 10 + T 01 * M ⊗ g 01 . Here we denote by T 10 x M and T 01 M the holomorphic and anti-holomorphic distributions of M and by g 10 , g 01 the subalgebras of g C (both isomorphic to g), which are generated by the vectors of type (1, 0) and (0, 1), respectively. Being each A x , x ∈ U, real, the projection of A x onto T 01 * x M ⊗ g 01 is the complex conjugate of the component in T 10 * x M ⊗ g 10 . So, the potential A is uniquely determined by the associated map A 10 : U → T 10 * M ⊗ g 10 , which we call (1, 0)-potential.
We finally remark that the covariant derivative D on an associated holomorphic vector bundle p E : E → M, determined by a J -invariant connection ω, is characterised by the property that it transforms sections of E C = E 10 ⊕ M E 01 , with values in E 10 or in E 01 , into sections which are still in E 10 or in E 01 , respectively. The covariant derivatives of (1, 0)type (or (0, 1)-type) sections and the change of (1, 0)-potentials under holomorphic gauge transformations behave exactly as in formulas (2.3) and (2.2).
We conclude this short section by recalling the definition of gauge fields. A gauge field with structure group G is a pair (E, D), formed by: (1) a vector bundle E, associated with a principal G-bundle p : P → M, (2) a covariant derivative D on E, induced by a connection ω on P.
In this case we say that (E, D) is a gauge field associated with the pair (P, ω). If G is complex, p : P → M admits a reduction to a G o -subbundle P o ⊂ P with G o compact real form of G and ω restricts to a connection ω o on P o , we say that (P, ω) is reducible to (P o , ω o ) and that (E, D) is the complexification of a gauge field with compact structure group G o .
If M is an oriented (pseudo-)Riemannian manifold, we may define the Hodge * -operator, * : p T * M → n− p T * M. Then, the gauge field (E, D) is called a Yang-Mills field if its curvature tensor F satisfies the Yang-Mills equation D * F = 0.

C k -norms and L p -norms of curvatures
Let G be a reductive complex Lie group and G o ⊂ G a compact real form of G. Further let g o = g s + z(g o ) be the decomposition of g o = Lie(G o ) into its semisimple part g s and centre z(g o ), and denote by ·, · any Ad G o -invariant Euclidean inner product on g o which, on g s ×g s , coincides with minus the Cartan-Killing form. This allows us to define the Hermitian inner product on g = Lie( (2.8) and the associated norm X + iY h = X 2 ·,· + Y 2 ·,· . If (M, g) is a Riemannian manifold, we can use the metric g to extend the Hermitian product h of g to a positive inner product, also denoted by h, on the space of tensor fields in ⊗ T * M ⊗ g over M. So, for any compact subset K ⊂ M, we have the usual sup-norm for g-valued C k functions on K Similarly, for each p ∈ [1, +∞), we may use the L p -space L p (U, g), the completion of the space of all C 0 -maps V : U → g with bounded values for the integral U V p h vol g , equipped with the usual L p -norm All such norms immediately generalise to spaces of g-valued r -forms.
Consider now a gauge field (E, D) associated with the pair (P, ω) and a trivialisation ϕ : P| U → U × G in which the curvature tensors of ω and D are F ϕ and F = F ϕ , respectively. Then, for any compact subset K ⊂ U, we define Now consider the cases when (P, ω) is reducible to a pair (P o , ω o ) with structure group given by the compact G o . Since the h-norms are Ad G o -invariant, if we consider only the gauges which determine such a reduction, then the norms (2.9) do not depend on ϕ and they coincide with the usual C kand Sobolev norms of curvatures for gauge fields with compact structure groups.

Lifting gauge fields
Let (E, D) be a gauge field on M associated with a pair (P, ω). If π : N →M is a principal H -bundle over M, the lift of (P, ω) is the pair (P , ω ) given by (a) the lifted G-bundle p : P := π * P → N , i.e. the submanifold π * P := {(y, U ) ∈ N × P such that p(U ) = π(y)} ⊂ N × P equipped with the natural projection p : π * P → N , p (y, u) := y (b) the pull-back connection ω := π * ω on P = π * P determined by the projection π : P ⊂ N × P → P.
The lift of the gauge field (E, D) is the gauge field (E = π * E, D = π * D) on N given by We now briefly discuss the problem of characterising the gauge fields (E , D ) on principal H -bundles π : N → M which are lifts of gauge fields on M. Given π : N → M and (E, D) as above, for each X ∈ h = Lie(H ), the corresponding 1-parameter subgroup exp(RX ) ⊂ H has clearly a natural right action on N and determines a natural 1-parameter group of diffeomorphisms on the lifted bundle P ⊂ N × P given by Note that each map R t is a bundle automorphism that preserves the connection ω and induces a bundle automorphism of the associated vector bundle E = π * E that commutes with the covariant derivative D = π * D. Therefore, the vector field X on P , whose flow is the 1-parameter group of automorphisms R t , is such that Note that (b) implies also that ı X = 0 and hence that ı X F = 0 for X ∈ k. (2.10) All this has the following converse.

Proposition 2.2 Let (E , D ) be a gauge field associated with a pair (P , ω ) on an H -bundle
π : N → M and, for any given gauge with domain U ⊂ N , denote by F , F the curvature tensors of ω and D .
If H is simply connected, then (E , D ) is the lift of a gauge field on M if and only if, for each X ∈ h, the associated infinitesimal transformation on N is such that a) ı X F = 0 or, equivalently, ı X F = 0 in any gauge and b) the unique ω -horizontal vector field X on P , which projects onto X , is complete.
Proof The necessity follows from previous remarks. Assume now that for each X ∈ h, conditions (a) and (b) hold. Since ı X ω = 0, we have that L X ω = ı X dω + d(ı X ω ) = ı X = 0, from which it follows that the flow of X commutes with the right G-action of P and preserves ω . We also have that, for any X , We therefore conclude that the collection of vector fields is a finite-dimensional Lie algebra of complete vector fields on P . By a classical theorem of Palais [32], this implies the existence of a right H -action on P whose infinitesimal transformations are precisely the ω -horizontal vector fields X ∈ h . All orbits of this action are regular and simply connected, because all of them are coverings of the simply connected H -orbits on N . Moreover, each transformation of this action is a bundle automorphism, which preserves the connection 1-form ω and commutes with the G-action. Hence, the orbit space P = P /H is a G-bundle over M := P /H × G = P/G and is equipped with the g-valued 1-form ω defined by One can directly check that ω is a connection and that ω is the pull-back of ω on P . The associated bundle E of P → M = P/G, equipped with the covariant derivation D determined by ω, is the desired gauge field, of which (E , D ) is a lift.

Harmonic spaces of (pseudo-)hyperkähler manifolds
In the rest of this paper, (M, g, J α ) denotes an hk manifold and (E, D) a complex gauge field on (M, g, J α ) associated with a pair (P, ω) with complex structure group G. We shall mostly assume that G is the complexification of a compact real form G o and that (P, ω) is the complexification of a pair (P o , ω o ), with structure group G o . We shall simply say that (E, D) is the "complexification" of a gauge field having compact structure group G o .

The twistor bundle of a (pseudo-)hyperkähler manifold
M, each equipped with the corresponding complex structure I (z) . Such complex structures combine with the classical complex structure of CP 1 and determine a natural almost complex structure on Z (M), which we denote by I . It was proved to be integrable in [34].
We remark that the complex structures on T x M, x ∈ M, coincide with the complex structures of the form I = u * (i) ∈ span(I α := J α | x ), given by adapted frames u of T x M as mentioned in Sect. 2.1. It follows that each of the 2n-tuples of complex vectors e +a (resp. e −a ), which are part of the complex bases (2.1), is actually a frame of holomorphic (resp. anti-holomorphic) vectors for a complex leaf M × {I (z) } ⊂ Z (M).

The harmonic space of a (pseudo-)hyperkähler manifold
The harmonic space of a 4n-dimensional hk manifold (M, g, J α ) is the trivial bundle is identified with sl 2 (C) by means of right invariant vector fields. Then let I (x,U ) be the unique complex structure on T (x,U ) H(M) given by where J o is the complex structure of sl 2 (C) given by the multiplication by i 0 0 i . From the above identification T U SL 2 (C) sl 2 (C), it follows that along each fibre {x} × SL 2 (C), the I-holomorphic distribution is generated by right invariant vector fields of SL 2 (C).
The collection I of such pointwise defined complex structures is a globally defined almost complex structure on H(M), which can be seen to be integrable as follows. The family of restricted complex structures I| M×{U } = I (z) | M×{U } on the manifolds M ×{U }, U ∈ SL 2 (C), is invariant under the natural left action of SL 2 (C) on H(M). Thus, the Lie derivative of an I-holomorphic vector field that is tangent to the (horizontal) slices M × {U } by means of an infinitesimal transformations of this SL 2 (C)-action always gives another I-holomorphic vector field, which is horizontal. On the other hand, the infinitesimal transformations of the left action of SL 2 (C) on each vertical fibre {x} × SL 2 (C) SL 2 (C) are nothing but the right invariant vector fields of SL 2 (C). This implies that the Lie bracket between a horizontal I-holomorphic vector field and a vertical I-holomorphic vector field is a horizontal I-holomorphic vector field. This property together with the fact that both the horizontal and vertical I-holomorphic distributions are involutive proves that the whole I-holomorphic distribution of H(M) is involutive, i.e. that I is globally integrable.

The complexified harmonic space
Consider an n-dimensional complex manifold (N , J ) and denote by A J the complete atlas of holomorphic coordinates, i.e. of systems of coordinates We define the complexification of (N , J ) as the pair (N C , ı) given by: Note that the complex structure J of N C is defined in such a way that the corresponding atlas A J of holomorphic coordinates is generated by coordinates of the form ξ = (z i , z j ) : U × V → C 2n for some (local) holomorphic coordinates (z i ), (z j ) of (N , J ). So, the antiholomorphic involution τ : N C → N , τ (x, y) = (y, x) has a fixed point set which is precisely the totally real submanifold ı(N ) N .
The above construction yields the following very convenient extension of harmonic spaces.

Instantons on hyperkähler manifolds
As usual, let (M, g, J α ) be an hk manifold and denote by Here ω H x and ω E x are the SL 2 (C)and Sp n (C)-invariant symplectic forms of H x and E x , respectively, and 2 0 E x is the irreducible Sp n (C)-submodule of 2 E x complementary to Cω E x . Since the isomorphism T C x M H x ⊗ E x is unique up to an action of an element in Sp 1 ·Sp p,q , the decomposition (4.1) is independent of the isomorphism chosen. Now, given a gauge field (E, D) on an hk manifold, we split the (C-linear extension of the) curvature tensor F x , x ∈ M as follows: (2) component of its curvature tensor vanishes everywhere. Such instantons provide minima of the Yang-Mills functional M |F| 2 ω g and are thus solutions of the Yang-Mills equations [41]. Such instanton equations have been studied by several authors [1,9,12,28,34,36].
Notice that the vanishing of F (2) corresponds to simple conditions on the components F ±a|±b = F(e ±a , e ±b ) with respect to the complex frames (e ±a ) defined in (2.1). In fact, F (2) = 0 if and only if In four dimensions these are precisely the well-known anti-self-duality equations.

Central and exponential central gauges on harmonic spaces
A gauge ϕ = (ϕ V , ϕ G ) : P| V →V × G for the G-bundle P naturally corresponds to a gauge for its lift P on the harmonic space H(M), namely the gauge ϕ defined on the restriction of P to V × SL 2 (C) ⊂ H(M) by Such a gauge is called the central gauge determined by ϕ [1,24].
Let us now introduce a very convenient special class of central gauges. Given x o ∈ M, for any unit vector v ∈ T x o M we denote by γ is called exponential if the corresponding potential A satisfies the following conditions: tangent to the radial geodesics.
We shall call the central gauges for P on H(M) determined by exponential gauges for P on M exponential central (or just exp-central). We recall that for any x o ∈ M there is always an exponential gauge on some appropriate neighbourhood of x o [37]. Thus, for any (x o , U ) ∈ H(M) there exists an exp-central gauge for P on a neighbourhood of (x o , U ).
It is also known that for all cases in which (E, D) is the complexification of a gauge field (E o , D o ) with compact structure group G o ⊂ G, if A is the potential in an exponential gauge ϕ for the bundle P o with domain U ⊂ M, then there exists a constant c U , which depends only on U, such that

Prepotentials for instantons on hk manifolds
We recall that any hk manifold (M, g, (J α )) has a natural structure of a real analytic manifold and that in such a structure, the tensors g and J α are real analytic [26]. Hence, if we lift g to H(M) as a tensor field with values in D * ⊗ D * , using real analyticity, for each point , to which g extends as a C-linear tensor field in D C * × D C * . We claim that an analogous extension property holds also for an instanton on an hk manifold provided that it is a complexification of an instanton with compact structure group. To prove this, let us introduce some additional convenient notation. Given a gauge field (E, D) associated with (P, ω), let H ⊂ T P and H C ⊂ T C P be, respectively, the real and complex horizontal distributions of P given by the kernels of the lifted connection ω on P . Further, for any (real or complex) vector field X on H(M), let us denote by X h the uniquely associated vector field on P with values in H or H C which projects onto X . Proof To prove these statements, we need the following simple lemma. It follows that the subbundle S 01 ⊂ T C P generated by the vectors X 01h and the antiholomorphic vertical distribution of P is an involutive complex distribution. The same holds for the subbundle S 10 = S 01 ⊂ T C P . Hence, the direct sum decomposition T C P = S 10 + S 01 corresponds to a G-invariant integrable complex structure J on P . Consequently, there is an atlas of complex charts for the complex manifold (P , J ) making P a holomorphic G-bundle over H(M). Moreover, the lift P o on H(M) of the bundle P o with compact structure group is necessarily a real analytic submanifold of P since it is the fixed point set of an appropriate real analytic involution. One can also check that the restricted distribution H| P o coincides with the distribution given by the J -invariant subspaces of the tangent spaces of P o . Since the latter is a real analytic distribution on P o and the distribution H of P is the unique G-invariant extension of H| P o , we conclude that also H is J -invariant on P . Consequently, the first claim follows immediately.

Lemma 4.2 The complex gauge field (E, D) is an instanton if and only if the curvature F of its lift (E , D ) on H(M) is such that
Concerning the second claim, the same arguments as above yield the existence of a complex structure J C on the extended bundle P (and, consequently, a corresponding complex structure on the associated bundle E ), which makes it a holomorphic bundle over the com-  Hence, by the complex Frobenius theorem, for each y o = (x o , x o , U , g) ∈ P | H(M) there is an SL 2 (C)×G-invariant neighbourhood P | U of y o , which is holomorphically foliated by integral leaves of D h − . Note that the union of the H h 0 -orbits of the points of one such integral leaf is an integral leaf of the larger complex distribution D h − ⊕ < H h 0 >. It follows that P | U is actually holomorphically foliated by the integral leaves of this larger distribution and we may consider a holomorphically parameterised family of integral leaves of this distribution, which fills a complex submanifold S transversal to the G-orbits at each point. Without loss of generality, we may also assume that such a submanifold S projects biholomorphically onto an SL 2 (x o , x o , U ) and hence it is a graph of a holomorphic section of the G-bundle P . Associated with such a section, there is a unique holomorphic gauge ϕ : P | U → U × G mapping S onto the trivial section U × {e} of the trivial bundle U × G. By construction, the (1, 0)-potential A in such a gauge satisfies (4.6).
Suppose now that ( E , D ) is a real analytic lifted instanton associated with the pair ( P , ω ) On the other hand, by Proposition 2.2, the vector fields H h 0 , H h ++ , H h −− generate a right holomorphic SL 2 (C)-action on P | U and the bundle P | U is foliated by regular orbits of this action. (Thus, each such orbit is identifiable with a copy of SL 2 (C).) We may therefore apply Lemma 5.3 in [16] (see also [24,Sect are eigenvectors with eigenvalue +1 for the same action. This implies that the e +a generate the distribution D + and that (e +a , e −b ) is a collection of generators for the distribution D 10 Setting A ±a :=A(e ±a ), A ±a := A(e ±a ) and recalling that, by hypothesis, A −a = A −a = 0, we have that (4.10) implies −e h −a ·A ++ = A +a and −e h −a · A ++ = A +a . Since we have proven that A ++ ≡ A ++ , this gives A ±a ≡ A ±a , as desired.  (E, D). The holomorphic gauges of the (extended) lifted bundle P in which the potential of ω satisfies (4.6) are called analytic.

Remark 4.5
The previous literature on the harmonic space formulation, e.g. [1,[22][23][24], used gauge conditions A 0 = A +a = 0 and prepotentials A ++ . Here we choose to reverse the role of the signs. This has the advantage that prepotentials are holomorphic rather than antiholomorphic with respect to the complex structure I of H(M) (see Remark 4.7). The maps g : U × SL 2 (C) → G, which give the gauge transformations h → g (x,U ) ·h from central gauges to analytic gauges are usually called bridges (e.g. [1,23,24]). Now, since G is the complexification of the compact Lie group G o , it is reductive and the exponential e (·) : g → G is a surjective local diffeomorphism. This means that any bridge g (x,U ) can be written as g (x,U ) = e ψ (x,U ) for some appropriate g-valued function ψ. We call ψ a g-bridge.

Analytic gauges, bridges and normalisations
In the next lemma we shall give the proof of existence of bridges and g-bridges, having the special property that the prepotentials determined in the newly built analytic gauges satisfy additional normalisation conditions, which drastically reduce their degrees of freedom. Such normalised analytic gauges can be considered as complex analogues of the Coulomb gauges for arbitrary gauge fields.
In order to properly state such a normalisation, we need to introduce some appropriate notation. Given x o ∈ M and a (sufficiently small) simply connected neighbourhood V ⊂ M of x o , let (e +a , e −b ) be a 4n-tuple of holomorphic vector fields of H C (M) that generate the distributions D ± ⊂ T C (V × V × SL 2 (C)) (⊂ T C H C (M)), constructed as in the proof of Theorem 4.3. Then, pick an element U o ∈ SL 2 (C), say U o = 1 0 0 1 , and let λ a , μ : V × V × SL 2 (C) → C 2n , a = 1, . . . , 2n, be a set of 2n + 1 holomorphic functions satisfying the following conditions Proof For simplicity, we use ϕ to identify P | V×SL 2 (C) with V × SL 2 (C) × G so that we may assume that the considered exp-central gauge is nothing but the identity map. Let us consider the integral leaves in V × V × SL 2 (C) × G of the holomorphic distribution D h − + < H h 0 >, which passes through the points of the manifold S × {e} ⊂ V × V × SL 2 (C) × {e}. Being horizontal, they are transversal to the G-orbits and, by dimension counting, they fill a submanifold S ⊂ P which projects diffeomorphically onto an SL 2 (4.12). We remark that these two conditions are satisfied by any prepotential, being merely consequences of the above properties of the curvature.
However, the prepotential A o −− does not necessarily satisfy the other conditions in (4.12) also. To get a prepotential with such additional properties, we need to further change ϕ o into a new (further restricted) gauge ϕ, which preserves the property of being analytic, i.e. with (1, 0)-potential components A 0 and A −a identically vanishing and with A −− satisfying the first pair of equations in (4.12). In order to determine this new analytic gauge ϕ, let us consider the (2n Observing that the matrix U o = I 2 has entries u 1 − , u 2 − equal to 0, 1, respectively, by appropriately choosing the component c 02 (x, y, g), the function can take any desired value at the points of such a residual degree of freedom for the h −− can be used to make the restriction h −− | S∩T (x,Uo,g) identically zero.
We combine the solutions along the leaves T (x,y,U o ,g) , (x, y, U o , g) ∈ S × G, into a global solution of (4.13) on V × V × SL 2 (C) × G and restrict such a globally defined g-valued map h −− to the submanifold V × V × SL 2 (C) × {e} V × V × SL 2 (C). Then, along each integral leaf in V × V × SL 2 (C) of the distribution spanned by H α and e −a , we may consider a new g-valued function ψ satisfying the differential problem The same argument as before shows the existence of such a solution. Moreover, the residual degree of freedom in the choice of the solution may be used to set it to 0 at each point of the form (x, y, U o ). Combining the solutions along all the considered integral leaves, we get a global solution ψ such that ψ (x, y, U o ) = 0 for all (x, y) ∈ V × V and satisfying the differential problem Along such an orbit, we may consider the unique g-valued connection for the G-bundle {x} × O × G, whose Since the space of all B-orbits {x}×O in V×SL 2 (C) is diffeomorphic to V × CP 1 and is therefore simply connected, all these new gauges combine into a globally defined gauge which maps each B×G-orbit into itself and satisfies (5.2) at all points. We may now consider the g-valued map on V × SL 2 (C) Combining We now need to show that the curvature of the associated covariant derivative D satisfies the following equalities for each X ± , Y ± ∈ D ± : we see that (5.13) is equivalent to the system of equations for the F +++a By [16,Lemma 5.3] (or, more precisely, by its analogue involving the vector field H −− in place of H ++ ) the restriction of (5.14) to each orbit O ⊂ U × G of the SL 2 (C)-action generated by the H h α , admits exactly one solution, namely the identically zero function. It follows that F +++a = F(H ++ , e +a ) ≡ 0 on V × SL 2 (C) × G.
Let us now focus on the components F(e +a , e +b ). By (5.12) and Bianchi identities (5.11) among the vector fields H 0 , e +a , e +b and H −− , e +a , e +b , we have that By the same argument as before, the unique G-equivariant extension of the g-valued function F ++ab := F(e +a , e +b ) is solution to the differential problem H h 0 ·F ++ab = 2F ++ab and H h −− ·F ++ab = 0. As above, by [16,Lemma 5.3], we get that F ++ab = F(e +a , e +b ) vanishes identically.

Bounds for normalised prepotentials
The existence of such a V is guaranteed by the fact that the family of integral leaves S (x,y,U ) is SL 2 (C)-invariant. The condition (5.18) is chosen to ensure that any point of ) be compact and denote by K ⊂ S the set of points (x, y, U ) ∈ S such that S (x,y,U ) ∩ K = ∅. The set K is compact. Indeed, it is the intersection between S and the compact set of the (regular) orbits of the points of K by the action on V × V × SL 2 (C) of the local group generated by the holomorphic vector fields H 0 and e −a .
Since A −− satisfies (4.12), by integration of the conditions e −a ·A −− = 0 and H 0 ·A −− = −2 A −− along each connected intersection S (x,y,U ) ∩ V × V × SL 2 (C), it follows that there exists a constant C V > 0, depending only on V, such that On the other hand, by the second line in (4.12), for some constant C K depending only on the set K or, equivalently, only on the compact set K . From this and the fact that A does not depend on the coordinates of SL 2 (C), we infer that A −− C 0 (K ,g) ≤ C V C K A C 0 ( p 1 (K ),g) , where p 1 : H(M) = M × SL 2 (C) → M is the projection onto the first factor. Since p 1 (K ) ⊂ V, the claim follows from (4.3).

The second prepotential and the curvature of instantons
Let In what follows, to keep a clear distinction between A −− and A ++ , we sometimes call A −− the first prepotential. Either function yields a complete local description of instantons on hk manifolds, since either one is completely determined by the other. However, in our framework, some features of the two descriptions are complementary: i.e. the non-trivial components of the curvature are just the second-order derivatives of A ++ along the anti-holomorphic directions e −a .
The above-mentioned nonlinear equation for A ++ has been used in various contexts in the physics literature, where it is known as the Leznov equation (see, for example, [12][13][14][15]27,35]). The next lemma provides useful relations between the C k -norms of the two types of prepotentials. We may now conclude the proof of (2). In fact, it suffices to observe that (5.25) is equivalent to saying that the section of E, given by the quadruple Since P is elliptic with ker P = {0}, classical Schauder estimates (see, for example, [7, Appendix H]) imply that, for each k ≥ 1 there are constants N k , M k > 0 such that This gives (5.21). The proof of (5.22) is similar.

The local compactness theorem
To conclude this paper, as an example of the utility of the harmonic space formulation, we present a streamlined proof of Uhlenbeck, Nakajima and Tian's celebrated local compactness theorem for Yang-Mills fields in the specific case of hk instantons. From the classical estimates in [30,37] (see also [36,43]), we know that for any geodesic ball B R = B R (x o ) of radius R of an m-dimensional Riemannian manifold (M, g), the