aa r X i v : . [ m a t h . DG ] D ec INSTANTONS ON HYPERK ¨AHLER MANIFOLDS
CHANDRASHEKAR DEVCHAND, MASSIMILIANO PONTECORVO,AND ANDREA SPIRO
Abstract.
An instanton (
E, D ) on a (pseudo-)hyperk¨ahler manifold M isa vector bundle E associated to a principal G -bundle with a connection D whose curvature is pointwise invariant under the quaternionic structures of T x M, x ∈ M , and thus satisfies the Yang-Mills equations. Revisiting a con-struction of solutions, we prove a local bijection between gauge equivalenceclasses of instantons on M and equivalence classes of certain holomorphicfunctions taking values in the Lie algebra of G C defined on an appropriateSL ( C )-bundle over M . Our reformulation affords a streamlined proof ofUhlenbeck’s Compactness Theorem for instantons on (pseudo-)hyperk¨ahlermanifolds. Introduction
Beginning in the mid-1970’s the self-duality equations for Yang-Mills fieldssuccessfully captured the imagination of theoretical physicists and mathem-aticians, epitomised by Donaldson’s flight into previously unforeseen realmsof four-manifold differential topology (reviewed for instance in [21, 29]). AYang-Mills field is a pair (
E, D ) on a Riemannian manifold (
M, g ), where E isa vector bundle associated to a principle G -bundle with a connection D whosecurvature satisfies the Yang-Mills equation D ∗ F = 0. The (anti-)self-dualityequations, requiring that the curvature F of a connection D over a Rieman-nian four-manifold ( M, g ) takes values in the (anti-)self-dual eigenspace of theHodge star-operator, implies the Yang-Mills equation in virtue of the Bian-chi identity DF = 0. Connections satisfying the (anti-)self-dual Yang-Mills(SDYM) equations are called (anti-) instantons. They are global minimisersof the Yang-Mills energy functional, S ( A ) = || F || = R M F ∧ ∗ F vol g .The quest for explicit instanton solutions [6] was initially physically motiv-ated, for instance by the mystery of the phenomenon of quark confinement [33],but the remarkable properties of instantons soon attracted powerful mathem-atical treatment. First, Ward showed that solutions of the self-duality equa-tions on R are encoded in certain holomorphic data on twistor space [40], Mathematics Subject Classification.
Key words and phrases.
Yang-Mills theory, instantons, hyperk¨ahler geometry, harmonicspace.This research was partially supported by
Ministero dell’Istruzione, Universit`a e Ricerca in the framework of the project “Real and Complex Manifolds: Geometry, Topology andHarmonic Analysis” and by GNSAGA of INdAM. effectively converting the problem to an algebro-geometric one. Then, Atiyah,Hitchin and Singer [4] obtained a correspondence between solutions of theSDYM equations on S and certain real algebraic bundles on the complex pro-jective 3-space C P . They thus established the relation between self-dualityand holomorphic structures, yielding in particular the dimension of the mod-uli space of solutions for any compact gauge group. This led to a sequenceof ans¨atze yielding SDYM solutions in terms of arbitrary solutions of linearequations [5, 10]. Subsequently, powerful algebro-geometric results were usedto obtain a complete construction of all SDYM fields on S [3, 20, 2].These developments were followed by fundamental analytical results on vari-ational methods for Yang-Mills theory. The moduli space of instantons is asubset of the quotient A / G of the space of all connections A with the group ofall gauge transformations G . Locally representing the connections in Coulombgauges Uhlenbeck [37, 38] developed analytical tools to study the singularitiesof the compact moduli space of instantons. Uhlenbeck’s work, together withthe novel variational methods introduced by Taubes to study gauge invarianttheories, prepared the path for Donaldson’s seminal work.These analytical results depended crucially on the fact that the Yang-Millsfunctional and therefore also the Yang-Mills equations are conformally invari-ant in four dimensions. Further, the above-mentioned constructions of SDYMsolutions crucially used the fact that R conformally compactifies to S . Allthis would seem to impede any generalisation to Yang-Mills fields in higherdimensions. Indeed, it is known that a connection over the sphere S d , d ≥ L -norm is necessarily flat [8]. However, the solvabilityof four dimensional SDYM equations relies in particular on the fact that, beinglinear algebraic constraints on the curvature, they are first-order equations forthe vector potential. This first-order property was partly responsible for thegood analytical properties of the SDYM equations. Indeed, an insistence uponthis familiar sight of partial-flatness conditions, requiring the vanishing of cer-tain linear combinations of the curvature components, which automaticallyimply the second-order Yang-Mills equations, yields the required instantonequations in dimensions greater than four. This idea was originally pursuedin [9], where it was shown that the required equations are restrictions of thecurvature F to an eigenspace of an endomorphism on the space of two-formsdefined by an appropriate co-closed four-form Ω, ∗ ( ∗ Ω ∧ F ) = λF , Ω ∈ Λ T ∗ M, λ ∈ R ∗ . (1.1)The co-closedness of Ω suffices to show that a Yang-Mills curvature field satis-fying (1.1) implies the second-order Yang-Mills equations. For d >
4, the four-form Ω is pointwise invariant only under some proper subgroup of SO d ( R ).The existence of Ω corresponds to some special holonomy on the manifold[19, 17, 39, 36, 25, 1]. NSTANTONS ON HYPERK ¨AHLER MANIFOLDS 3
The above-mentioned compactness results for the moduli space of Yang-Mills fields were already generalised to higher dimensions by Uhlenbeck andNakajima [30]. For the higher-dimensional generalisations of the self-dualityequations the analytical programme in the spirit of Uhlenbeck and Taubes wasbegun by Tian [36]. The investigation of (local) solutions of higher dimensionalequations of the form (1.1) began [41, 11, 22, 1] with the case of instantons onspaces having hyperk¨ahler (hk) structure (i.e. with holonomy in Sp n ), thesebeing natural generalisations of R = H . Some global results on generalinstantons on quaternionic K¨ahler (qk) manifolds (with holonomy in Sp n · Sp )also exist; e.g. [31, 28].The twistor formulation of SDYM, which led to the ADHM construction,has a (local) field theory variant, the harmonic space formulation . This wasoriginally developed as a tool to study supersymmetric harmonic maps [22, 24].The harmonic space formulation enlarges the C P fibre of the twistor bundleto SL ( C ), yielding a total space more amenable to standard field theoreticaltreatment. In the process (gauge equivalent classes of) local solutions of self-dual theories are parametrised by a prepotential, much as the K¨ahler potentialparametrises K¨ahler metrics. In a previous paper by two of us [16], we havegiven a differential geometric description of the corresponding constructionof (pseudo-)hyperk¨ahler metrics. In the current paper we investigate prop-erties of Yang-Mills instantons on (pseudo-)hyperk¨ahler manifolds using theharmonic space formulation, presenting a differential geometric formulation ofthe method based on the work of [1].The harmonic space of an hk manifold ( M, g ) is the trivial bundle H ( M ) =SL ( C ) × M → M , equipped with a certain (non-product) complex structure.The space H ( M ) fibres naturally over the quotient Z ( M ) = SL ( C ) /B × M ≃ C P × M , where B is the Borel subgroup of upper triangular matrices. Z ( M )is the twistor bundle of ( M, g ) and has a well defined complex structure, ca-nonically determined by the hypercomplex structure of M . Now, the complexstructure of the harmonic space H ( M ) is the unique complex structure suchthat the projection p : H ( M ) → Z ( M ) is holomorphic.A gauge field ( E, D ) on a complex hyperk¨ahler manifold (
M, g ) is an instan-ton if the curvature of D is pointwise invariant under the quaternionic struc-ture of T p M, p ∈ M . Its pull-back field ( E ′ , D ′ ) over H ( M ) admits an analyticgauge condition , by which we mean a special class of local trivialisations (=gauges) of E ′ . This class has the crucial feature that its gauge transformationsare holomorphic, supplemented by some other conditions. In such a trivial-isation, the potential A ′ of D ′ is completely determined by just one of itscomponents, which is moreover a holomorphic function on H ( M ). This com-ponent is called the prepotential of the gauge field ( E ′ , D ′ ). A freely-specifiableholomorphic prepotential, satisfying an appropriate first-order linear equationon H ( M ), encodes all local properties of the corresponding instanton solution C. DEVCHAND, M. PONTECORVO, AND A. SPIRO on M and may be used to reconstruct the associated Yang-Mills field ( E, D ).This construction, together with complete proofs of an essentially bijectivecorrespondence between normalised prepotentials on H ( M ) and moduli oflocally defined instantons on M takes up the bulk of the content of this paper.The existence of the analytic gauge condition and the resulting holomorphicprepotential allows the use of classical results on holomorphic functions, suchas Montel’s Theorem or Hartogs’ Removable Singularity Theorem, to invest-igate the moduli spaces of instantons. Thus, in this formulation, holomorphyprovides very useful tools. This is analogous to Uhlenbeck’s Coulomb gaugecondition, which allows the use of the machinery of elliptic equations. Asan example, we establish some simple estimates relating C k -norms of prepo-tentials to those of curvatures. These estimates, combined with the classicalMontel Theorem of Complex Variable Theory, lead to a new direct proofof Uhlenbeck’s Strong Compactness Theorem for instantons on hk manifolds[37, 38, 18, 30, 36, 42, 43].The paper is structured as follows. After the preliminary section §
2, weintroduce the notion of harmonic space and discuss its relation with the twistorbundle of a (pseudo-)hyperk¨ahler manifold M in §
3. In §
4, we discuss theanalytic gauge condition of the pull-back of the instanton over H ( M ) and theconstruction of the instanton field over M from the corresponding holomorphicprepotential on H ( M ). Our main new contributions appear in § § Acknowledgements.
One of us (CD) thanks Hermann Nicolai and the Al-bert Einstein Institute for providing an excellent research environment.2.
Preliminaries
Basics of hyperk¨ahler manifolds
Given a 4 n -dimensional real vector space W , we recall that a hypercomplexstructure on W is a triple ( I , I , I ) of endomorphisms satisfying the multi-plicative relations of the imaginary quaternions, I α = − Id W and I α I β = I γ forall cyclic permutations ( α, β, γ ) of (1 , , hypercomplex struc-ture on a 4 n -dimensional real manifold M is a triple ( J , J , J ) of integrablecomplex 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 3-dimensionalsubspace Q W of End ( W ), which is the span Q W = span R ( I , I , I ) of theendomorphisms of a hypercomplex structure ( I α ). A (pseudo-)quaternionicK¨ahler structure on W is an inner product h· , ·i on W which is hermitian with NSTANTONS ON HYPERK ¨AHLER MANIFOLDS 5 respect to Q W , that is, every element J ∈ Q W is skew-symmetric with respectto h· , ·i . For what concerns manifolds, we have instead the following Definition 2.1.
A 4 n -dimensional (pseudo-)Riemannian manifold ( M, g ) ofsignature (4 p, q ), p + q = n , is a (pseudo-)quaternionic K¨ahler ( qk ) manifold if it admits a subbundle Q ⊂ End (
T M ) of quaternionic structures on thetangent 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-)hyperk¨ahler (hk) manifold if there exist threeglobal ∇ -parallel sections J , J , J of Q (which are thus integrable complexstructures) 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 andonly 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 , , 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 n → T x M mapping the standard triple ofcomplex structures ( i , j , k ) of H n ≃ R n into the triple of complex structures( J α | x ) α =1 , , . Note that any adapted frame u = ( e A ) allows the identifica-tion of the standard Sp - and Sp p,q -actions on H n with uniquely associatedactions of Sp - and Sp p,q -actions on T x M . The family O g ( M, J α ) of all adap-ted frames is a principle bundle over M with structure group Sp · Sp p,q and isinvariant under the parallel transport of the Levi-Civita connection. Hence, ifthe bundle O g ( M, J α ) admits a global section, the above actions of Sp · Sp p,q on the tangent spaces T x M combine to yield a global action of Sp · Sp p,q onto T M .We now fix a few technical details, which we shall need.As the standard representation of Sp · Sp p,q on H n = C n , we choose theone for which the element ( i − i ) ∈ sp acts on ( H n ) C ≃ C ⊗ C n as theleft multiplication by the complex structure i . Hence, for any given adaptedframe u ∈ O g ( M, J α ) | x , the corresponding action of ( i − i ) on T x M coincideswith the action of the complex structure I = u ∗ ( i ). This action of Sp · Sp p,q extends by C -linearity to the standard action of SL ( C ) · Sp n ( C ) on ( H n ) C ≃ C ⊗ C n . Consequently, the C -linear extensions u : ( H n ) C → T C x M of adaptedframes give SL ( C ) · Sp n ( C )-equivariant isomorphisms T C x M ≃ H x ⊗ E x forcomplex vector spaces H x ≃ C and E x ≃ C n . These isomorphisms arenecessarily related to each other by the action of some element of Sp · Sp p,q ⊂ SL ( C ) · Sp n ( C ). C. DEVCHAND, M. PONTECORVO, AND A. SPIRO
Furthermore, as standard bases for C and C n , we shall use (cid:0) h o = (1 , , h o = (0 , (cid:1) , (cid:0) e oa = (0 , . . . , a -th entry , . . . , (cid:1) . For any ( C -linearly extended) adapted frame u : ( H n ) C = C ⊗ C n → T C x M ,we then denote by e ia := u ( h oi ⊗ e oa ), i = 1 ,
2, the complex vectors in T C x M corresponding to the basis h oi ⊗ e oa of C ⊗ C n . If u : H n → T x M is changedinto another adapted frame u ′ = u ◦ U with U = (cid:18) u u − u u − (cid:19) ∈ Sp ⊂ SL ( C ) = Sp ( C ) , the corresponding complex basis ( e ja ) of T C x M is changed into e + a = u j + e ja , e − a = u i − e ia . (2.1)Similarly, if u ′ = u ◦ U is further transformed by some A = ( A ba ) ∈ Sp p,q ⊂ Sp n ( C ), the basis ( e ± a ) is then transformed into e ′± a = A ba e ± b .Note that, since the vectors h ⊗ e oa and h ⊗ e oa are ± i -eigenvectors of theelement ( i − i ) ≃ i ∈ Sp , the corresponding elements e + a , e − a ∈ T C x M arevectors of type (1 , and (0 , with respect to the complex structure I = u ( i ) of T x M .2.2. Connections, gauges, potentials and Yang-Mills fields
In this section we briefly review certain basic facts about gauge theorieswhich we shall need in what follows.Given a manifold M and a (principal or vector) bundle π : E → M , wedenote by X ( M ) the space of all vector fields of M and by Γ( E ) the set ofall 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 anassociated vector bundle with fibre V ≃ R N , determined by a faithful linearrepresentation ρ : G → GL( V ). We shall refer to an open subset U ⊂ M onwhich 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 ofthe language). Given two gauges ϕ, ϕ ′ with overlapping domains U , U ′ , thetransition function ϕ ′ ◦ ϕ − : ( U ∩ U ′ ) × G → ( U ∩ U ′ ) × G is a map of the form ϕ ′ ◦ ϕ − ( x, h ) = ( x, g x · h ) for a smooth map g : U → G .
The family of the automorphisms of G defined by h g x · h is what is usuallycalled gauge transformation between the gauges ϕ and ϕ ′ . Finally, we recallthat if a principal G -bundle p : P → M admits a collection of gauges, whosedomains 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 p o : P o → M of P , called G o -reduction . NSTANTONS ON HYPERK ¨AHLER MANIFOLDS 7
A connection 1-form ω on P induces a unique covariant derivative on theassociated bundle E . We recall that a covariant derivative on E is an operator D : X ( M ) × Γ( E ) → Γ( E ), which is R -linear in both arguments, C ∞ ( M ; R )-linear in X ( M ) and satisfies the Leibniz rule D X ( f σ ) = ( X · f ) σ + f D X σ foreach f ∈ C ∞ ( M ; R ).Now, consider a gauge P | U ∼ → U × G with domain U . In this gauge, anyvector 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 expressedas a sum of the form ω | ( x,g ) = − A ( x,g ) + ω Gg , 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 withrespect 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 haveoverlapping domains U , U ′ , the corresponding potentials A , A ′ are relatedthrough the gauge transformation h g x · h by means of A ′ = Ad g − A + g − · dg . (2.2)We now recall that a section σ : U → E | U ≃ U × V of the associated bundle hasthe form σ ( x ) = ( x, s i ( x )) for a smooth map ( s i ) : U → V = R N . Hence, foreach vector field X ∈ X ( U ), it is possible to consider the section of E | U ≃ U × VD X σ ( x ) = (cid:0) x, ( X · σ i ) | x + A ij ( X ) σ j | x (cid:1) , (2.3)where A ij := ρ ∗ ◦ A with ρ ∗ : g → gl ( V ) the Lie algebra representation determ-ined by the linear representation ρ : G → GL( V ) that gives the associatedvector bundle E = P × G,ρ V . The gl ( V )-valued 1-form A ij is called the poten-tial of D induced by the connection ω . The operator (2.3) is gauge independentdue to (2.2), thus globally well defined as a covariant derivative on the sectionsof E .We recall that the curvature -form of ω is the g -valued 2-form on P definedby Ω = dω + [ ω, ω ]. Given a gauge ϕ : P | U → U × g and the associated gauge b ϕ for E b ϕ : E | U → U × V , b ϕ (cid:0) [ ϕ − ( x, e ) , v )] (cid:1) := ( x, v ) , 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 F ϕx ( v, w ) := 2Ω ( x,e ) ( v, w ) , F b ϕx ( v, w ) := 2 ρ ∗ ◦ F ϕx ( v, w ) , for v, w ∈ T x U . (2.4)Note that F ϕ can be recovered from the potential A of ω by the formula F ϕ ( X, Y ) = X · ( A ( Y )) − Y · ( A ( X )) + [ A ( X ) , A ( Y )] − A ([ X, Y ]) . (2.5)and that, if h g x · h is the gauge transformation between ϕ , ϕ ′ , one has that F ϕ ′ x = Ad g x F ϕx , F b ϕ ′ x = F b ϕx . (2.6) C. DEVCHAND, M. PONTECORVO, AND A. SPIRO
This shows that the curvature tensor of ω does depend on the gauge, whilethe curvature tensor of D does not . Due to this, the curvatures F b ϕx combineand determine a globally defined gl ( V )-valued 2-form F on M . It can be alsochecked that it satisfies the identity F ( X, Y ) s = [ D X , D Y ] s − D [ X,Y ] s for sections s ∈ Γ( E ) , (2.7)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 holo-morphic bundles, which we now briefly recall.If ( M, J ) is a complex manifold, G is a complex Lie group and V is a complexvector space, a principal G -bundle p : P → M (resp. a complex vector bundle p : E → M with fiber V ) is called holomorphic if it is equipped with a complexstructure b J , such that the right action of G on P (resp. the vector bundlestructure on E ) is b J -holomorphic and the projection p is a ( b J , J )-holomorphicmapping. In this case, a trivialisation is called holomorphic if it is a localholomorphic map from ( P, b J ) (resp. ( E, b 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, b J ) is called b J -invariant if the corresponding horizontal spaces H ( x,g ) = ker ω ( x,g ) are invariant underthe complex structure b J . This is equivalent to say that, in any holomorphictrivialisation ϕ : P | U → U × G , the corresponding potential A : U → T ∗ U ⊗ g takes actually values in in T ∗ M ⊗ g + T ∗ M ⊗ g . Here we denote by T x M and T M the holomorphic and anti-holomorphic distributions of M and by g , g the subalgebras of g C (both isomorphic to g ), which are generatedby the vectors of type (1 ,
0) and (0 , A x , x ∈ U ,real, the projection of A x onto T ∗ x M ⊗ g is the complex conjugate of thecomponent in T ∗ x M ⊗ g . So, the potential A is uniquely determined by theassociated map A : U → T ∗ M ⊗ g , which we call (1 , -potential .We finally remark that the covariant derivative D on an associated holo-morphic vector bundle p E : E → M , determined by a b J -invariant con-nection ω , is characterised by the property that it transforms sections of E C = E ⊕ M E , with values in E or in E , into sections which arestill in E or in E , respectively. The covariant derivatives of (1 , , , 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 NSTANTONS ON HYPERK ¨AHLER MANIFOLDS 9 with G o compact real form of G and ω restricts to a connection ω o on P o , wesay 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, ∗ : Λ p T ∗ M → Λ n − p T ∗ M . Then, the gauge field ( E, D ) is calleda
Yang-Mills field if its curvature tensor F satisfies the Yang-Mills equation D ∗ F = 0 .2.3. C k -norms and L p -norms of curvatures Let G be a reductive complex Lie group and G o ⊂ G a compact real formof G . Further let g o = g s + z ( g o ) be the decomposition of g o = Lie ( G o ) into itssemisimple part g s and center z ( g o ), and denote by h· , ·i any Ad G o -invariantEuclidean inner product on g o which, on g s × g s , coincides with minus theCartan-Killing form. This allows us to define the Hermitian inner product on g = Lie ( G ) = g o + i g o h : g × g → C , h ( X + iY, X ′ + iY ′ ) := h X + iY, X ′ − iY ′ i (2.8)and the associated norm k X + iY k h = q k X k h· , ·i + k Y k h· , ·i . If ( M, g ) isa Riemannian manifold, we can use the metric g to extend the Hermitianproduct h of g to a positive inner product, also denoted by h , on the space oftensor fields in ⊗ ℓ T ∗ M ⊗ g over M . So, for any compact subset K ⊂ M , wehave the usual sup-norm for g -valued C k functions on K k V k C k ( K, g ) := k X j =0 sup x ∈ K k∇ j V | x k h . Similarly, for each p ∈ [1 , + ∞ ), we may use the L p -space L p ( U , g ), the com-pletion of the space of all C -maps V : U → g with bounded values for theintegral R U k V k ph vol g , equipped with the usual L p -norm k · k L p ( U ) := (cid:18)Z U k · k ph vol g (cid:19) p . 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 atrivialisation ϕ : P | U → U × G in which the curvature tensors of ω and D are F ϕ and F = F b ϕ , respectively. Then, for any compact subset K ⊂ U , wedefine k F k ( ϕ ) C k ( K ) := k F ϕ k C k ( K, g ) , k F k ( ϕ ) L p ( U ) := k F ϕ k L p ( U , g ) . (2.9)Now consider the cases when ( P, ω ) is reducible to a pair ( P o , ω o ) with struc-ture group given by the compact G o . Since the h -norms are Ad G o -invariant, ifwe 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 k - and Sobolevnorms 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 bya) 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 theprojection 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 byi) the lifted vector bundle q ′ : E ′ = π ∗ E = P ′ × G,ρ V → N over N ii) the covariant derivative D ′ = π ∗ D on E ′ induced by ω ′ .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 corresponding1-parameter subgroup exp( R X ) ⊂ H has clearly a natural right action on N and determines a natural 1-parameter group of diffeomorphisms on the liftedbundle P ′ ⊂ N × P given by R ′ : R −→ Diff( P ′ ) , R ′ t ( y, u ) = ( R exp( t ) ( y ) , u ) . 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, thevector field X ′ on P ′ , whose flow is the 1-parameter group of automorphisms R t , is such thata) p ′∗ ( X ′ ) = X and π ′∗ ( X ′ ) = 0; here X is identified with the verticalvector field of the H -bundle N b) ı X ′ ω ′ = 0 and 0 = L X ′ ω ′ = ı X ′ dω ′ + d ( ı X ′ ω ′ ) = ı X ′ dω ′ ;c) if A is the potential of ω ′ in a gauge π ∗ P | U ≃ U × G , the vector field X ′ has the form X ′ ( y,g ) = X y + X g ( y ) | g where X g ( y ) is the left-invariantvector field of G defined by ω G ( X g ( y ) ) = A ( X y ).Note that (b) implies also that ı X ′ Ω ′ = 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 ′ . NSTANTONS ON HYPERK ¨AHLER MANIFOLDS 11 If H is simply connected, then ( E ′ , D ′ ) is the lift of a gauge field on M ifand 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) theunique ω ′ -horizontal vector field X ′ on P ′ , which projects onto X , is complete.Proof. The necessity follows from previous remarks. Assume now that foreach 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 ′ Ω ′ = 0, from which it follows that the flowof X ′ commutes with the right G -action of P ′ and preserves ω ′ . We alsohave that, for any X, Y ∈ h , their ω ′ -horizontal lifts X ′ , Y ′ are such that ω ′ ([ X ′ , Y ′ ]) = Ω ′ ( X ′ , Y ′ ) = 0. This means that [ X ′ , Y ′ ] is the ω ′ -horizontal liftof the Lie bracket [ X, Y ]. We therefore conclude that the collection of vectorfields h ′ := { X ′ is ω ′ -horizontal lift of some X ∈ h } is a finite dimensional Lie algebra of complete vector fields on P ′ . By a classicaltheorem 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, becauseall of them are coverings of the simply connected H -orbits on N . Moreover,each transformation of this action is a bundle automorphism, which preservesthe connection 1-form ω ′ and commutes with the G -action. Hence, the orbitspace P = P ′ /H is a G -bundle over M := P ′ /H ′ × G = P/G and is equippedwith the g -valued 1-form ω defined by ω [ u ] ( v ) := ω ′ u ( v ′ ) for some v ′ ∈ T u P ′ that projects onto v ∈ T [ u ] P ′ /H ′ . 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 thecovariant derivation D determined by ω , is the desired gauge field, of which( E ′ , D ′ ) is a lift.3. Harmonic spaces of (pseudo-)hyperk¨ahler manifolds
In the rest of this paper (
M, g, J α ) denotes an hk manifold and ( E, D ) acomplex gauge field on (
M, g, J α ) associated with a pair ( P, ω ) with complexstructure group G . We shall mostly assume that G is the complexificationof 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 .3.1. The twistor bundle of a (pseudo-)hyperk¨ahler manifold
Let (
M, g, J α ) be a 4 n -dimensional hk manifold. It is well known that fora point z = ( a, b, c ) ∈ S ( ≃ C P ), the tensor field I ( z ) := aJ + bJ + cJ is an integrable complex structure on M and that the twistor bundle Z ( M ) of M is simply the trivial C P -bundle Z ( M ) := M × { I ( z ) , z ∈ S } → M .Since each z ∈ S corresponds to a distinct integrable complex structure, thetwistor bundle Z ( M ) is foliated by the complex submanifolds M × { z } ≃ M ,each equipped with the corresponding complex structure I ( z ) . Such complexstructures combine with the classical complex structure of C P and determinea natural almost complex structure on Z ( M ), which we denote by b I . It wasproved to be integrable in [34].We remark that the complex structures on T x M, x ∈ M , I ( z ) | x := aJ | x + bJ | x + cJ | x , z = ( a, b, c ) ∈ S ≃ C P , 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 § n -tuples of complex vectors e + a (resp. e − a ), which are part of the complexbases (2.1), is actually a frame of holomorphic (resp. anti-holomorphic) vectorsfor a complex leaf M × { I ( z ) } ⊂ Z ( M ).3.2. The harmonic space of a (pseudo-)hyperk¨ahler manifold
The harmonic space of a 4 n -dimensional hk manifold ( M, g, J α ) is the trivialbundle H ( M ) = M × SL ( C ) → M , endowed with an integrable complexstructure I defined as follows. For each point ( x, U ) ∈ H ( M ) consider thenatural direct sum decomposition T ( x,U ) H ( M ) = T x M + T U SL ( C ) ≃ T x M + sl ( C ), where T U SL ( C ) is identified with sl ( C ) by means of right invariantvector fields. Then let I ( x,U ) be the unique complex structure on T ( x,U ) H ( M )given by I ( x,U ) | T x M := I ( z ) | x , z = U · [0 : 1] ∈ C P ( ≃ S ) and I ( x,U ) | sl ( C ) = J o , (3.1)where J o is the complex structure of sl ( C ) given by the multiplication by( i i ). From the above identification T U SL ( C ) ≃ sl ( C ), it follows that alongeach fiber { x } × SL ( C ), the I -holomorphic distribution is generated by rightinvariant vector fields of SL ( C ).The collection I of such pointwise defined complex structures is a globallydefined almost complex structure on H ( M ), which can be seen to be integrableas follows. The family of restricted complex structures I | M ×{ U } = I ( z ) | M ×{ U } on the manifolds M × { U } , U ∈ SL ( C ), is invariant under the natural leftaction of SL ( C ) on H ( M ). Thus, the Lie derivative of an I -holomorphicvector field that is tangent to the (horizontal) slices M × { U } by meansof an infinitesimal transformations of this SL ( C )-action always gives an-other I -holomorphic vector field, which is horizontal. On the other hand,the infinitesimal transformations of the left action of SL ( C ) on each ver-tical fiber { x } × SL ( C ) ≃ SL ( C ) are nothing but the right invariant vec-tor fields of SL ( C ). This implies that the Lie bracket between a horizontal I -holomorphic vector field and a vertical I -holomorphic vector field is a hori-zontal I -holomorphic vector field. This property together with the fact that NSTANTONS ON HYPERK ¨AHLER MANIFOLDS 13 both the horizontal and vertical I -holomorphic distributions are involutiveproves that the whole I -holomorphic distribution of H ( M ) is involutive, i.e.that I is globally integrable. Remark 3.1.
Note that H ( M ), considered as a principal SL ( C )-bundle over M , can be (locally) identified with a bundle of vertical complex frames ( e + , e − )for the fibers of the rank 2 holomorphic vector bundle π : H → M introducedby Salamon in [34] for general quaternionic K¨ahler manifolds. (Note that thefibers of H are the spaces H x defined in § H ( M ) and H directly yields that the projection p : H ( M ) → H \ { } , p (( x, ( e + , e − )) = e + ∈ H x , x ∈ M is holomorphic. Thus H ( M ) fibers holomorphically over the twistor space Z ( M ) = P ( H ) ≃ M × C P with typical fiber given by the subgroup B ⊂ SL ( C ) of upper triangular matrices. Thus, Z ( M ) ≃ H ( M ) /B = M × SL ( C ) /B .By construction, the harmonic space H ( M ) is equipped with the I -invariantintegrable distribution D given by the tangent spaces of the leaves M × { U } .The two complex subdistributions of D C , spanned by the holomorphic andanti-holomorphic vector fields, will be denoted by D + , D − ⊂ D C . By theremarks at the end of § x, U ) ∈ H ( M ) there is at least one adaptedframe u : H n → T x M , more precisely, a frame with u ( i ) = I ( z ) x , z = U · [0 : 1],such that the corresponding 2 n -tuples ( e + a ) and ( e − a ) are bases for the vectorspaces D + | ( x,U ) and D − | ( x,U ) , respectively.3.3. The complexified harmonic space
Consider an n -dimensional complex manifold ( N, J ) and denote by A J thecomplete atlas of holomorphic coordinates, i.e., of systems of coordinates ξ =( z i ) : U ⊂ N → C n in which the integrable complex structure J has thestandard form J = i ∂∂z j ⊗ dz j − i ∂∂z k ⊗ dz k . We define the complexification of( N, J ) as the pair ( N C , ı ) given by:a) the complex manifold N C := N × N having complex structure e J definedat each point ( x, y ) ∈ N × N by e J ( x,y ) ( v, w ):= J x ( v ) − J y ( w )b) the standard diagonal embedding ı : N → N C , ı ( x ) = ( x, x ).Note that the complex structure e J of N C is defined in such a way that the cor-responding atlas A e J of holomorphic coordinates is generated by coordinatesof the form e ξ = ( z i , z ′ j ) : U × V → C n for some (local) holomorphic coordin-ates ( z i ), ( z ′ j ) of ( N, J ). So, the anti-holomorphic involution τ : N C → N , τ ( x, y ) = ( y, x ) has a fixed point set which is precisely the totally real sub-manifold ı ( N ) ≃ N .The above construction yields the following very convenient extension ofharmonic spaces. Let H C ( M ) be the cartesian product H C ( M ) = M × M × SL ( C ), equipped with the unique (integrable) complex structure I ( C ) , whichcoincides with the right invariant complex structure along the (vertical) leaves { x } × { y } × SL ( C ) ≃ SL ( C ) (see (3.1)) and with the complex structureof the complexification of ( M, I ( z ) ), z = U · [0 : 1], along each horizontal leaf M × M × { U } . In other words, H C ( M ) is the union of the complexifications ofthe manifolds ( M, I ( z ) ), z ∈ S , and not the complexification of the harmonicspace H ( M ). Nevertheless, we call H C ( M ) the complexified harmonic space .Both of the plus and minus distributions D ± ⊂ T C H ( M ) naturally extendto holomorphic distributions on H C ( M ). In order to see this, just considerthe distribution D ′ on H C ( M ), determined by the tangent spaces to all leaves M × M ×{ U } and the associated complex subdistribution D I ( C ) ⊂ D ′ C spannedby I ( C ) -holomorphic vector fields. The subdistributions of D I ( C ) that projectisomorphically onto the tangent spaces either on the first or the second copyof M coincide with the distributions D ± at the points of the real submanifold H ( M ) ⊂ H C ( M ). For simplicity, we shall denote these subdistributions of D I ( C ) also by D ± .4. Instantons on hk manifolds and prepotentials
Instantons on hyperk¨ahler manifolds
As usual, let (
M, g, J α ) be an hk manifold and denote by T C x M ≃ H x ⊗ E x , x ∈ M , the isomorphisms described in § T ∗ C x M splits into three irreducible SL ( C ) · Sp n ( C ) moduli:Λ T ∗ C x M ≃ C ω H x ⊗ S E ∗ x + S H ∗ x ⊗ C ω E x + S H ∗ x ⊗ Λ E ∗ x . (4.1)Here ω H x and ω E x are the SL ( C )- and Sp n ( C )-invariant symplectic forms of H x and E x , respectively, and Λ E x is the irreducible Sp n ( C )-submodule ofΛ E x complementary to C ω E x . Since the isomorphism T C x M ≃ H x ⊗ E x isunique up to an action of an element in Sp · Sp p,q , the decomposition (4.1) isindependent of the isomorphism chosen.Now, given a gauge field ( E, D ) on an hk manifold, we split the ( C -linearextension of the) curvature tensor F x , x ∈ M as follows: F x = F (1) x + F (2) x with F (1) x ∈ C ω H x ⊗ S E ∗ x ⊗ End ( E x ) F (2) x ∈ ( S H ∗ x ⊗ ( C ω E x + Λ E ∗ x )) ⊗ End ( E x ) . A gauge field (
E, D ) is called instanton if the F (2) component of its curvaturetensor vanishes everywhere. Such instantons provide minima of the Yang-Mills functional R M | F | ω g and are thus solutions of the Yang-Mills equations[41]. Such instanton equations have been been studied by several authors[34, 28, 9, 36, 1, 12].Notice that the vanishing of F (2) corresponds to simple conditions on thecomponents F ± a |± b = F ( e ± a , e ± b ) with respect to the complex frames ( e ± a ) NSTANTONS ON HYPERK ¨AHLER MANIFOLDS 15 defined in (2.1). In fact, F (2) = 0 if and only if F + a | + b = F − a |− b = 0 , F + a |− b = − F − a | + b , F + a |− b = F + b |− a . (4.2)In four dimensions these are precisely the well-known anti-self-duality equa-tions.4.2. 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 corres-ponds to a gauge for its lift P ′ on the harmonic space H ( M ), namely thegauge ϕ defined on the restriction of P ′ to V × SL ( C ) ⊂ H ( M ) by ϕ : P ′ | V × SL ( C ) → V × SL ( C ) × G , ϕ ( u, U ) := ( ϕ V ( u ) , U, ϕ G ( u )) . Such a gauge is called the central gauge determined by ϕ [24, 1].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 ( v ) t = exp x o ( tv ) the radialgeodesic determined by v . Now let V ⊂ T x o M be a neighbourhood of x o suchthat the exponential map exp x o : V ⊂ T x o M → M is a diffeomorphism ontoits image U = exp x o ( V ). A gauge ϕ : P | U → U × G on the domain U = exp( V )is called exponential if the corresponding potential A satisfies the followingconditions:(a) A | x o = 0 and(b) A ( ˙ γ ( v ) t ) = 0 for all vectors ˙ γ ( v ) t tangent to the radial geodesics.We shall call the central gauges for P ′ on H ( M ) determined by exponentialgauges for P on M exponential central (or just exp-central ). We recall thatfor any x o ∈ M there is always an exponential gauge on some appropriateneighbourhood of x o [37]. Thus, for any ( x o , U ) ∈ H ( M ) there exists anexp-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 complexificationof a gauge field ( E o , D o ) with compact structure group G o ⊂ G , if A is thepotential 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 k A k C ( K, g ) ≤ c U k F k ( ϕ ) C ( K, g ) for K ⊂⊂ U , (4.3)(see e.g. [37, Lemma 2.1]). Similar estimates clearly hold for potentials andcurvatures of the lifted gauge fields ( E ′ , D ′ ) in exp-central gauges.4.3. Prepotentials for instantons on hk manifolds
We recall that any hk manifold ( M, g, ( J α )) has a natural structure of areal analytic manifold and that in such a structure, the tensors g and J α arereal-analytic [26]. Hence, if we lift g to H ( M ) as a tensor field with valuesin D ∗ ⊗ D ∗ , using real-analyticity, for each point ( x, U ) ∈ H ( M ) ⊂ H C ( M ) we may determine a tubular SL ( C )-invariant neighbourhood W ⊂ H C ( M ) of( x, U ), 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 instantonon an hk manifold provided that it is a complexification of an instanton withcompact structure group. To prove this, let us introduce some additionalconvenient 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 horizontaldistributions 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 whichprojects onto X . Proposition 4.1. If ( E, D ) is an instanton on ( M, g, ( J α )) , which is the com-plexification of a gauge field with compact structure group G o , all data of thelifted pairs ( E ′ , D ′ ) and ( P ′ , ω ′ ) on H ( M ) are real analytic. Moreover, thereis a complex structure J ′ on P ′ , invariant under G = ( G o ) C , which makes p ′ : P ′ → H ( M ) and the associated vector bundle p E ′ : E ′ → H ( M ) holo-morphic bundles over H ( M ) with J ′ -invariant connections.It follows that, for any ( x o , U ) ∈ H ( M ) ⊂ H C ( M ) , there are unique real-analytic extensions of ( E ′ , D ′ ) and ( P ′ , ω ′ ) to some SL ( C ) -invariant tubularneighbourhood W ⊂ H C ( M ) of the real submanifold U := W ∩ H ( M ) contain-ing ( x o , U ) . Further, both the extended bundles E ′ , P ′ have naturally extendedcomplex structures that make them holomorphic bundles over ( H C ( M ) , I ( C ) ) and the extended connection ω ′ is J ′ C -invariant with respect to the complexstructure J ′ C of P ′ .Finally, the components in holomorphic gauges of the extended connection ω ′ and of its (1 , -potential are holomorphic functions of any set of com-plex coordinates ( z ℓ , w m , ( u ir )) of the complex manifold ( H C ( M ) = M × M × SL ( C ) , I C ) that correspond to complex coordinates ( z ℓ , z m = w m , ( u ir )) of thesubmanifold H ( M ) ⊂ H C ( M ) ,Proof. To prove these statements, we need the following simple lemma.
Lemma 4.2.
The complex gauge field ( E, D ) is an instanton if and only ifthe curvature F ′ of its lift ( E ′ , D ′ ) on H ( M ) is such that F ′ ( X + , Y + ) = 0 = F ′ ( X + , Y + ) for X + , Y + ∈ D + . (4.4) Proof of Lemma.
As observed above, each space D + | ( x,U ) or (more precisely,its isomorphic projection onto T C x M ) is spanned by the vectors e + a of theadapted frames u : H n → T x M . Hence the curvature of ( E, D ) satisfies thefirst condition in (4.2) if and only if F ′ ( X + , Y + ) = 0 and F ′ ( X − , Y − ) = 0 for X ± , Y ± ∈ D ± . The other conditions in (4.2) are direct consequences of thedecomposition of D C | ( x,U ) ≃ T C x M into irreducible SU -moduli. NSTANTONS ON HYPERK ¨AHLER MANIFOLDS 17
Due to this lemma and (2.4), the curvature 2-form Ω ′ of the lifted connection ω ′ identically vanishes on any pair of horizontal lifts X h , Y h ∈ H C of vectorfields X , Y in the anti-holomorphic distribution D − + V = T H ( M )of ( H ( M ) , I ), where D − and V are the I -anti-holomorphic horizontal andvertical distributions of H ( M ) described in § S ⊂ T C P ′ generated by the vectors X h and the anti-holomorphic verticaldistribution of P ′ is an involutive complex distribution. The same holds for thesubbundle S = S ⊂ T C P ′ . Hence the direct sum decomposition T C P ′ = S + S 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 areal analytic submanifold of P ′ since it is the fixed point set of an appropriatereal analytic involution. One can also check that the restricted distribution H | P o ′ coincides with the distribution given by the J ′ -invariant subspaces ofthe 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 claimfollows immediately.Concerning the second claim, the same arguments as above yield the exist-ence of a complex structure J ′ C on the extended bundle P ′ (and, consequently,a corresponding complex structure on the associated bundle E ′ ), which makesit a holomorphic bundle over the complex manifold ( H C ( M ) , I ( C ) ) and leavesinvariant the horizontal distribution H determined by the extended connection1-form ω ′ . The final claim is a consequence of the fact that the extension to W ⊂ H C ( M ) of every real analytic datum on H ( M ) is obtained by consider-ing the power series of such a datum in the variables ( z ℓ , w m := ¯ z m , ( u iℓ )) asa power series in the independent complex variables ( z ℓ , w m , ( u iℓ )), thus holo-morphic in both the variables z ℓ and w m . The holomorphy in the u iℓ followsfrom the fact that ω ′ is the lift of a connection form of p : P → M to thebundle p ′ : P ′ → H ( M ).Consider the basis ( H o , H o ++ , H o −− ) of sl ( C ) defined by H o := (cid:18) − (cid:19) , H o ++ := (cid:18) (cid:19) , H o −− := (cid:18) (cid:19) (4.5)and denote by H , H ++ , H −− the associated holomorphic vector fields on H ( M ), determined as holomorphic parts of the infinitesimal transformationsof the right actions of the one-parameter groups generated by H o , H o ++ , H o −− .The restrictions of such vector fields H α to each vertical fiber { x } × SL ( C ) ≃ SL ( C ) are left invariant and generate a Lie algebra isomorphic to sl ( C ).We may now prove the theorem, on which all our results are based (see also[1], Prop. 7 and Thm.4). Theorem 4.3.
Given a real analytic complex instanton ( E, D ) and a point ( x o , x o , U ) ∈ H ( M ) , there exists an SL ( C ) -invariant neighbourhood U ⊂ H C ( M ) of ( x o , U ) and a holomorphic gauge ϕ : P ′ | U → U × G of the H C ( M ) -extension of the lifted bundle of P with associated (1 , -potential A satisfyingthe conditions A ( H ) = A ( X − ) = 0 , X − ∈ D − . (4.6) Such a potential A is uniquely determined by the g -valued function A −− := A ( H −− ) in the following sense: given the lift ( b E ′ , b D ′ ) on H ( M ) of a realanalytic complex instanton ( b E, b D ) on M , if b E ′ | U coincides with E ′ | U and iffurthermore the (1 , -potential b A of ( b E ′ , b D ′ ) in a holomorphic gauge satisfies (4.6) , then b D ′ | U ∩ H ( M ) = D ′ | U ∩ H ( M ) ⇐⇒ b A ( H −− ) | U ∩ H ( M ) = A ( H −− ) | U ∩ H ( M ) . (4.7) Remark 4.4.
Note that by b D ′ | U ∩ H ( M ) = D ′ | U ∩ H ( M ) we mean that the differen-tial operators D ′ , b D ′ are identical , not just equivalent up to an automorph-ism (gauge transformation) of b E ′ | U . However, (4.7) implies a bijective cor-respondence between covariant derivatives and their potential components A −− = A ( H −− ). This induces a natural bijective correspondence betweenequivalence classes of instantons up to local automorphisms and equivalenceclasses of (1 , A −− = A ( H −− ) up to appropriate gaugetransformations (2.2). In fact, the required gauge transformations are preciselythose leaving condition (4.6) unchanged. Proof.
Let us extend ( P ′ , ω ′ ) to an SL ( C )-invariant tubular neighbourhood W ⊂ H C ( M ) of H ( M ) and consider the two G -invariant complex distributions D h − and D h − ⊕ < H h > on the extended bundle P ′ , the former generated by thehorizontal lifts X h of the complex vector fields in D − , the latter generated bythe complex vector fields in D h − and the horizontal lift H h of the holomorphicvector field H of H C ( M ). We recall that the distribution D − is spanned by thecomplex vector fields e − a described in § H ( M ) and the frames for thefibers H x of the bundle H , the distribution D − is seen to be invariant underthe flow of the vector field H . This implies that the horizontal distribution D h − is invariant under the flow of the horizontal vector field H h .By Lemma 4.2, the curvature 2-form Ω ′ of ω ′ vanishes identically on thedistribution D h − , meaning that D h − is involutive. Moreover, from the lastclaim in Proposition 4.1, D h − is generated by holomorphic vector fields of theholomorphic bundle π : P ′ → W ⊂ H C ( M ). Hence, by the complex FrobeniusTheorem, for each y o = ( x o , x o , U, g ) ∈ P ′ | H ( M ) there is an SL ( C ) × G -invariantneighbourhood P ′ | U of y o , which is holomorphically foliated by integral leavesof D h − . Note that the union of the H h -orbits of the points of one such integralleaf is an integral leaf of the larger complex distribution D h − ⊕ < H h > . Itfollows that P ′ | U is actually holomorphically foliated by the integral leaves of NSTANTONS ON HYPERK ¨AHLER MANIFOLDS 19 this larger distribution and we may consider a holomorphically parameterisedfamily of integral leaves of this distribution, which fills a complex submanifold S ′ transversal to the G -orbits at each point. Without loss of generality, wemay also assume that such a submanifold S ′ projects biholomorphically ontoan SL ( C )-invariant neighbourhood U ⊂ H C ( M ) of ( x o , x o , U ) and hence it isa graph of a holomorphic section of the G -bundle P ′ . Associated with such asection, 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 , A in such a gauge satisfies (4.6).Suppose now that ( b E ′ , b D ′ ) is a real analytic lifted instanton associatedwith the pair ( b P ′ , b ω ′ ), and assume that b A ( H −− ) | U ∩ H ( M ) = A ( H −− ) | U ∩ H ( M ) . Byreal analyticity, we may assume that U is such that b A ( H −− ) | U = A ( H −− ) | U .Thus, the horizontal lifts H hα , b H hα of the holomorphic vector fields H α , α ∈{ , ++ , −−} , determined by the connection forms ω ′ and b ω ′ , respectively, havethe forms b H h = H h , b H h −− = H −− + b A −− = H −− + A −− = H h −− , b H h ++ = H ++ + b A ++ , H h ++ = H ++ + A ++ , (4.8)where we denote by A ±± := A ( H ±± ), b A ±± := b A ( H ±± ) the components alongthe vector fields H ±± of the corresponding (1 , B ++ := b A ++ − A ++ , the expansions (4.8) can be written as b H h = H h , b H h ++ = H h ++ + B ++ , b H h −− = H h −− . (4.9)Since the considered principal bundles P ′ , b P ′ are lifts to H ( M ) of bundles over M , the Lie brackets among the horizontal lifts of vector fields H A coincide withthe Lie brackets among their projections (see (2.10)) so that[ b H h , b H h ++ ] = 2 b H h ++ , [ b H h ++ , b H h −− ] = b H h . This and (4.9) imply that the g -valued holomorphic function B ++ is such that H h · B ++ = 2 B ++ , H h −− · B ++ = 0 . On the other hand, by Proposition 2.2, the vector fields H h , H h ++ , H h −− gener-ate a right holomorphic SL ( C )-action on P ′ | U and the bundle P ′ | U is foliatedby regular orbits of this action (thus, each such orbit is identifiable with acopy of SL ( C )). We may therefore apply Lemma 5.3 in [16] (see also [24], § H , H −− in place of the pair H , H ++ – to the restrictions of B ++ to each such orbit. This lemma implies that B ++ is identically vanishing along each such orbit. It follows that B ++ ≡ b A ++ ≡ A ++ , and that b H h ++ = H h ++ .In order to conclude that b D ′ = D ′ , it is now sufficient to prove the existenceof a collection of holomorphic vector fields ( e + a , e − b ) on U ⊂ H C ( M ) with thefollowing two properties: a) they span a distribution which is complementary to the one generated by the vector fields H A and b) the remaining compon-ents b A ± a := A ( e ± a ) and A ± a = A ( e ± a ) of the potentials of b D ′ and D ′ areidentical. Let us consider a set of (locally defined) holomorphic vector fields( e − a ) generating the distribution D − and projecting pointwise onto vectors e − a ∈ T C M determined by adapted frames as in § e − a are eigenvectors with eigenvalue − H , the vectors e + a := [ H ++ , e − a ] 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 I C complementary to the distribution spanned by the H A . Moreover, their cor-responding lifts e h ± a , b e h ± a , determined by the two connection forms ω ′ , b ω ′ , aresuch that [ H h ++ , e h − a ] = e h + a , [ b H h ++ , b e h − a ] = b e h + a . (4.10)Setting A ± a := A ( e ± a ), b A ± a := b A ( e ± a ) and recalling that, by hypothesis, A − a = b A − a = 0, we have that (4.10) implies − e h − a · A ++ = A + a and − e h − a · b A ++ = A + a .Since we have proven that b A ++ ≡ A ++ , this gives b A ± a ≡ A ± a , as desired.We call the g -valued map A −− | U ∩ H ( M ) := A ( H −− | U ∩ H ( M ) ), which uniquelydetermines the extended holomorphic function A −− | U and thereby the gaugefield ( E ′ | U , D ′ ), a prepotential on U ∩ H ( M ) for the instanton ( E, D ). Theholomorphic gauges of the (extended) lifted bundle P ′ in which the potentialof ω ′ satisfies (4.6) are called analytic . Remark 4.5.
The previous literature on the harmonic space formulation e.g.[23, 22, 24, 1] used gauge conditions A = A + a = 0 and prepotentials A ++ .Here we choose to reverse the role of the signs. This has the advantage thatprepotentials are holomorphic rather than anti-holomorphic with respect tothe complex structure I of H ( M ) (see Remark 4.7).4.4. Analytic gauges, bridges and normalisations
Assume that our instanton (
E, D ) is the complexification of an instantonwith compact structure group G o over the hk manifold ( M, g, ( J α )). Aroundeach ( x o , U ) ∈ H ( M ) there are two very important classes of gauges to beconsidered: the exp-central gauges and the analytic gauges. Let us brieflycompare their main features:– If ϕ : P ′ | V × SL ( C ) → V × SL ( C ) × G is (the restriction to H ( M ) ⊂ H C ( M )of) an analytic gauge, the corresponding holomorphic potential A for ω is such that A := A ( H ) is identically vanishing and A ( X − ) = 0 for anyvector field X − ∈ D − . In contrast with this, the functions A ±± := A ( H ±± )and the functions A ( X + ), X + ∈ D + , are in general non-trivial.– If e ϕ : P ′ | V × SL ( C ) → V × SL ( C ) × G is an exp-central gauge, the cor-responding potential e A is such that the components e A α = e A ( H α ), α ∈{ , ++ , −−} , identically vanish, while all functions e A ( X ± ), X ± ∈ D ± , are NSTANTONS ON HYPERK ¨AHLER MANIFOLDS 21 in general non-trivial, being nevertheless constrained by the conditions (a),(b) in § g : U × SL ( C ) → G , which give the gauge transformations h → g ( x,U ) · h from central gauges to analytic gauges are usually called bridges (e.g.[23, 24, 1]). 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 localdiffeomorphism. 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 .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 thenewly built analytic gauges satisfy additional normalisation conditions , whichdrastically reduce their degrees of freedom. Such normalised analytic gaugescan be considered as complex analogues of the Coulomb gauges for arbitrarygauge fields.In order to properly state such a normalisation, we need to introduce someappropriate notation. Given x o ∈ M and a (sufficiently small) simply connec-ted neighbourhood V ⊂ M of x o , let ( e + a , e − b ) be a 4 n -tuple of holomorphicvector fields of H C ( M ) that generate the distributions D ± ⊂ T C ( V × V × SL ( C )) ( ⊂ T C H C ( M )), constructed as in the proof of Theorem 4.3. Then,pick an element U o ∈ SL ( C ), say U o = ( ), and let λ a , µ : V × V × SL ( C ) → C n , a = 1 , . . . , n , be a set of 2 n + 1 holomorphic functions satisfying thefollowing conditions H ++ · λ a = 0 , H ++ · µ = − , H · λ a = H · µ = 0 ,e + b · λ a = δ ab , e + b · µ = 0 , λ a | ( x o ,x o ,U o ) = µ | ( x o ,x o ,U o ) = 0 . (4.11)By [16, Lemma 5.2], functions satisfying the first line of these conditions surelyexist and are determined up to addition of a holomorphic function constantalong each SL ( C )-orbits. Using the commutation relations between H ++ , H and e + a , we can see that also the second line of these conditions canbe satisfied, fixing the λ a and µ completely. Now consider the holomorphicdistribution e D ⊂ T C ( V × V × SL ( C )) generated by the vector fields e + a , H ++ , e H −− := H −− + λ a e − a + µH , which can be directly checked to be involutive. Finally let b S and S be twocomplex submanifolds of V × V × SL ( C ) ⊂ H C ( M ), one included in theother, which are integral leaves of the holomorphic distributions D + and e D ,respectively, and both passing through ( x o , x o , U o ). Note that each tangentspace of S is complementary to D − | ( x,y,U ) + < H | ( x,y,U ) > .We may now state and prove the advertised existence result. Lemma 4.6.
Let e ϕ : P ′ | V × SL ( C ) → V × SL ( C ) × G be an exp-central gauge,determined by an exponential gauge for ( P, ω ) on a simply connected neigh-bourhood V ⊂ M around x o ∈ M , and e A the associated potential for ω . Let also b S ⊂ S ⊂ V × V × SL ( C ) be the pair of complex submanifolds passingthrough ( x o , U o ) described above. If V is sufficiently small, there exists an ana-lytic gauge ϕ : P ′ | V × V × SL ( C ) → V × V × SL ( C ) × G , in which the prepotential A −− is such that H · A −− = − A −− , e − a · A −− = 0 , H −− · A −− = 0 ,A −− | b S = − λ a e A − a | b S , H ++ · A −− | b S = − λ a e A + a | b S . (4.12) Proof.
For simplicity, we use e ϕ to identify P ′ | V × SL ( C ) with V × SL ( C ) × G so that we may assume that the considered exp-central gauge is nothing butthe identity map. Let us consider the integral leaves in V × V × SL ( C ) × G of the holomorphic distribution D h − + < H h > , which passes through thepoints of the manifold S × { e } ⊂ V × V × SL ( C ) × { e } . Being horizontal,they are transversal to the G -orbits and, by dimension counting, they filla submanifold S ′ ⊂ P ′ which projects diffeomorphically onto an SL ( C )-invariant neighbourhood V ′ × V ′ × SL ( C ) of ( x o , x o , U o ). Thus there is a gauge ϕ o : P ′ | V × V × SL ( C ) → V × V × SL ( C ) × G , which maps S ′ into the submani-fold V × V × SL ( C ) × { e } and which we may assume to satisfy the condition ϕ o | S ×{ e } = Id S ×{ e } (see also the proof of Theorem 4.3). By construction, ϕ o is an analytic gauge and the bridge g ( x,y,U ) from e ϕ = Id to ϕ o is such that h g o ( x,y,U ) · h = e · h for each ( x, y, U ) ∈ S . Hence, writing this bridge in theform g o ( x,y,U ) = e ψ o ( x,y,U ) for an appropriate g -bridge ψ o , we have that ψ o | S ≡ ψ o satisfies the equations H · ψ o = 0 and e − a · ψ o + e A ( e − a ) = 0 be-cause of the following three properties: a) in any central gauge the potential e A satisfies e A ( H ) = 0; , b) the potential A o in the analytic gauge ϕ o satisfies(4.6) and c) the potentials e A and A o are related by (2.2).Let us now denote by A o −− the prepotential of the given instanton in thisgauge ϕ o . Expanding the identities F ϕ o ( H , H −− ) = F ϕ o ( e − a , H −− ) = 0 interms of the potential A o in this gauge and recalling that, being an ana-lytic gauge, we have A o = A o − a = 0, we find that H · A o −− = − A o −− and e − a · A o −− = 0, i.e. the first two conditions of (4.12). We remark that these twoconditions are satisfied by any prepotential , being merely consequences of theabove properties of the curvature.However, the prepotential A o −− does not necessarily satisfy the other condi-tions in (4.12) also. To get a prepotential with such additional properties, weneed to further change ϕ o into a new (further restricted) gauge ϕ , which pre-serves the property of being analytic, i.e. with (1 , A and A − a identically vanishing and with A −− satisfying the first pair of equa-tions in (4.12). In order to determine this new analytic gauge ϕ , let us considerthe (2 n + 2)-dimensional involutive distribution b D ⊂ T C ( V × V × SL ( C ) × G )generated by the holomorphic vector fields H , H ±± + A o ±± and e − a . Then,for each point ( x, y, U o , g ) of the manifold b S × G , consider the unique integral NSTANTONS ON HYPERK ¨AHLER MANIFOLDS 23 leaf through ( x, y, U o , g ) of this distribution T ( x,y,U o ,g ) ⊂ V × V × SL ( C ) × G .Now we may determine a G -equivariant g -valued holomorphic function h −− on T ( x,y,U o ,g ) (the G -equivariance being with respect to the standard right actionof G on V × V × SL ( C ) × G and the adjoint G -action on g ) satisfying theequations H · h −− = − h −− , H −− · h −− = − H −− · A o −− − [ h −− , A o −− ] , e − a · h −− = 0 . (4.13)Once again, the existence of such an h can be checked using [16, Lemma 5.3].Indeed, one can construct a solution to the first two conditions as follows.Along the image of T ( x,y,U o ,g ) under the inverse gauge transformation e ϕ ◦ ϕ o − (= ϕ o − ) (note that, by G -equivariance of the trivialisation, each intersection ofthe submanifold ϕ o − ( T ( x,y,U o ,g ) ) with a vertical set { ( e x, e y ) } × SL ( C ) × G isentirely included in a submanifold of the form { ( e x, e y ) }× SL ( C ) ×{ g } ) we mayconsider a G -equivariant g -valued solution e h −− to the differential problem H · e h −− = − e h −− , H −− · e h −− = − H −− · (cid:0) A o −− ◦ ϕ o (cid:1) , e o − a · e h −− = 0 . (4.14)By [16, Lemma 5.3], applied to the pair of vector fields H , H −− in place ofthe pair H , H ++ , such a solution e h −− exists. The corresponding function h −− = e h −− ◦ ϕ o − is therefore a solution to (4.13). Note that the solution e h −− to (4.14) is uniquely determined up to addition of a solution e k −− of theassociated system H · e k −− = − e k −− , H −− · e k −− = 0 , e o − a · e k −− = 0and that such a solution e k −− can be assumed to take any desired valued atthe points of V × V × { U o } × G . This can be checked by observing that anexplicit expression for any holomorphic solution e k −− is actually determined in[16, Lemma 5.3] as e k −− ( x, y, U, g ) = X m,n ≥ m + n =2 c mn ( x, y, g )( u − ) m ( u − ) n , with U = ( u i ± ). Observing that the matrix U o = I has entries u − , u − equalto 0, 1, respectively, by appropriately choosing the component c ( x, y, g ), thefunction can take any desired value at the points of V × V × { U o } × G . Since ϕ o (( V × V × { U o } × G ) ∩ b S ) ⊂ ( V × V × { U o } × G ) ∩ b S , such a residual degreeof freedom for the e h −− can be used to make the restriction h −− | b 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 ) ∈ b S × G ,into a global solution of (4.13) on V × V × SL ( C ) × G and restrict such aglobally defined g -valued map h −− to the submanifold V × V × SL ( C ) × { e } ≃ V × V × SL ( C ). Then, along each integral leaf in V × V × SL ( C ) of thedistribution spanned by H α and e − a , we may consider a new g -valued function ψ ′ satisfying the differential problem H · ψ ′ = 0 , H −− · ψ ′ = h −− , e − a · ψ ′ = 0 . 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 itto 0 at each point of the form ( x, y, U o ). Combining the solutions along all theconsidered integral leaves, we get a global solution ψ ′ such that ψ ′ ( x, y, U o ) = 0for all ( x, y ) ∈ V × V and satisfying the differential problem H · ψ ′ = 0 , H −− · ( H −− · ψ ′ ) = − H −− · A o −− − [ H −− · ψ ′ , A o −− ] , e − a · ψ ′ = 0 . (4.15)Let us now consider the new gauge ϕ , obtained by applying to the analyticgauge ϕ o the gauge transformation g ( x,y ) = e ψ ′ ( x,y ) , ( x, y ) ∈ V × V . Recall-ing that A o = A o − a = 0, we see that the potential in the new gauge ϕ hascomponents A , A −− , A − a given by A = e − ad ψ ′ ( A o + H · ψ ′ ) = 0 , A − a = e − ad ψ ′ ( A o − a + e − a · ψ ′ ) = 0 ,A −− = e − ad ψ ′ ( A o −− + H −− · ψ ′ ) . (4.16)From this and (4.15), it follows that H −− · A −− = e ad ψ ′ (cid:0) [ H −− · ψ ′ , A o −− ] + H −− · A o −− + H −− · ( H −− · ψ ′ ) (cid:1) = 0and, with a similar computation, that H · A −− = − A −− and e − a · A −− = 0.In other words, the new prepotential A −− satisfies all three conditions in thefirst line of (4.12). To check the last two normalising conditions, we recall atfirst that in any central frame, the corresponding potential e A satisfies H · e A − a = − e A − a , H ++ · e A − a = e A + a , H −− · e A − a = 0 . Hence, if we differentiate the identity ψ o | S = 0 in directions tangent to S ,using (4.11), (4.12) and commutation relations we get e + a · ψ o | S = H ++ · ψ o | S = H −− · ψ o | S + λ a e A − a | S = 0 . (4.17)Since e ψ o | S = Id, we conclude that A o −− | S = e A −− | S + H −− · ψ o | S = − λ a e A − a | S .Now, from (4.15), the property that b S ⊂ S and ψ ′ | b S = H −− · ψ ′ | b S = 0, thesecond line in (4.12) follows immediately.The g -bridge ψ , the corresponding analytic gauge ϕ and the associated pre-potential A −− established by Lemma 4.6 are called normalised at ( x o , x o , U o ). Remark 4.7.
The proof of the previous theorem shows that any prepotential A −− (not just the normalised ones) is such that X − · A −− = 0 for any horizontalvector field in the distribution D − . On the other hand, we also have that:a) F ( V, · ) = 0 for any vertical vector field V of H C ( M ), i.e. for any V which is tangent to the vertical fibers { ( x, y ) } × SL ( C ) of H C ( M ). NSTANTONS ON HYPERK ¨AHLER MANIFOLDS 25 b) The vertical anti-holomorphic distribution of H C ( M ) is spanned byright-invariant vector fields along the fibers { x } × SL ( C ), which there-fore commute with the left-invariant holomorphic vector field H −− .c) The holomorphic potential A in an analytic gauge vanishes identicallyalong the anti-holomorphic vector fields of H C ( M ). This is due to Pro-position 4.1 and the fact that the analytic gauges are holomorphic withrespect to the complex structures of the extended P ′ over ( H C ( M ) , I C ).From (a), (b), (c) and the explicit expression of F in terms of a potentialit follows that, for any anti-holomorphic vertical vector field V of H C ( M ),we have V · A −− = 0. This and the above property X − · A −− = 0 provethat for any prepotential A −− defined on some open set W of the complexifiedharmonic space ( H C ( M ) , I C ) , the restriction A −− | H ( M ) ∩ W is also holomorphicwith respect to the complex structure I of the harmonic space H ( M ).5. Existence, uniqueness and compactness theorems
Existence and uniqueness of an instanton with a given prepo-tentialTheorem 5.1.
Let V ⊂ M be open and simply connected. For any map A −− : V × SL ( C ) ⊂ H ( M ) → g , which is holomorphic (i.e. with X − · A −− = 0 for any X − ∈ D − and holomorphic in the complex coordinates of SL ( C ) ) andsatisfies H · A −− = − A −− , (5.1) there is a unique instanton ( E | V , D ) on V and an analytic gauge ϕ : P | V × SL ( C ) → ( V × SL ( C )) × G , for which A −− is the prepotential in that gauge.Proof. In order to prove the existence of an associated instanton, we proceedas in the proof of [1, Thm. 4] and we consider an orbit { x } × O ⊂ V × SL ( C )of the Borel subgroup B ⊂ SL ( C ), generated by h H o −− , H o i ⊂ sl ( C ), i.e. B = (cid:26) (cid:18) ζ z ζ − (cid:19) , ( ζ , z ) ∈ C ∗ × C (cid:27) ⊂ SL ( C ) . Along such an orbit, we may consider the unique g -valued connection for the G -bundle { x } × O × G , whose (1 , A ( H −− ) = A −− and A ( H ) := 0. Due to (5.1) and the fact that [ H , H −− ] = − H −− , such aconnection has zero curvature. Hence there exists a new holomorphic gauge ϕ : { x } × O × G → { x } × O × G which fixes all points of { x } × O × { e } and transforms A into the identically vanishing potential. By (2.2), this istantamount to saying that the associated gauge transformation g ( x,U ) , ( x, U ) ∈{ x } × O , is a solution to the differential problem H · g = 0 , H −− · g + A −− = 0 , g | ( x,U ) = e . (5.2) Since the space of all B -orbits { x }× O in V × SL ( C ) is diffeomorphic to V × C P and is therefore simply connected, all these new gauges combine into a globallydefined gauge ϕ : V × SL ( C ) × G → V × SL ( C ) × G , (5.3)which maps each B × G -orbit into itself and satisfies (5.2) at all points. Wemay now consider the g -valued map on V × SL ( C ) A ++ : V × SL ( C ) → g , A ++ := − ( H ++ · g ) g − . Combining this function with the g -valued maps A −− and A = 0, we mayconstruct the G -invariant vector fields H h , H h ±± on V × SL ( C ) × G , which onthe submanifold V × SL ( C ) × { e } are H h | V × SL ( C ) ×{ e } := H | V × SL ( C ) ×{ e } ,H h ±± | V × SL ( C ) ×{ e } := H ±± | V × SL ( C ) ×{ e } + A ±± . (5.4)By real analyticity, there is an open U ⊂ H C ( M ) with U ∩ H ( M ) = V × SL ( C ),where these vector fields extend as holomorphic fields on U × G . Moreover,by the construction of A −− and of the map (5.3), along each fiber { x } × SL ( C ) × G , the functions A := 0, A ±± can be considered as the threecomponents of the holomorphic potential of a connection for the G -bundle π ( x ) : { x }× SL ( C ) × G → { x }× SL ( C ), which is transformed by the gauge ϕ into the trivial potential. This means that the associated covariant deriv-ative has identically vanishing curvature, i.e. F ( H α , H β ) ≡ α, β ∈{ , ++ , −−} .These connections on the submanifolds { x }× G of the open set U × SL ( C ) × G can be considered as restrictions of a G -connection on the (trivial) bundle p : U × G → U , with associated holomorphic potential A : U → T ∗ U ⊗ g satisfying A ( H α ) := A α , α ∈ { , ++ , −−} ,A ( X − ) := 0 , for any X − ∈ D − , (5.5) A ( e + a ) := − e − a · A ++ for any frame field ( e + a , e − b ) generating D + ⊕ D − and projecting onto the complex frames ( e ± a ) of M described in § . We now need to show that the curvature of the associated covariant derivative D ′ satisfies the following equalities for each X ± , Y ± ∈ D ± : F ( X + , Y + ) = 0 = F ( X − , Y − ) ,F ( H α , X + ) = 0 = F ( H α , X − ) , α ∈ { , ++ , −−} . (5.6)This would conclude the proof. Indeed, by Proposition 2.2 and Lemma 4.2,it would imply that the corresponding gauge field ( E ′ | U , D ′ ) is the extensionto U ⊂ H C ( M ) of the lift of an instanton ( E | V , D ) on V and with A −− asprepotential. Further, by the above definition of the potential, the trivial NSTANTONS ON HYPERK ¨AHLER MANIFOLDS 27 gauge ϕ = Id V × V × SL ( C ) × G would be an analytic gauge for such an instanton.Hence, by Theorem 4.3, any instanton having prepotential A −− in such agauge would necessarily coincide with ( E | V , D ).In order to prove (5.6), for each given point ( x o , y o , U o ) ∈ U ⊂ H C ( M ) = M × M × SL ( C ) we select a (locally defined) collection of I ( C ) -holomorphicvector fields ( e + a , e − a ), which generate the distribution D + ⊕ D − and projectto the complex vectors in T C M determined by adapted frames of M and de-scribed in § e ± a of this kind can be constructedthrough the following three step procedure:1) Pick a local section σ : U ′ → O g ( V × V , J α ) of the bundle of adapted framesover V × V .2) Consider the restrictions to the section σ ( U ′ ) of the canonical horizontalvector fields ∇ e ± a := e ± a + Γ ± aJI determined by the Levi-Civita connection(here, the Γ ± aJI are the Christoffel symbols).3) Take the projections ( e ± a ) of the vector fields ∇ e ± a := e ± a + Γ ± aJI onto theunderlying open set U ′ .Since the Levi-Civita connection has vanishing torsion, the vector fields e ± a constructed in this way are such that[ e + a , e + b ] = 0 = [ e − a , e − a ] . (5.7)Further, just by looking at the standard action of the H oα ∈ sl ( C ) on theelements ( h oi ⊗ e oa ) ⊂ C ⊗ C n ≃ ( H n ) C , the actions of the vector fields H α of H C ( M ) = M × M × SL ( C ) on the e ± a are[ H , e ± a ] = ± e ± a , [ H ±± , e ± a ] = 0 , [ H ±± , e ∓ a ] = e ± a . (5.8)Hence, by (2.5) and the assumption A ( H ) = A ( e − a ) = 0, F ( H , e − a ) = F ( e − a , e − b ) = 0 . (5.9)From condition A ( e + a ) = − e − a · A ( H ++ ), we also have F ( H ++ , e − a ) = − e − a · A ( H ++ ) − A ( e + a ) = 0 ,F ( H −− , e − a ) = − e − a · A ( H −− ) = − e − a · A −− (5.1) = 0 . (5.10)Finally, from (5.9), (5.10), the property F ( H α , H β ) = 0 and Bianchi identities X cyclic permutationsof (1 , , D ′ X i F ( X i , X i ) + F ( X i , [ X i , X i ]) = 0 (5.11)with X , X , X equal to the triple H ++ , H −− , e − a or to the triple H ++ , H , e − a , we get that F ( H −− , e + b ) = 0 = F ( H , e + b ). All this shows that F ( e − a , e − b ) = 0 ,F ( H α , e − a ) = 0 , α ∈ { , ++ , −−} ,F ( H , e + a ) = F ( H −− , e + a ) = 0 . (5.12) Now, in order to conclude the proof we need the following
Lemma 5.2.
The components F ( H ++ , e + a ) , F ( e + a , e + b ) , ≤ a, b ≤ n , ofthe above defined covariant derivative D ′ are identically vanishing.Proof of Lemma. We recall that, by real analyticity, F ( H ++ , e + a ), F ( e + a , e + b )are extended as holomorphic functions on an SL ( C )-invariant neighbourhood U × { e } ⊂ H C ( M ) × { e } of V × SL ( C ) × { e } . Note also that, due to (5.12)and Bianchi identities amongst the vector fields H , H ++ , e + a or H ++ , H −− , e + a , respectively, we have that D ′ H F ( H ++ , e + a ) = 3 F ( H ++ , e + a ) ,D ′ H −− F ( H ++ , e + a ) = 0 . (5.13)If we consider the (holomorphic extensions of the) G -invariant vector fields H h , H h ±± on V × SL ( C ) × G defined by (5.4) and the G -equivariant function F +++ a : U × G → g , F +++ a | U ×{ e } := F ( H ++ , e + a ) , we see that (5.13) is equivalent to the system of equations for the F +++ a H h · F +++ a = 3 F +++ a ,H h −− · F +++ a = 0 . (5.14)By [16, Lemma 5.3] (or, more precisely, by its analogue involving the vectorfield H −− in place of H ++ ) the restriction of (5.14) to each orbit O ⊂ U × G ofthe SL ( C )-action generated by the H hα , admits exactly one solution, namelythe identically zero function. It follows that F +++ a = F ( H ++ , e + a ) ≡ V × SL ( C ) × G .Let us now focus on the components F ( e + a , e + b ). By (5.12) and Bianchiidentities (5.11) amongst the vector fields H , e + a , e + b and H −− , e + a , e + b , wehave that D ′ H F ( e + a , e + b ) = 2 F ( H ++ , e + a ) ,D ′ H −− F ( e + a , e + b ) = F ( e + a , e − b ) − F ( e + b , e − a ) = 0 , (5.15)where the last equality is a consequence of the fact that A − a = 0 and of F ( e + a , e − b ) = e − b · ( e − a · A ++ ) [ e − a ,e − b ]=0 = e − a · ( e − b · A ++ ) = F ( e + b , e − a ) . (5.16)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 · F ++ ab = 2 F ++ ab and H h −− · F ++ ab = 0. As above, by [16, Lemma 5.3], weget that F ++ ab = F ( e + a , e + b ) vanishes identically.From (5.12) and Lemma 5.2, all conditions in (5.6) are satisfied. NSTANTONS ON HYPERK ¨AHLER MANIFOLDS 29
Bounds for normalised prepotentials
Let V × SL ( C ) be an SL ( C )-invariant open subset of H ( M ) with V ⊂ M relatively compact and simply connected and such that V × SL ( C ) isa domain for both an exp-central gauge e ϕ and an analytic gauge ϕ for aninstanton ( E, D ) which is normalised around ( x o , x o , U o := I ) ∈ V × V × SL ( C ),with x o ∈ V . Let us also denote by A −− : V × SL ( C ) → g the correspondingnormalised prepotential. Theorem 5.3.
There exists a relatively compact simply connected neighbour-hood V ′ ⊂ V of x o , such that for each compact subset K ⊂ V ′ × SL ( C ) , thereis a constant c K, V > , depending just on the compact sets K and V , such that k A −− k C ( K, g ) ≤ c K, V k F k ( e ϕ ) C ( V , g ) . (5.17) Proof.
Let b S ⊂ V × V × SL ( C ) be the complex submanifold passing through( x o , x o , I ) and defined in § x, y, U ) ∈ b S , let S ( x,y,U ) be theunique integral leaf through ( x, y, U ) of the complex distribution generated bythe holomorphic vector fields e − a and H of H C ( M ). Let also V ′ be a relativelycompact connected open subset of V which contains ( x o , x o , I ) and such thatthe SL ( C )-invariant set V ′ × V ′ × SL ( C ) is included in V ′ × V ′ × SL ( C ) ⊂ [ ( x,y,U ) ∈ b S S ( x,y,U ) ∩ ( V × V × SL ( C )) . (5.18)The existence of such a V ′ is guaranteed by the fact that the family of integralleaves S ( x,y,U ) is SL ( C )-invariant. The condition (5.18) is chosen to ensure thatany point of V ′ × V ′ × SL ( C ) ⊂ H C ( M ) lies in some (connected) intersection S ( x,y,U ) ∩ ( V × V × SL ( C )).Let K ⊂ V ′ × SL ( C ) ⊂ H ( M )( ⊂ H C ( M )) be compact and denote by K ′ ⊂ b S the set of points ( x, y, U ) ∈ b S such that S ( x,y,U ) ∩ K = ∅ . The set K ′ is compact. Indeed, it is the intersection between b S and the compact set ofthe (regular) orbits of the points of K by the action on V × V × SL ( C ) of thelocal group generated by the holomorphic vector fields H and e − a .Since A −− satisfies (4.12), by integration of the conditions e − a · A −− = 0 and H · A −− = − A −− along each connected intersection S ( x,y,U ) ∩ V × V × SL ( C ),it follows that there exists a constant C V >
0, depending only on V , such that k A −− k C ( K, g ) ≤ C V sup ( x,y,U ) ∈ K ′ ⊂ b S k A −− k ! = C V k A −− k C ( K ′ , g ) . On the other hand, by the second line in (4.12), k A −− k C ( K ′ , g ) ≤ C ′ K ′ k e A k C ( K ′ , g ) for some constant C ′ K ′ depending only on the set K ′ or, equivalently, onlyon the compact set K . From this and the fact that e A does not depend onthe coordinates of SL ( C ), we infer that k A −− k C ( K, g ) ≤ C V C ′ K ′ k e A k C ( p ( K ′ ) , g ) , where p : H ( M ) = M × SL ( C ) → M is the projection onto the first factor.Since p ( K ′ ) ⊂ V , the claim follows from (4.3).5.3. The second prepotential and the curvature of instantons
Let V × SL ( C ) be an SL ( C )-invariant open subset of H ( M ) with simplyconnected V ⊂ M , which is domain for both an exp-central gauge e ϕ and a (notnecessarily normalised) analytic gauge ϕ . Further, let A −− : V × SL ( C ) → g be the prepotential of an instanton ( E ′ , D ′ ) in the analytic gauge ϕ . Definition 5.4.
The second prepotential in the analytic gauge ϕ is the g -valued map A ++ := A ( H ++ ) : V × SL ( C ) → g , determined by the evaluationof the holomorphic (1 , A of the extension on H C ( M ) of the pair( P ′ , ω ′ ) along the vector field H ++ .In what follows, to keep a clear distinction between A −− and A ++ , wesometimes call A −− the first prepotential. Either function yields a completelocal description of instantons on hk manifolds, since either one is completelydetermined by the other. However, in our framework, some features of thetwo descriptions are complementary:1) A −− is holomorphic on ( H ( M ) , I ) and is a solution to the simple firstorder linear equation H · A −− = − A −− . If no normalisation is taken,there are no further restrictions. However, there is no direct way tocompute the curvature tensor from A −− .2) A ++ satisfies a (set of) second order nonlinear equations, but in termsof it the curvature is given by the simple formula, F ϕ ( e + a , e − b ) = e − a · ( e − b · A ++ ) , (5.19)i.e. the non-trivial components of the curvature are just the secondorder derivatives of A ++ along the anti-holomorphic directions e − a .The above-mentioned nonlinear equation for A ++ has been used in variouscontexts in the physics literature, where it is known as the Leznov equation(see e.g. [27, 14, 35, 13, 15, 12]). The next lemma provides useful relationsbetween the C k -norms of the two types of prepotentials. Proposition 5.5.
Let A −− : V × SL ( C ) → g be the (first) prepotential foran instanton and A ++ the corresponding second prepotential. Then:1) A −− is the unique solution to the differential problem for the unknown B −− , H ++ · B −− = H −− · A ++ − [ A ++ , B −− ] , H · B −− = − B −− . (5.20) A similar claim holds for A ++ , provided appropriate sign changes aremade. NSTANTONS ON HYPERK ¨AHLER MANIFOLDS 31
2) For each k ≥ , there exist constants M k, , M k ′ > , depending only on k ,such that for x ∈ V , k A −− | { x }× SU k C k (SU , g ) ≤ M k k A ++ | { x }× SU k C k (SU , g ) , (5.21) k A ++ | { x }× SU k C k (SU , g ) ≤ M k ′ k A −− | { x }× SU k C k (SU , g ) . (5.22) Proof. (1) Let ( E ′ | V × SL ( C ) , D ′ ) be the lifted instanton, for which A ±± , A (=0) are components of the potential in some analytic gauge ϕ . The expressionin terms of curvature components of the identity F ϕ ( H ++ , H −− ) = 0 showsthat A −− solves (5.20). This solution is unique. Indeed, if there were anothersolution to (5.20), say A ′−− , the g -valued map e B −− := Ad Ψ ( A ′−− − A −− ) (hereΨ is the gauge transformation from ϕ to an exp-central gauge) would be asolution to the differential problem H ++ · e B −− = 0, H · e B −− = − e B −− . Thisfact and [16, Lemma 5.3] would imply that e B −− = 0. Interchanging the signs,the corresponding claim for A ++ follows.(2) Consider the real basis for su ⊂ sl ( C ) given by the elements G o := H o ++ − H o −− = (cid:18) − (cid:19) , G o := iH o ++ + iH o −− := (cid:18) ii (cid:19) ,G o = iH o = (cid:18) i − i (cid:19) , (5.23)and let G = iH , G = H ++ − H −− , G = iH ++ + iH −− be the correspondingvector fields on H ( M ) = M × SL ( C ). Note that, for each ( x, U ) ∈ M × SU ,the real vectors G α | ( x,U ) , 1 ≤ α ≤
3, give a frame for the tangent space of the totally real submanifold { x } × SU of { x } × SL ( C ).Let ( E ′ | V × SL ( C ) , D ′ ) be the lifted instanton, for which A ±± , A are compon-ents of the potential in some analytic gauge ϕ . Since A = 0 and the curvature F ′ of ( E ′ | V × SL ( C ) , D ′ ) vanishes identically along any vector field that is tangentto the fibres of H ( M ), at each ( x, U ) ∈ { x } × SU we have that0 = iF ′ ( H , H −− ) = iH · A −− + 2 iA −− = G · A −− + 2 iA −− , F ′ ( H ++ , H −− ) = 2 H ++ · A −− − H −− · A ++ + 2[ A ++ , A −− ] == ( G − iG ) · A −− + ( G + iG ) · A ++ + 2[ A ++ , A −− ] . (5.24)Hence, the restrictions V −− := A −− | { x }× SU , W ++ := A ++ | { x }× SU solve thesystem G · V −− + 2 iV −− = 0 , ( G − iG ) · V −− + 2 ad W ++ ( V −− ) = − ( G + iG ) · W ++ . (5.25)We claim that if (5.25) is considered as a system of equations for V −− withcoefficients determined by W ++ , then it is equivalent to a system given by anappropriate first order elliptic operator P with trivial kernel (for definition andfirst properties of elliptic operators, we refer to [7, Appendix G]). To check this, consider the subbundle E of the bundle P := SU × ( g × g × g × g ) → SU ,defined by E = { ( x, U ; X, Y, Z, W ) ∈ P : X = Z, Y = W } . We consider E equipped with the Hermitian product along the fibres determ-ined by (2.8). Then, define P : C ∞ ( M × SU , E ) −→ C ∞ ( M × SU , E ) , P (cid:18) XYXY (cid:19) := G +2 iI G + iG +2 ad W ++ G − iG +2 ad W ++ G − iI − G − iI G + iG +2 ad W ++ G − iG +2 − G +2 iI · (cid:18) XYXY (cid:19) (5.26)One can directly check that P is a first order elliptic operator. Moreover,ker P = { } . Indeed, a quadruple ( X, Y, X, Y ) is in ker P if and only if X and Y are both solutions on SU to the system of equations for g -valued maps fH · f = − f ,H ++ · f + ad W ++ ( f ) = 0 . (5.27)By applying the inverse of a gauge transformation from a central to the ana-lytic gauge ϕ , the (flat) connection on SU , determined by the potential( A = 0 , A ++ , A −−− ), is transformed into the (flat) connection determinedby a potential with trivial components ( A ′ = 0 , A ′ ++ = 0 , A ′−− = 0). Inparticular, the solution f to (5.27) is transformed into a solution e f of thesystem H · e f = − e f ,H ++ · e f = 0 . (5.28)By [16, Lemma 5.3] (applied to each component of the matrix valued function e f ), we get that e f is equal to 0. This shows that (5.28) has ( X, Y, X, Y ) =(0 , , ,
0) as the unique solution.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( V −− , V −− , V −− , V −− ), is a solution to the differential problem P V −− V −− V −− V −− ! = − ( G − iG ) · W ++ − ( G + iG ) · W ++ − ( G − iG ) · W ++ − ( G + iG ) · W ++ ! (5.29)Since P is elliptic with ker P = { } , classical Schauder estimates (see e.g. [7,Appendix H]) imply that, for each k ≥ N k , M k > k V −− k C k (SU , g ) ≤ N k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − ( G − iG ) · W ++ − ( G + iG ) · W ++ − ( G − iG ) · W ++ − ( G + iG ) · W ++ !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C k − (SU , g ) ≤ M k k W ++ k C k (SU , g ) . This gives (5.21). The proof of (5.22) is similar.
NSTANTONS ON HYPERK ¨AHLER MANIFOLDS 33
The local compactness theorem
To conclude this paper, as an example of the utility of the harmonic spaceformulation, we present a streamlined proof of Uhlenbeck, Nakajima andTian’s celebrated local compactness theorem for Yang-Mills fields in the spe-cific case of hk instantons.From the classical estimates in [37, 30] (see also [36, 43]), we know that forany geodesic ball B R = B R ( x o ) of radius R of an m -dimensional Riemannianmanifold ( M, g ), the C -norms of curvatures of Yang-Mills fields are controlledby their corresponding L - or L m -norms and R . In fact, there are constants ε , C, c m > k F k L ( B R ) < εR m − implies k F k C ( B R/ , g ) ≤ CR m k F k L ( B R ) (5.30) k F k L m ( B R ) < c m implies k F k C ( B R , g ) ≤ m CR n k F k L ( B R ) (5.31)Combining these estimates with Theorem 5.3 yields Theorem 5.6 (Local Compactness Theorem for instantons on hk manifolds) . Let B R = B R ( x o ) ⊂ M be a geodesic ball in a n -dimensional hk manifold ( M, g, J α ) , which is included in a relatively compact neighbourhood V ′ of x o where Theorem 5.3 holds. Further let ( E | B R = B R × V, D ( k ) ) be a sequence of(trivialised) instantons, each with the same compact structure group G o andcorresponding normalised prepotential A ( k ) −− on B R × SL ( C ) .If the curvatures are such that k F ( k ) k L ( B R ) < εR n − for all k , with ε > as in (5.30) , then there exists a subsequence ( E | B R , D ( k n ) ) , whose curvatures F ( k n ) converge uniformly to the curvature of a limit instanton ( E | B R , D ( ∞ ) ) .The same conclusion holds if the curvatures are such that k F ( k ) k L n ( B R ) < c n with constant c n > as in (5.31) .Proof. By (5.30), (5.31), (5.17), the sequence of normalised holomorphic pre-potentials A ( k ) −− is uniformly bounded on any compact subset K of B R × SL ( C ).It follows from Montel’s Theorem that there is a subsequence A ( k n ) −− converginguniformly on compacta to a holomorphic map A ( ∞ ) −− , which is the prepotentialof some instanton due to Theorem 5.1. Using (5.22), we may also assume thatthe second prepotentials A ( k n )++ and all their derivatives converge uniformly oncompacta to the second prepotential A ( ∞ )++ and its derivatives corresponding tothe instanton determined by A ( ∞ ) −− . Thus, by (5.19), the curvatures convergeuniformly on compacta as well. References [1] D. V. Alekseevsky, V. Cortes and C. Devchand,
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E-mail : [email protected] Massimiliano PontecorvoDipartimento di Matematica e FisicaUniversit`a di Roma IIILargo San Leonardo Murialdo 1I-00146 RomaItaly
E-mail : [email protected] Andrea SpiroScuola di Scienze e TecnologieUniversit`a di CamerinoVia Madonna delle CarceriI-62032 Camerino (Macerata)Italy
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