Yang-Mills instantons in Kaehler spaces with one holomorphic isometry
IIFT-UAM/CSIC- - October nd , Yang-Mills instantons in Kähler spaceswith one holomorphic isometry
Samuele Chimento a , Tomás Ortín b and Alejandro Ruipérez c , Instituto de Física Teórica UAM/CSICC/ Nicolás Cabrera, – , C.U. Cantoblanco, E- Madrid, Spain
Abstract
We consider self-dual Yang-Mills instantons in -dimensional Kählerspaces with one holomorphic isometry and show that they satisfy a gen-eralization of the Bogomol’nyi equation for magnetic monopoles on certain -dimensional metrics. We then search for solutions of this equation in -dimensional metrics foliated by -dimensional spheres, hyperboloids orplanes in the case in which the gauge group coincides with the isometrygroup of the metric (SO ( ) , SO (
1, 2 ) and ISO ( ) , respectively). Using ageneralized hedgehog ansatz the Bogomol’nyi equations reduce to a simpledifferential equation in the radial variable which admits a universal solu-tion and, in some cases, a particular one, from which one finally recoversinstanton solutions in the original Kähler space. We work out completely afew explicit examples for some Kähler spaces of interest. a E-mail:
Samuele.Chimento [at] csic.es b E-mail:
Tomas.Ortin [at] csic.es c E-mail: alejandro.ruiperez [at] uam.es a r X i v : . [ h e p - t h ] O c t Introduction
There is a well-known relation between Yang-Mills instantons in four flat Euclideandimensions and static magnetic monopoles of the Yang-Mills-Higgs theory in four flatLorentzian dimensions. A particular case that has focused most of the research is therelation between (anti-) selfdual instantons and monopoles satisfying the Bogomol’nyiequation [ ] in E . These are usually known as BPS magnetic monopoles because theSU ( ) ’t Hooft-Polyakov monopole [ , ] satisfies it in the Prasad-Sommerfield limit[ ]. Both the (anti-) selfduality condition and the Bogomol’nyi equation are first-orderequations that imply that the action is extremized locally and the second-order Euler-Lagrange equations are automatically satisfied. On the one hand, first-order equationsare easier to solve than second-order ones, and, for instance, this allowed Protogenovto construct all the spherically-symmetric SU ( ) BPS magnetic monopoles using the so-called hedgehog ansatz , that exploits the relation between the isometry group of the solu-tion and the gauge group [ ]. On the other, these, as many other interesting first-orderequations, naturally arise in the context of supersymmetric theories, when one searchesfor field configurations preserving some unbroken supersymmetries. Supersymmetricsolutions have many interesting properties, which makes them worth studying fortheir own sake. The possibility of constructing solutions, such as the one in Ref. [ ],describing an arbitrary number of magnetic monopoles in static equilibrium is one ofthe most remarkable ones, and has been exploited to construct dyonic, non-Abelianmulti-black-hole solutions in -dimensional Super-Einstein-Yang-Mills theories [ ].In Ref. [ ] Kronheimer showed that the above relation between (anti-) selfdual in-stantons in E and monopoles satisfying the Bogomol’nyi equation in E could beextended to a relation between (anti-) selfdual instantons in -dimensional hyperKäh-ler spaces admitting a triholomorphic isometry, usually known as Gibbons-Hawkingspaces [ , ], and, again, monopoles satisfying the Bogomol’nyi equation in E . Thismap between instantons and monopoles is surjective and from a given BPS monopolesolution one can construct an instanton solution in every Gibbons-Hawking space,which is characterized by an additional function H , harmonic in E , which is not partof the monopole fields. Thus, one can use all the spherically-symmetric SU ( ) BPSmagnetic monopoles found by Protogenov to construct instantons with SO ( ) symme-try in any Gibbons-Hawking space, for instance. This mechanism has been used in the context of the construction of timelike super-symmetric solutions of -dimensional Super-Einstein-Yang-Mills theories (non-Abelian The time coordinate is irrelevant in this problem because of the restriction to static monopoles. For a review on field configurations and solutions with global or local unbroken supersymmetriessee, for instance, Ref. [ ]. The relation does not preserve the regularity of the solutions in either sense. In particular, thewell-known BPST SU ( ) instanton [ ] on E , characterized by the choice H = r , gives rise to theProtogenov solution known as coloured monopole , which is singular [ ]. On the other hand, the rest ofthe spherically-symmetric BPS monopoles (including the globally regular ’t Hooft-Polyakov one) giverise to badly-behaved instantons solutions in the same GH space and one must consider other GH spaces[ ]. lack holes and rings, microstate geometries and global instantons [ , , , , ])because the spacetime metrics of these -dimensional supersymmetric solutions areconstructed using a -dimensional hyperKähler metric (often called “base space met-ric”) in terms of which a piece of the -form field strengths of the theory is forced bysupersymmetry to be selfdual [ ].In -dimensional supergravities with Abelian gaugings via Fayet-Iliopoulos termsthe base-space metric is forced to be Kähler in supersymmetric solutions [ ] and, ifthere are additional non-Abelian gaugings of the isometries of the real Special scalarmanifold ( i.e. we are dealing with a Super-Einstein-Yang-Mills theory with an ad-ditional Abelian gauging that introduces a non-trivial scalar potential, among otherthings), one faces the problem of finding selfdual instantons in Kähler spaces. Just as in the hyperKähler case, it is convenient to have a “parametrization” of theclass of metrics under consideration in order to find a set of differential equations forthe problem. The space of hyperKähler metrics is very large and finding a generic onefor a hyperKähler metric, in terms of a small number of functions satisfying some rela-tions is too complicated or impossible. The restriction to GH metrics, which depend onjust one independent function, transforms the selfduality condition into a set of differ-ential equations which, in the end, can be identified with the Bogomol’nyi equations.Alternatively, one can just view the requirement of the existence of a triholomorphicisometry as a condition necessary to dimensionally reduce the equations along theisometric direction preserving the hyperKähler structure.In the Kähler case it is natural to assume the existence of a holomorphic isometryalong which a dimensional reduction can be performed preserving the Kähler struc-ture (and supersymmetry as well, in the supersymmetric context). In Ref. [ ] andreferences therein, it was shown that these metrics can be written in terms of essen-tially two real functions related by a differential equation (see Eq. ( . )) and in thispaper we are going to make use of this result to transform the selfduality equations ofa Yang-Mills field on these metrics into a set of differential equations which ultimatelycan be seen as a generalization of the Bogomol’nyi equations in some -dimensionalspace (Section ).While the physical interpretation of these -dimensional equations is not as trans-parent as those obtained in the hyperKähler space (in particular, it is not clear that theycorrespond to BPS magnetic monopole solutions in general), they provide an excellentstarting point to construct instanton solutions. Thus, in Section we are going to usethe hedgehog ansatz and generalizations thereof adequate for other gauge groups inorder to simplify the equations and obtain explicit instanton solutions in some simpleKähler spaces of interest in gauged -dimensional supergravity, such as CP . Section contains our conclusions. This is not just a very twisted academic problem: this kind of theories arise naturally in Type IISuperstring compactifications to dimensions [ ]. Generalized Bogomol’nyi equations
Any -dimensional Kähler metric admitting a holomorphic isometry can be written as[ ] ds = H − ( dz + χ ) + H (cid:110) ( dx ) + W ( (cid:126) x )[( dx ) + ( dx ) ] (cid:111) , ( . )with the functions H and W , and the -form χ , independent of z and satisfying theconstraint ˘ (cid:63) d χ = dH + H ∂ log W dx , ( . )where ˘ (cid:63) is the Hodge dual in the -dimensional manifold d ˘ s = ( dx ) + W ( (cid:126) x )[( dx ) + ( dx ) ] . ( . )The integrability condition of this constraint is a W -dependent deformation of theLaplace equation for H on E ∂ ∂ H + ∂ ∂ ( W H ) + ∂ ∂ H = . )Thus, we can construct Kähler metrics with a holomorphic isometry by choosingsome function W , solving the above integrability condition for H and then solving theconstraint Eq. ( . ) for the -form χ . Observe that the choice W = , ].We are interested in Yang-Mills fields A I in the above space which are z -independent(at least in some gauge) and whose -form field strengths F I = dA I + g f JK I A J ∧ A K , ( . )are self-dual F I = + (cid:63) F I . ( . )Here (cid:63) is the Hodge operator in the full -dimensional metric Eq. ( . ), with theorientation (cid:101) z = + Underlined indices refer to the coordinate basis. We can also construct metrics of this kind starting with an arbitrary real function K ( x , x , x ) andcomputing directly H = ∂ K , W = − H − (cid:16) ∂ + ∂ (cid:17) K , χ = − ∂ ∂ K , χ = ∂ ∂ K , ( . )which solve all the above equations in coordinates in which χ = ollowing Kronheimer, who considered the hyper-Kähler case ( W =
1) in Ref. [ ],we decompose A I as A I = − H − Φ I ( dz + χ ) + ˘ A I , ( . )and substituting into the self-duality equation ( . ) one finds that it is equivalent to thefollowing generalization of the Bogomol’nyi equation [ ] ˘ (cid:63) ˘ F I − ˘ D Φ I = Φ I ∂ log W dx , ( . )where the ˘ sign in the field strength and the covariant derivative refers to the -dimensional Yang-Mills connection ˘ A I .In general, the right-hand side of this equation does not have a clear geometric orfield-theoretic meaning and, therefore, there is no obvious relation between the equa-tion and the Yang-Mills-Higgs action in some -dimensional spacetime: unlike whathappens with the usual Bogomol’nyi equation, it does not seem to be related to theextremization of an action of this kind and it does not guarantee that the correspond-ing second order Yang-Mills-Higgs equations of motion are satisfied. As explainedin the introduction, this does not make them completely useless or meaningless, be-cause they can arise in more complex theories such as - and -dimensional gaugedsupergravities.Many of the most interesting Kähler metrics in this class are characterized by afunction H that only depends on x , which we will denote by (cid:36) from now on. Equation( . ) implies that W is of the form [ ] W ( (cid:126) x ) = (cid:36) H Φ ( x , x ) + H Φ ( x , x ) . ( . )We will consider, for the sake of simplicity, the case in which either Φ = Φ = Φ , W can be written as W = Ψ ( (cid:36) ) Φ ( x , x ) , where Ψ ( (cid:36) ) ≡ (cid:36) (cid:101) H ( (cid:36) ) . ( . )Thus, the metrics in this class are completely determined by an arbitrary function of (cid:36) (either Ψ or H ) and an arbitrary function Φ of x , x .For these metrics, the Bogomol’nyi equation ( . ) takes the more geometric expres-sion Ψ ˘ (cid:63) ˇ F I − ˘ D ( ΨΦ I ) = . )which can be derived from the Yang-Mills-Higgs action with a Higgs field ˜ Φ I = ΨΦ I in a 1 + d ˜ s + = g tt dt − Ψ ( (cid:36) ) d ˘ s , ( . ) The standard Bogomol’nyi equation is defined in Euclidean -dimensional space. or any time-independent g tt by the usual squaring of the action arguments.We are not going to follow this line of reasoning any further here. Instead, wewill just try to find some explicit solutions to the above Bogomol’nyi equation and thecorresponding instantons for some interesting Kähler metrics. Generalized hedgehog ansatz and solutions
As a further simplification, we are going to restrict ourselves to the case in which the -dimensional metric Φ ( x , x )[( dx ) + ( dx ) ] is maximally symmetric. The threedistinct possibilities, namely the round sphere S , the hyperbolic plane H and theEuclidean plane E , are encompassed by the function Φ ( k ) ( x , x ) = { + k [( x ) + ( x ) ] } , ( . )for the values of the parameter k being +
1, 0 and −
1, respectively. For this particularkind of metrics, the -form χ that occurs in the metric Eq. ( . ) is given by χ = (cid:101)χ ( k ) , with χ ( k ) = ( x dx − x dx ) + k [( x ) + ( x ) ] . ( . )The generic coordinate change x = k − tan ( k θ /2 ) cos ϕ , x = (cid:36) , x = k − tan ( k θ /2 ) sin ϕ , ( . )brings the -dimensional metric ( . ) into the form d ˘ s = d (cid:36) + Ψ ( (cid:36) ) d Ω ( k ) , where d Ω ( k ) = d θ + k − sin ( k θ ) d ϕ , ( . )and χ ( k ) to the form χ ( k ) = k − [ cos ( k θ ) − ] d ϕ . ( . )It is natural to search for monopole solutions in gauge groups which coincide withthe isometry group of the maximally symmetric -dimensional spaces that foliate the -dimensional metrics that we are considering here: SO ( ) , ISO ( ) and SO (
1, 2 ) , re-spectively. Observe that, while the non-semisimple group ISO ( ) is just a mere curios-ity, the non-compact group SO (
1, 2 ) actually occurs in supergravity theories withoutany of the pathologies that arise in Yang-Mills(-Higgs) theories because these theo-ries have scalar-dependent kinetic matrices which make compatible SO (
1, 2 ) symmetrywith positive-defined kinetic energies. The k = k → or instance, in N = d = ∼ a I J ( φ ) F I ∧ (cid:63) F J . ( . )In theories with SO (
1, 2 ) symmetry the scalar fields and, hence, the kinetic matrix a I J ( φ ) transform under that group so that this kinetic term is positive definite andinvariant. For the kind of field configurations that we are considering (selfdual in-stantons living in some -dimensional Euclidean submanifold) the contribution to theaction of this term would be a positive-definite generalization of the usual instantonnumber ∼ a I J ( φ ) F I ∧ F J .In these theories, there is another term relevant for this discussion: the r.h.s. of theequations of motion of the vector fields contains a term of the form ∼ C I JK F I ∧ F J , ( . )where C I JK is a constant, symmetric tensor, invariant under the gauge group. This termwill contain the SO (
1, 2 ) Killing metric and will be proportional to the non-definite pos-itive “instanton number”, but the sign of these terms is irrelevant for the consistencyof the theory and the SO (
1, 2 ) selfdual instanton solutions are of potential interest inconsistent theories.For the gauge group SO ( ) , the so-called hedgehog ansatz leads to the constructionof all the magnetic monopole solutions with this geometry [ ]. Here we propose a gen-eralization of this ansatz that encompasses the three cases we are considering. In termsof the structure constants f I J K of these groups and the coordinates y I = y I ( (cid:36) , θ , ϕ ) y = (cid:36) k − sin ( k θ ) cos ϕ , y = (cid:36) cos ( k θ ) , y = (cid:36) k − sin ( k θ ) sin ϕ , ⇒ ( y ) + k [( y ) + ( y ) ] = (cid:36) , ( . )the generalized hedgehog ansatz for the Higgs and Yang-Mills fields can be written inthe form Φ I = F ( (cid:36) ) y I (cid:36) ,˘ A I = J ( (cid:36) ) f JK I y J (cid:36) d (cid:18) y K (cid:36) (cid:19) , ( . )where F ( (cid:36) ) and J ( (cid:36) ) are two functions to be determined.Plugging the ansatz into the Bogomol’nyi equation ( . ), we get the followingsystem of first-order differential equations involving F , J and the metric function Ψ : For k = ±
1, 0 the structure constants are given by f IJK = ε IJL η LK , where ( η IJ ) ≡ (cid:16) k
00 0 1 (cid:17) . Ψ F ) (cid:48) = k J ( + g J ) , J (cid:48) = F ( + g J ) . ( . )where primes stand for derivatives with respect to (cid:36) .In Section . we are going to search for explicit solutions of this system, but, before,we are going to show the form of the -dimensional instanton fields in terms of thefunctions that appear in these equations and in the Kähler metric. . Instanton fields
Given a solution of Eqs. ( . ), F ( (cid:36) ) , J ( (cid:36) ) for a Kähler metric characterized by thefunction Ψ ( (cid:36) ) and the parameters (cid:101) =
0, 1 and k = ±
1, 0 ds = Ψ (cid:36) (cid:101) ( dz + (cid:101) χ ( k ) ) + (cid:36) (cid:101) Ψ d (cid:36) + (cid:36) (cid:101) d Ω ( k ) , ( . )with d Ω ( k ) and χ ( k ) given by Eq. ( . ) and Eq. ( . ), respectively, the instanton field isgiven by A = − Ψ F (cid:36) (cid:101) k − sin ( k θ ) cos ϕ [ dz + (cid:101)χ ( k ) ]+ J (cid:104) sin ϕ d θ + k − sin ( k θ ) cos ( k θ ) cos ϕ d ϕ (cid:105) , A = − Ψ F (cid:36) (cid:101) cos ( k θ )[ dz + (cid:101)χ ( k ) ] − k J (cid:104) k − sin ( k θ ) (cid:105) d ϕ , A = − Ψ F (cid:36) (cid:101) k − sin ( k θ ) sin ϕ [ dz + (cid:101)χ ( k ) ] − J (cid:104) cos ϕ d θ − k − sin ( k θ ) cos ( k θ ) sin ϕ d ϕ (cid:105) . ( . )This is our main result, but we can elaborate it a bit more.For (cid:101) = k (cid:54) =
0, one can also write the instanton fields in a more compact formby using a generalization of the Maurer-Cartan forms v = − sin ϕ d θ + k sin ( k θ ) cos ϕ d ˜ z , v = d ϕ + k cos ( k θ ) d ˜ z , v = cos ϕ d θ + k sin ( k θ ) sin ϕ d ˜ z , ( . ) Here we are using a shifted coordinate ˜ z = z − k − ϕ . hich satisfy dv I = − f JK I v J ∧ v K and in terms of which the instanton field reads A I = − k Ψ F (cid:36) v I + (cid:18) J − k Ψ F (cid:36) (cid:19) u I , ( . )where the u I ’s are given by u = sin ϕ d θ + k − sin ( k θ ) cos ( k θ ) cos ϕ d ϕ , u = − k ( k − sin ( k θ )) d ϕ , u = − cos ϕ d θ + k − sin ( k θ ) cos ( k θ ) sin ϕ d ϕ . ( . )and satisfy du I = f JK I u J ∧ u K .Before we give an expression for the associated field strengths, we find convenientto introduce a basis of three self-dual -forms B i = e (cid:93) ∧ e i + ε jki e j ∧ e k , ( . )where e a is the Vierbein basis of the four-dimensional metric e (cid:93) = Ψ (cid:36) (cid:101) /2 [ dz + (cid:101)χ ( k ) ] , e = (cid:36) (cid:101) /2 d θ , e = (cid:36) (cid:101) /2 Ψ d (cid:36) , e = (cid:36) (cid:101) /2 k − sin ( k θ ) d ϕ . ( . )Then, in terms of these three self-dual three forms the field strengths are = Ψ J (cid:48) (cid:36) (cid:101) cos ( k θ ) cos ϕ B + (cid:18) Ψ F (cid:36) (cid:101) (cid:19) (cid:48) k − sin ( k θ ) cos ϕ B − Ψ J (cid:48) (cid:36) (cid:101) sin ϕ B , F = − k Ψ J (cid:48) (cid:36) (cid:101) k − sin ( k θ ) B + (cid:18) Ψ F (cid:36) (cid:101) (cid:19) (cid:48) cos ( k θ ) B , F = Ψ J (cid:48) (cid:36) (cid:101) cos ( k θ ) sin ϕ B + (cid:18) Ψ F (cid:36) (cid:101) (cid:19) (cid:48) k − sin ( k θ ) sin ϕ B + Ψ J (cid:48) (cid:36) (cid:101) cos ϕ B . ( . )For later use, it is interesting to have an explicit expression for Tr F ∧ F ∼ η I J F I ∧ F J ,even if in most cases there is no well-defined notion of instanton number (density).One has η I J F I ∧ F J = d x (cid:113) | g | (cid:34)(cid:18) Ψ F (cid:36) (cid:101) (cid:19) (cid:48) (cid:35) + k Ψ ( J (cid:48) ) (cid:36) (cid:101) = d x (cid:113) | g | g (cid:36) (cid:101) (cid:34)(cid:18) K (cid:48) − (cid:101) K (cid:36) (cid:19) + K Ψ ( K (cid:48) + k ) (cid:35) = d x (cid:113) | g | g (cid:36) (cid:101) (cid:34)(cid:18) G − − (cid:101) k K (cid:36) (cid:19) + k K G Ψ (cid:35) = g d (cid:20) k ( G − ) K (cid:36) (cid:101) − (cid:101) K (cid:36) (cid:101) + (cid:21) ∧ k − sin ( k θ ) dz ∧ d θ ∧ d ϕ . ( . )where we have defined the functions K ≡ g Ψ F , G ≡ ( + g J ) , ( . )for reasons that will become clear in the next section, and where, in the second line, ithas been assumed that k (cid:54) = or k = (cid:90) F I ∧ F I = π Tg (cid:20) ( G − ) K (cid:36) (cid:101) − (cid:101) K (cid:36) (cid:101) + (cid:21) (cid:36) F (cid:36) , ( . )where T is the period of z and (cid:36) , (cid:36) F the limits of integration of (cid:36) , which depend onthe chosen Kähler space. . Solutions
Let us now go back to the solutions of the system Eqs. ( . ). We consider the k = k (cid:54) = . . The k = case In this case, the first equation of ( . ) can be integrated directly for arbitrary Ψ ( (cid:36) ) ,giving F ( (cid:36) ) = K g Ψ ( (cid:36) ) , ( . )where K is an integration constant. Plugging this result into the second equation weget J ( (cid:36) ) = C e I ( (cid:36) ) − g , ( . )where C is another integration constant and I ( (cid:36) ) ≡ K (cid:90) (cid:36) du Ψ ( u ) . ( . )For instance, the metric of CP can be written in the k = Ψ = (cid:36) / (cid:96) and H = (cid:36) / Ψ , ( i.e. (cid:101) =
1) [ ] and, therefore, we have F ( (cid:36) ) = λ g (cid:36) , J = Ce − λ (cid:36) − g . ( . ) . . The k (cid:54) = case The system Eqs. ( . ) can be simplified with the change of variables ( . ), after whichit takes the form K (cid:48) = k ( G − ) , Ψ G (cid:48) = KG . ( . ) he first equation in ( . ) can be used to eliminate G in the second one, whichleads to a second order equation that only involves the variable K : Ψ K (cid:48)(cid:48) − KK (cid:48) − kK = . )Given the function Ψ ( (cid:36) ) corresponding to a Kähler metric in the class we are con-sidering, this equation determines K . Observe that we can turn around the problemand choose some arbitrary K ( (cid:36) ) and then find the Kähler manifold in which it definesa selfdual instanton by computing directly Ψ = K ( K (cid:48) + k ) K (cid:48)(cid:48) . ( . )There is a simple solution to Eq. ( . ) which is valid for any Ψ , with K (cid:48)(cid:48) = K = K − k (cid:36) , G =
0. The functions F and J that appear in the hedgehog ansatzEq. ( . ) are given by F ( (cid:36) ) = K − k (cid:36) g Ψ ( (cid:36) ) , J = − g . ( . )This solution corresponds to a fixed point of the system Eqs. ( . ).In order to find more solutions we need to know Ψ ( (cid:36) ) . In many interesting cases Ψ ( (cid:36) ) is a polynomial of order N , and, in particular, with N =
3. If Ψ ( (cid:36) ) is a polynomialof order N we can assume that K is also a polynomial whose order must be N − . ) to have solutions, in general. The differential equation becomes a setof algebraic equations relating the coefficients of the polynomial K to those of thepolynomial Ψ . For N =
3, if Ψ ( (cid:36) ) = Ψ + Ψ (cid:36) + Ψ (cid:36) + Ψ (cid:36) , K ( (cid:36) ) = K + K (cid:36) + K (cid:36) , ( . )one readily finds the following relations ( Ψ (cid:54) =
0, by assumption) K = Ψ K = Ψ − k K = Ψ Ψ − ( Ψ + k )( Ψ − k ) Ψ , ( . )and one constraint for the coefficients of Ψ ( (cid:36) ) Ψ Ψ ( Ψ + k ) − ( Ψ + k ) ( Ψ − k ) = Ψ Ψ , ( . ) This solution is also available in the k = C = In this case k = k and, in the form in which we are giving this general solution, it is automaticallyvalid for the k = hich has to be understood just as the condition that Ψ has to satisfy in order forEq. ( . ) to admit a solution in which K is a second order polynomial.Since J is a real function, the second of Eqs. ( . ) G must be a positive definitefunction. The first of Eqs. ( . ) and Eqs. ( . ) tell us that it is given by G = + kK (cid:48) = + kK + kK (cid:36) = + k Ψ + k Ψ (cid:36) , ( . )so that (cid:36) is restricted to the interval (cid:36) > − k ( + k Ψ ) Ψ for k Ψ > (cid:36) < − k ( + k Ψ ) Ψ for k Ψ < . )The metric of CP can be written in the k = ± Ψ which is the cubicpolynomial Ψ = (cid:36) ( k + (cid:36) / (cid:96) ) ( . )and, since Ψ must also be positive in the metric, the variable (cid:36) is restricted to (cid:36) > − k (cid:96) /4 , ( . )which is compatible with the restriction found above only for k = Conclusions
We have completed our program of finding simple equations that selfdual instantonsolutions in Kähler spaces with one holomorphic isometry have to satisfy, general-izing Kronheimer’s work in the hyperKähler case. We have also constructed someexplicit solutions in some Kähler spaces of particular interest from the point of viewof -dimensional Abelian-gauged supergravity. In passing, we have generalized thehedgehog ansatz to some non-spherical symmetries and gauge groups different fromSU ( ) . There is little work in the literature on non-compact gaugings and we thinkthese results will allow us to find interesting solutions in those cases.We have not analyzed the regularity of the solutions we have obtained because, ul-timately, they are going to be part of a complicated -dimensional gauge field definedin a -dimensional spacetime whose regularity does not depend on the regularity ofeach of its building blocks. There are perfectly regular -dimensional solutions (mi-crostate geometries) built over singular base spaces like ambipolar GH spaces [ , ].And most regular, charged, extremal black holes use Coulomb-like -form fields which re singular at a point which, in the end, turns out to be not a point but a regular hori-zon. The same mechanism saves the regularity of the -dimensional non-Abelian blackholes which bear BPS magnetic monopole fields different from the ’t Hooft-Polyakovone. Thus, one should not be worried about the possible singularities of the instantonsuntil the full -dimensional supergravity solution is constructed. For the same reason(and also because the Kähler spaces that we are considering are not compact) we havenot computed the instanton number.In a forthcoming publication [ ] we will put to use the results obtained in thiswork, which we hope will also be useful in other contexts. Acknowledgments
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