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MONOPOLES AND THE GIBBONS-MANTON METRIC ROGER BIELAWSKI Abstract. Weshowthat,intheregionwheremonopolesarewellseparated, 8 the L2-metric on the moduli space of n-monopoles is exponentially close to 9 the Tn-invariant hyperka¨hler metricproposed by Gibbons and Manton. The 9 proof is based on a description of the Gibbons-Manton metric as a metric 1 on a certain moduli space of solutions to Nahm’s equations, and on twistor n methods. In particular, we show how the twistor description of monopole a metricsdeterminestheasymptoticmetric. J 4 1 The modulispaceM of(framed)staticSU(2)-monopolesofchargen,i.e. solu- n tions to Bogomolny equations dAΦ= F, carries a natural hyperkähler metric [1]. 1 ∗ Thegeodesicmotioninthismetric isagoodapproximationtothe dynamicsoflow v 1 energy monopoles [26, 33]. For the charge n = 2 the metric has been determined 9 explicitly by Atiyah and Hitchin [1], and it follows from their explicit formula that 0 when the two monopoles are well separated, the metric becomes (exponentially 1 fast)the EuclideanTaub-NUT metric with a negativemass parameter. It was also 0 shownbyN.Manton[27]thatthisasymptoticmetriccanbedeterminedbytreating 8 well-separatedmonopoles as dyons. The equations of motion for a pair of dyons in 9 / R3 are found to be equivalent to the equations for geodesic motion on Taub-NUT h space. t - Foranarbitrarychargen,itwasshownin[3]that,whentheindividualmonopoles p are well-separated, the L2-metric is close (as inverse of the separation distance) to e h the flat Euclidean metric. Gibbons and Manton [14] have then calculated the La- : grangian for the motion of n dyons in R3 and shown that it is equivalent to the v i Lagrangianfor geodesic motionin a hyperkähler metric on a torus bundle overthe X configurationspaceC˜ (R3). ThismetricisTn-invariantandhasasimplealgebraic n r form. Gibbons andMantonhaveconjectured,by analogywiththe n=2case,that a the exact n-monopole metric differs from their metric by an exponentially small amount as the separation gets large. We shall prove this conjecture here. Ourstrategyisasfollows. WeconstructcertainmodulispaceM˜ ofsolutionsto n Nahm’s equations which carries a Tn-invariant hyperkähler metric. Using twistor methods we identify this metric as the Gibbons-Manton metric. Finally, we show that the metrics onM˜ and M are exponentially close. This proofadapts equally n n well to the asymptotic behaviour of SU(N)-monopole metrics with maximal symmetry breaking, as will be shown elsewhere. The asymptoticpicture canbe explainedinthe twistorsetting. We recallthata monopoleisdetermined(uptoframing)byacurveS -thespectralcurve-inTCP1, which satisfies certain conditions [16]. One of these is triviality of the line bundle L−2 over S, and a nonzero section of this bundle is the other ingredient needed to determine the metric [19, 1]. Asymptotically we have now the following situation. When the individual monopoles become well separated the spectral curve of the 1991 Mathematics Subject Classification. 53C25,81T13. 1 2 ROGER BIELAWSKI n-monopoledegenerates(exponentially fast)into the unionof spectralcurves S of i individualmonopoles, while the sectionof L−2 becomes (alsoexponentially fast) n meromorphic sections of L−2 over the individual S . The zeros and poles of these i sections occur only at the intersection points of the curves S . This information i (andthetopologyoftheasymptoticregionofM )is,asweshowinthelastsection, n sufficient to conclude that the asymptotic metric is the Gibbons-Manton metric. Thearticleisorganizedasfollows. Insections1and2werecallthedefinitionsof the Gibbons-Manton and monopole metrics. In section 3 we introduce the moduli space M˜ of solutions to Nahm’s equations and give heuristic arguments why the n metric on M˜ should be exponentially close to the monopole metric. In section n 4, as a preliminary step to study M˜ we introduce yet another moduli space of n solutionstoNahm’sequations,somewhatsimplerthanM˜ . Inthatsectionwealso n discuss the relation with Kronheimer’s metrics on GC/TC, where G is a compact semisimple Lie group and T G is a maximal torus. In section 5 we identify M˜ as a differential, complex,≤and finally complex-symplectic manifold. In section n 6 we calculate the twistor space of M˜ and identify its hyperkähler metric as the n Gibbons-Mantonmetric. Insection7wefinallyshowthatthemonopolemetricand the metric on M˜ are exponentially close. Short section 8 shows how one can read n off the Gibbons-Manton metric, as the asymptotic form of the monopole metric, from the twistor description of the latter. 1. The Gibbons-Manton metric The Gibbons-Manton metric [14] is an example of 4n-dimensional (pseudo)- hyperkähler metric admiting a tri-Hamiltonian (hence isometric) action of the n- dimensionaltorusTn. Suchmetricshaveparticularlynicepropertiesandwerestud- iedby severalauthors[25, 18,32]. The Gibbons-Mantonmetric wasdescribedas a hyperkählerquotientofaflatquaternionicvectorspacebyGibbonsandRychenkova in [15]. We recall here this description, which we slightly modify to better suit our purposes. We start with flat hyperkähler metrics g and g on M = S1 R3 n 1 2 1 × and M = Hn(n−1)/2. We consider a pseudo-hyperkähler metric on the product 2 (cid:0) (cid:1) manifold M = M M given by g = g g . The complex structures on H are 1 2 1 2 × − given by the right multiplication by quaternions i,j,k. The metric g is invariant 1 under the obvious action (by translations) of Tn = (S1)n and the metric g is in- 2 variant under the left diagonal action of Tn(n−1)/2. We consider a homomorphism φ:Tn(n−1)/2 Tn given by → n i−1 (t ) t t−1 . ij i<j 7→ ij ji  j=i+1 j=1 Y Y i=1,...,n   ThisdefinesanactionofTn(n−1)/2 onM =M M byt (m ,m )=(φ(t) m ,t 1 2 1 2 1 × · · · m ). GibbonsandRychenkovahaveshownthatthehyperkählerquotientof(M,g) 2 by this action of Tn(n−1)/2 is the Gibbons-Manton metric. Weremarkthat,ifwechoosecoordinates(t ,x )onM ,t S1andx R3,and i i 1 i i quaternionic coordinates q , i < j, on Hn(n−1)/2, then the m∈oment map∈equation ij are: 1 q iq¯ =x x . (1.1) ij ij i j 2 − MONOPOLES AND THE GIBBONS-MANTON METRIC 3 As long as x = x for i = j, the torus Tn(n−1)/2 acts freely on the zero-set of i j 6 6 the moment map. The quotient of this set by Tn(n−1)/2 is a smooth hyperkähler manifold which we denote by M . The action of Tn on M induces a free tri- GM 1 Hamiltonian action on M for which the moment map is just (x ,... ,x ). This GM 1 n makes M into a Tn-bundle over the configuration space C˜ (R3) of n distinct GM n points in R3. We shall now determine this bundle. We recall that a basis of H C˜ (R3),Z is given by the n(n 1)/2 2-spheres 2 n − (cid:0)Si2j ={(x(cid:1)1,...,xn)∈R3⊗Rn;|xi−xj|=const, xk =const if k 6=i,j} (1.2) where i<j. We have Proposition 1.1. The hyperka¨hler moment map for the action of Tn makes M GM into a Tn-bundle over C˜ (R3) determined by the element (s ,... ,s ) of n 1 n H2 C˜ (R3),Zn given by n (cid:0) (cid:1) 1 if k =i − s (S2)= 1 if k =j k ij  0 otherwise.  Proof. From the formula (1.1) it follows that restricting the bundle to a fixed S2 ij is equivalent to considering the case n = 2. In other words s (S2) = 0 if k = i,j k ij 6 and we have to consider only one quaternionic coordinate q . The zero-set of the ij moment map is 1q iq¯ = x x and the circle S1 by which we quotient acts by 2 ij ij i− j t q ,(t ,x ),(t ,x ) = tq ,(tt ,x ),(t−1t ,x ) . The quotient can be obtained ij i i j j ij i i j j · bysettingt =1andthe inducedactionofthei-thgenerators ofTn isthengiven i i by(cid:0)left multiplication(cid:1)by (cid:0)s−1 on q . Since the m(cid:1)ap q 1q iq¯ with the left i ij ij → 2 ij ij actionofS1 on q H; q =1 isthe Hopfbundle, itfollowsthats (S2)= 1. { ij ∈ | ij| } i ij − A similar argument shows that s (S2)=1. j ij Inparticular,(t,x)=(t ,x )formlocalcoordinatesonM . The metrictensor i i GM can be then written in the form [32]: g =Φdx dx+Φ−1(dt+A)2, · where the matrix Φ and the 1-form A depend only on the x and satisfy certain i linear PDE’s. In particular, Φ determines the metric. For the Gibbons-Manton metric 1 1 if i=j 4Φ = − k6=i kxi−xkk ij (kxi−1Pxjk if i6=j. 2. Nahm’s equations and monopole metrics We shall recall in this section the description of the L2-metric on the moduli space of charge n SU(2)-monopoles in terms of Nahm’s equations. A proof that the Nahm transform [30, 16] between the two moduli spaces is an isometry was given by Nakajima in [31]. Onestartswiththe space ofquadruples(T ,T ,T ,T )ofsmoothu(n)-valued 0 1 2 3 A functions on ( 1,1) such that T ,T ,T have simple poles at 1 with residues 1 2 3 − ± 1ρ(σ ),i=1,2,3,whereρ:su(2) u(n)isthe standardirreduciblen-dimensional 2 i → representationof su(2) and σ are the Pauli matrices. Equipped with the L2-norm i 4 ROGER BIELAWSKI (givenbyabiinvariantinnerproductonu(n)), becomesaflatquaternionicaffine A space. There is an isometric and triholomorphic action of the gauge group of G U(n)-valued functions g :[ 1,1] U(n) which are 1 at 1: − → ± T Ad(g)T g˙g−1 0 0 7→ − T Ad(g)T , i=1,2,3. (2.1) i i 7→ The zero-set of the hyperkähler moment map for this action is then described by Nahm’s equations [30]: 1 T˙ +[T ,T ]+ ǫ [T ,T ]=0, i=1,2,3. (2.2) i 0 i ijk j k 2 j,k=1,2,3 X The quotient of the space of solutions by is the a smooth hyperkähler manifold G M of dimension 4n. By the above mentioned result of Nakajima, M is the n n modulispaceof(framed)chargenSU(2)-monopoles. Withrespecttoanycomplex structure M is biholomorphic to the space of based rational maps of degree n on n CP1 [13]. If we replace U(n) by = SU(n) (resp. by PSU(n)) in the above description, we obtain the moduli space of strongly centered (resp. centered) SU(2)-monopoles of charge n. Remark 2.1. Asimilarconstructioncanbe doneforanycompactLiegroupG. We require ρ : su(2) g to be a Lie algebra homomorphism whose image lies in the → regularpartof g. We obtain a smooth hyperkählermanifold of dimension 4rankG whichcanbe identifiedwithatotallygeodesicsubmanifoldofcertainmodulispace of SU(N)-monopoles (with a minimal symmetry breaking). Alternatively, as a complexmanifold,itisadesingularizationof hC TC /W whereTC isamaximal torus in GC, hC its Lie algebra, and W the corres×ponding Weyl group [6]. (cid:0) (cid:1) The tangent space to M can be described as the space of solutions to the n linearized Nahm’s equations and satisfying the condition of being orthogonal (in theL2-metric)tovectorsarisingfrominfinitesimalgaugetransformations. Inother words the tangent space to M at a solution (T ,T ,T ,T ) can be identified with n 0 1 2 3 the set of solutions (t ,t ,t ,t ) to the following system of linear equations: 0 1 2 3 t˙ +[T ,t ]+[T ,t ]+[T ,t ]+[T ,t ]=0, 0 0 0 1 1 2 2 3 3 t˙ +[T ,t ] [T ,t ]+[T ,t ] [T ,t ]=0, 1 0 1 − 1 0 2 3 − 3 2 (2.3) t˙ +[T ,t ] [T ,t ] [T ,t ]+[T ,t ]=0, 2 0 2 1 3 2 0 3 1 − − t˙ +[T ,t ]+[T ,t ] [T ,t ] [T ,t ]=0. 3 0 3 1 2 2 1 3 0 − − The metric is defined by 1 1 3 (t ,t ,t ,t ) 2 = t 2. (2.4) 0 1 2 3 i k k 2 k k Z−1 0 X The three anti-commuting complex structures can be seen by writing a tangent vector as t +it +jt +kt . 0 1 2 3 MONOPOLES AND THE GIBBONS-MANTON METRIC 5 3. The asymptotic moduli space We shall now construct a one-parameter family of moduli spaces M˜ (c), c R, n ∈ of solutions to Nahm’s equations carrying (pseudo-)hyperkähler metrics. We shall see later on that these metrics are the Gibbons-Manton metric with different mass parameters. WeconsiderthesubspaceΩ ofexponentiallyfastdecayingfunctionsinC1[0, ], 1 ∞ i.e.: Ω = f :[0, ] u(n); sup eηt f(t) +eηt df/dt <+ . 1 η>0 ∞ → ∃ k k k k ∞ (cid:26) t (cid:27) (3.1) (cid:0) (cid:1) Asintheprevioussection,ρ:su(2) u(n)isthestandardirreduciblen-dimensional → representationof su(2) (in particular, ρ(σ ) is a diagonalmatrix). We denote by h 1 the (Cartan) subalgebra of u(n) consisting of diagonal matrices. Let ˜ be the space of C1-functions (T ,T ,T ,T ) defined on (0,+ ] and satis- n 0 1 2 3 A ∞ fying (cf. [23]): (i) T ,T ,T have simple poles at 0 with resT = 1ρ(σ ); 1 2 3 i 2 i (ii) T (+ ) h for i=0,... ,3; i ∞ ∈ (iii) (T (+ ),T (+ ),T (+ )) is a regular triple, i.e. its centralizer is h; 1 2 3 ∞ ∞ ∞ (iv) (T (t) T (+ )) Ω for i=0,1,2,3. i i 1 − ∞ ∈ Next we shall define the relevant gauge group. The Lie algebra of our gauge group (c) is the space of C2-paths ρ:[0,+ ) u(n) such that G ∞ → (i) ρ(0)=0 and ρ˙ has a limit in h at + ; ∞ (ii) (ρ˙ ρ˙(+ )) Ω , and [τ,ρ] Ω for any regular element τ h; 1 1 − ∞ ∈ ∈ ∈ (iii) cρ˙(+ )+lim (ρ(t) tρ˙(+ ))=0. t→+∞ ∞ − ∞ It is the Lie algebra of the Lie group (c)= g :[0,+ ) U(n); g(0)=1, s(g):=limg˙g−1 h, (τ Ad(g)τ) Ω , 1 G { ∞ → ∈ − ∈ (g˙g−1 s(g)) Ω , exp(cs(g))lim(g(t)exp( ts(g)))=1 . 1 − ∈ − } Remark . Thelastconditioninthedefinitionof (c)meansthatg(t)isasymptotic G to exp(ht ch) for some diagonal h. − We introduce a family of metrics on ˜ . Let (t ,t ,t ,t ) be a tangent vector n 0 1 2 3 A to the space ˜ at a point (T ,T ,T ,T ). The functions t are now regular at 0, n 0 1 2 3 i A i=0,... ,3. We put 3 +∞ 3 (t ,t ,t ,t ) 2 =c t (+ ) 2+ t (s) 2 t (+ ) 2 ds. k 0 1 2 3 kc k i ∞ k k i k −k i ∞ k 0 Z0 0 (3.2) X X(cid:0) (cid:1) Weobservethatthegroup (c)actingby(2.1)preservesthemetric andthe c G k·k three complex structure of the flat hyperkähler manifold ˜ . We define M˜ (c) as n n A the (formal)hyperkählerquotientof ˜ by (c) (with respectto the metric ). n c A G k·k Thezerosetofthemomentmapis givenbythe equations(2.2)(herethe condition (iii)inthedefinitionofLie( (c))isessential)andsoM˜ (c)isdefinedasthemoduli n G space of solutions to Nahm’s equations: M˜ (c)= solutions to (2.2) in ˜ / (c). n n A G n o 6 ROGER BIELAWSKI Remark . If c > 0, then the metric (3.2) on M˜ (c) will be seen to be positive n definite if (T (+ ),T (+ ), T (+ )) is sufficiently far from the walls of Weyl 1 2 3 ∞ ∞ ∞ chambers. On the other hand, if c < 0, then the metric will be shown to be everywhere negative definite. Therefore, for c < 0 we should really replace c k·k with its negative; it is, however more convenient to consider the metrics . c k·k We observe that sending a solution T to the solution rT (rt) for any r > 0 in- i i duces a homothety of factor r between M˜ (c) and M˜ (rc). n n Before we begin the detailed study of M˜ (c), let us explain why we expect this n metric to be exponentially close to the monopole metric. It is known [4] that the solutionstoNahm’sequationson(0,2)correspondingtoawell-separatedmonopole are exponentially close to being constant away from the boundary points (i.e. on any[ǫ,2 ǫ]). Thesameistrueforsolutionsonthehalfline(0,+ ): aslongasthe − ∞ triple (T (+ ),T (+ ),T (+ )) is regular, the solutions are exponentially close 1 2 3 ∞ ∞ ∞ to being constant away from 0 [23] (it is helpful to notice that the space of regular triplesisthesameasthe spaceC˜ R3 ofdistinctpointsinR3). Ourstrategyisto n taketwosolutions,onhalf-lines(0, )and( ,2)withthesamevaluesat ,cut (cid:0)∞ (cid:1) −∞ ±∞ themoffatt=1andusethisnon-smoothsolutionon(0,2)(withcorrectboundary behaviour) to obtain an exact solution to the monopole Nahm data. The exact solution will differ from the approximate one by an exponentially small amount. Furthermorethepartofthehalf-linesolutionswhichwehavecutoffisexponentially close to being constant and, for c = 1, contributes an exponentially small amount to the metric (all estimates are uniform and can be differentiated). This can c k · k be seen from the fact that we can rewrite (3.2) as c 3 +∞ 3 (t ,t ,t ,t ) 2 = t (s) 2+ t (s) 2 t (+ ) 2 ds. k 0 1 2 3 kc k i k k i k −k i ∞ k Z0 0 Zc 0 (3.3) X X(cid:0) (cid:1) The first term, together with the corresponding term for the solution on ( ,2), −∞ is exponentially close to the monopole metric (for c=1). 4. Moduli space of regular semisimple adjoint orbits In order to obtain information about M˜ (c) we need to consider first another n modulispaceofsolutionstoNahm’sequations,definedanalogously,exceptthatwe requirethesolutionstobesmoothatt=0. Thisspace,whichcanbedefinedforan arbitrarycompact Lie groupG, is of some interest as allhyperkähler structures on GC/TC (here TC is a maximaltorus)due to Kronheimer[23]can be obtainedfrom it as hyperkähler quotients (see Theorem 4.3 below). A reader who is primarily interested in monopoles should think of G as U(n). Let us first recall how Kronheimer constructs hyperkähler metrics on GC/TC. Let h be the Lie algebra of TC and let (τ ,τ ,τ ) h3 be a regular triple, i.e. one 1 2 3 ∈ whose centralizer is h. For a fixed η >0, consider the Banach space Ωη = f :[0, ] g;sup eηt f(t) +eηt df/dt <+ 1 ∞ → k k k k ∞ (cid:26) t (cid:27) (cid:0) (cid:1) with the norm f = sup (eηt f(t) +eηt df/dt ). Define η(τ ,τ ,τ ) as the k k t k k k k A 1 2 3 space of C1-functions (T ,T ,T ,T ):(0,+ ] g which satisfy: 0 1 2 3 ∞ → T (t),(T (t) τ );i=1,2,3 Ωη. { 0 i − i }⊂ 1 MONOPOLES AND THE GIBBONS-MANTON METRIC 7 Define also η by replacing Ω with Ωη in the definition of given in the previous G 1 1 G section. Kronheimer shows then that for small enough η M(τ ,τ ,τ )= solutions to (2.2) in η(τ ,τ ,τ ) / η 1 2 3 1 2 3 { A } G equipped with the L2 metric is a smooth hyperkähler manifold, diffeomorphic to GC/TC. Futhermore,if (τ ,τ ) is regular,thenM(τ ,τ ,τ )is biholomorphic,with 2 3 1 2 3 respect to the complex structure I, to the complex adjoint orbit of τ +iτ . 2 3 WeobservethattheunionofallM(τ ,τ ,τ )hasanaturaltopologyanditis,in 1 2 3 fact,asmoothmanifold. WeshallshownowthatthereisaT-bundleoverthisunion which carries a (pseudo)-hyperkähler metric. We define the space by omitting G A the condition(i) in the definition of ˜ in the previous section. Insteadwe require n that the T are smooth at t = 0 forAi = 0,1,2,3. We define M (c), c R, as the i G ∈ (formal) hyperkähler quotient of by (c) with respect to the metric (3.2). We G A G have: Proposition 4.1. M (c) equipped with the metric (3.2) is a smooth hyperka¨hler G manifold. The tangent space at a solution (T ,T ,T ,T ) is described by the equa- 0 1 2 3 tions (2.3). We remark that the metric 3.2 may be degenerate at some points. However the hypercomplex structure is defined everywhere. Proof. Define Mη(c) by replacing Ω with Ωη in the definition of M (c). By the G G exponential decay property of solutions to Nahm’s equations ([23], Lemma 3.4), a neighbourhood of a particular element in M (c) is canonically identified with its G neighbourhoodinMη(c)forsmallenoughη. Thereforewecanusethetransversality G arguments of [23], Lemma 3.8 and Proposition 3.9 (with a slight modification due tocondition(iii)inthe definition ofLie( (c))) todeduce the smoothness. Thefact G that the metric is hyperkähler is, formally, the consequence of the fact that M (c) G is a hyperkähler quotient. One can, in fact, check directly that the three Kähler forms are closed. We shall also, later on, identify the complex structures and the complex symplectic forms proving their closedness. We observe now that the action on of gauge transformations which are G asymptotic to exp( th + λh), h h, λA R, induce a free isometric action of − ∈ ∈ T =exp(h)onM (c). Infactthisactionistri-Hamiltonianandasimplecalculation G shows Proposition 4.2. The hyperka¨hler moment map µ=(µ ,µ ,µ ) for the action of 1 2 3 T on M (c) is given by µ (T ,T ,T ,T )=T (+ ) for i=1,2,3. G i 0 1 2 3 i ∞ As an immediate corollary we have: Theorem 4.3. Let (τ ,τ ,τ ) be a regular triple in h3. The hyperka¨hler quotient 1 2 3 µ−1(τ ,τ ,τ )/T ofM (c)bythetorusT isisometrictoKronheimer’sM(τ ,τ ,τ ). 1 2 3 G 1 2 3 We have also a tri-Hamiltonian action of G on M (c) given by the gauge trans- G formations with arbitrary values at t = 0. The hyperkähler moment map for this action is (T (0),T (0),T (0)). 1 2 3 WehavetwoothergroupactionsonM (c). Thereisafreeisometricandtriholo- G morphicactionoftheWeylgroupW =N(T)/T givenbythegaugetransformations which become constant (and in W) exponentially fast. 8 ROGER BIELAWSKI FinallythereisafreeisometricSU(2)-actionwhichrotatesthecomplexstructures. As a consequence it has a globally defined Kähler potential for each Kähler form (cf. [18]). The potential for ω (or ω ) is given by the moment map for the action 2 3 of a circle in SU(2) which preserves I. This is easily seen to be 3 +∞ 3 K =c T (+ ) 2+ T (s) 2 T (+ ) 2 ds. J i i i k ∞ k k k −k ∞ k Xi=2 Z0 Xi=2(cid:16) (cid:17) Remark 4.4. There is a similar (pseudo)-hyperkähler manifold with a torus action such that the hyperkähler quotients by this torus are isometric to Kronheimer’s ALE-metrics on the minimal resolution of a given Kleinian singularity C2/Γ [24]. This manifold is defined as M except that the T have poles at t = 0 with the G i residues defined by a subregular homomorphism su(2) g (cf. [6, 5]). → Remark 4.5. One can observe that M (0) is a cone metric (with the R -action G >0 givenby T (t) rT (rt)) and in fact, it is anH∗-bundle overa pseudo-quaternion- i i 7→ Kähler manifold (cf. [34]). 5. M˜ (c) as a manifold n We now return to the space M˜ (c) defined in section 3. Our first task is to n showthatthis spaceis smooth. We shallshowthatM˜ (c)is asmoothhyperkähler n quotientoftheproductofthespaceM (c 1)consideredintheprevioussection U(n) − and of another moduli space of solutions to Nahm’s equations. This latter space, denoted by N , is given by u(n)-valued solutions to Nahm’s equations defined on n (0,1] smooth at t = 1 and with the same poles as M˜ (c) at t = 0. The gauge n group consists of gauge transformations which are identity at t = 0,1. Equipped withthe metric (2.4)this is a smoothhyperkählermanifold[6,11]. Itadmits a tri- Hamiltonian action of U(n) given by gauge transformations with arbitrary values at t = 1. In addition, we consider the space M (c 1) defined in the previous U(n) − section. We identify itthis time withthe spaceofsolutionson[1,+ ]viathe map ∞ T (t) T (t +1) (so that the gauge transformations behave now, near + , as i i 7→ ∞ elements of (c)). G It is easy to observe that the space M˜ (c) is the hyperkähler quotient of N n n × M (c 1)bythediagonalactionofU(n)(cf. [6];themomentmapequationssim- U(n) − plymatchthefunctionsT ,T ,T att=1;afterthat,quotientingbyGmeansthat 1 2 3 the remaining gauge transformations are smooth at t = 1). Using this description of M˜ (c) we can finally show n Proposition 5.1. M˜ (c) equipped with the metric (3.2) is a smooth hyperka¨hler n manifold. The tangent space at a solution (T ,T ,T ,T ) is described by the equa- 0 1 2 3 tions (2.3). Proof. Since the metric (3.2) may be degenerate, we still have to show that the momentmapequationsonN M (c 1)areeverywheretransversal. Considera n U(n) × − particularpointinM (c 1)whichwerepresentbyasolutionm=(T ,T ,T ,T ) U(n) 0 1 2 3 − with T (+ )=0 and T (+ )=τ , i=1,2,3. Let µ be the hyperkähler moment 0 i i ∞ ∞ map for the action of G on N M . We observe that the image of dµ n × U(n) |m containsthe imageofdµ′ ,µ′ beingthehyperkählermomentmapfortheactionof |m GonN M(τ ,τ ,τ )(Kronheimer’sdefinitionofM(τ ,τ ,τ )wasrecalledinthe n 1 2 3 1 2 3 × MONOPOLES AND THE GIBBONS-MANTON METRIC 9 previoussection). ThemetriconN M(τ ,τ ,τ )isnon-degenerateand,asGacts n 1 2 3 × freely, dµ′ is surjective. Thus dµ is surjective at each point in N M (c 1) |m n× U(n) − and M˜ (c) is smooth. n We observe that, as in the case of M (c), M˜ (c) has isometric actions of the U(n) n torus Tn (defined as the diagonal subgroup of U(n)), of the symmetric group S , n and of SU(2). In particular, the hyperkähler moment map for the action of Tn is still given by the values of T ,T ,T at infinity (cf. Proposition 4.2). 1 2 3 We can describe the topology of M˜ (c): n Proposition 5.2. M˜ (c) is a principal Tn-bundle over the configuration space n C˜ (R3) of n distinct points in R3. n We postpone identifying this bundle until the next section (Proposition 6.3). Proof. ThespaceC˜ (R3)isthespaceofregulartriplesinthesubalgebraofdiagonal n matricesandthemomentmapµfortheactionofTn givesusaprojectionM˜ (c) n → C˜ (R3). Let us consider a fixed regular triple (τ ,τ ,τ ) and all elements of M˜ (c) n 1 2 3 n with T (+ ) = τ , i = 1,2,3, i.e. µ−1(τ ,τ ,τ ). For each such solution we can i i 1 2 3 ∞ makeT identically0bysomegaugetransformationgwithg(0)=1. Thisisdefined 0 uniquely up to the action of Tn and so the set of Tn-orbits projecting via µ G × to (τ ,τ ,τ ) can be identified with the set of solutions to Nahm’s equations with 1 2 3 T 0, T ,T ,T having the appropriate residues at t = 0 and being conjugate 0 1 2 3 ≡ to τ ,τ ,τ at infinity. By the considerations at the beginning of this section this 1 2 3 space is the hyperkählerquotient ofN M(τ ,τ ,τ ) by U(n). The arguments of n 1 2 3 × [6]showthatthecorrespondingcomplex-symplecticquotientcanbeidentifiedwith the intersection of a regular semisimple adjoint orbit of GL(n,C) with the slice to the regular nilpotent orbit. This intersection is a single point. Finally, in order to identify in this case the hyperkähler quotient with the complex-symplectic one we can adapt the argument in the proof of Proposition 2.20 in [20]. Our next task is to describe the complex structure of M˜ (c) (because of the action n of SU(2) all complex structures are equivalent). As usual (cf. [13]), if we choose a complex structure, say I, we can introduce complex coordinates on the moduli space of solutions to Nahm’s equations by writing α=T +iT and β =T +iT . 0 1 2 3 The Nahm equations can be then written as one complex and one real equation: dβ =[β,α] (5.1) dt d (α+α∗)=[α∗,α]+[β∗,β]. (5.2) dt By the remark made at the beginning of this section, M˜ (c) is the hyperkähler n quotientoftheproductmanifoldN M (c 1). Weshallshowthatasacomplex n U(n) × − symplecticmanifoldM˜ (c)isthecomplex-symplecticquotientofN M (c 1). n n U(n) × − Let us recall the complex structure of N [13, 19, 6, 12]. Let e ,... ,e denote the n 1 n standard basis of Cn. There is a unique solution w of the equation 1 dw = αw (5.3) dt − with lim t−(n−1)/2w (t) e =0. (5.4) 1 1 t→0 − (cid:16) (cid:17) 10 ROGER BIELAWSKI Setting w (t)=βi−1(t)w (t), we obtain a solution to (5.3) with i 1 lim ti−(n+1)/2w (t) e =0. i i t→0 − (cid:16) (cid:17) The complex gauge transformation g(t) with g−1 = (w ,... ,w ) makes α identi- 1 n cally zero and sends β(t) to the constant matrix 0 ... 0 ( 1)n+1S n − B(β1,... ,βn)=1 ...... ... (−1)n...Sn−1. (5.5)   0 ... 1 S   1    Here β denote the (constant) eigenvalues of β(t) and S is the i-th elementary i i symmetric polynomial in β ,... ,β . 1 n { } The mapping (α,β) (g(1),B) gives a biholomorphism between (N ,I) and n Gl(n,C) Cn [6]. → × We describe the complex structure of M˜ (c) as follows: n Proposition 5.3. There exists a Tn-equivariant biholomorphism between M˜ (c) n and an open subset of [g,b] Gl(n,C) (d+n);gbg−1 is of the form (5.5) , N n ∈ × !(cid:30)∼ a(cid:8) (cid:9) where d denotes diagonal matrices, the union is over unipotent algebras n (with respect to d) and N = expn. Furthermore, the relation is given as follows: ∼ [g,d+n] [g′,d′ +n′] if and only if n n,n′ n′, and either n′ n and there ∼ ∈ ∈ ⊂ exists an m N such that gm−1 = g′,Ad(m)(d+n) = d′+n′ or vice versa (i.e. ∈ n n′ etc.). ⊂ Remark . It will follow from the description of the twistor space that this biholomorphism is actually onto. Proving this right now would require showing that the Tn-action on M˜ (c) extends to the global action of C∗ n. This, in turn, requires n showing existence of solutions to a mixed Dirichlet-Robin problem on the half-line (cid:0) (cid:1) - something that seems quite tricky. Proof. Fix a unipotent algebra n and consider the set of all solutions (α,β) = (T +iT ,T +iT )on[1,+ )suchthattheintersectionofthesumofpositiveeigen- 0 1 2 3 ∞ valuesofad(iT (+ ))withC(β(+ ))iscontainedinn. LetM(n;c 1)bethecor- 1 ∞ ∞ − respondingsubsetofM (c). Weobservethat,since(T (+ ),T (+ ),T (+ )) U(n) 1 2 3 is a regular triple, the projection of T (+ ) onto dC ∞C(β(+ ∞)) is a re∞gu- 1 ∞ ∩ ∞ lar element, and so n contains the unipotent radical of a Borel subalgebra of C(β(+ )) for any element of M(n;c 1). Using gauge freedom, we always make ∞ − T (+ )=0 and, by Proposition4.1 of Biquard [8], such a representative is of the 0 ∞ form g α(+ ),β(+ )+Ad(exp α(+ )t )n , where n n and g is a bounded Gl(n,C)-val∞ued gau∞ge transforma{t−ion. ∞The}transformatio∈n g is defined modulo (cid:0) (cid:1) exp α(+ )t g exp α(+ )t with g P =exp(d+n). Since T (+ )=0 and 0 0 0 T i{s−decay∞inge}xponen{tially∞fas}t,g hasa∈limit(inTC)at+ . Ifwerep∞laceg(t)by 0 ∞ g′(t) = g(t)g(+ )−1exp α(+ )t+cα(+ ) , then (α,β) = g′(0,β(+ )+n′) ∞ {− ∞ ∞ } ∞ for an n′ n. The transformation g′, which satisfies (at infinity) the boundary condition o∈f an element of (c 1)C, is now defined modulo constant gauge trans- G − formations in N. Moreover g′(1) is independent of (c 1) and we obtain a map G −