Scalar--flat K"ahler metrics with conformal Bianchi V symmetry

D AMTP-2010-79 Scalar–flat K¨ ahler me trics with conformal Bianc hi V symm etry Maciej Duna jski ∗ Department of Appli ed Mathema tics a nd T heoreti cal Physics, Universit y of Cambridge, Wilb er force Ro a d, Cambridge CB3 0W A, UK. and Institut e of Mathema tics o f the Polish Accademy of Sciences, ´ Sniadeckic h 8, 00-95 6 W arsaw, P oland. Prim Pla nsangk ate † Cen tre de Recherc hes Math´ emati ques (CRM), Universit ´ e de Montr ´ eal, C. P . 612 8, Mon tr´ eal (Qu ´ eb ec) H3C 3J7, Canada . De dic ate d to Maciej Przanowski on the o c c asion o f his 65th birthday. Abstract W e pro vide an affirmativ e answ er to a question p osed by T o d [19], and construct all four–dimensional K¨ ahler metrics with v anish- ing scalar curv atur e w hic h are in v arian t u nder th e conformal action of Bianc h i V group. The construction is based on the com b in ation of t wistor theory and the isomono dromic prob lem with t w o doub le p oles. The resulting metrics are non–diagonal in the left–in v arian t basis and are exp licitly given in terms of Bessel fun ctions and their in tegrals. W e also mak e a connection w ith the LeBrun ansatz, and c h aracterise the asso ciated solutions of the S U ( ∞ ) T o d a equation b y the existence a n on–ab elian tw o–dimensional group of p oin t sy m metries. ∗ Email: M.Duna jski@da mt p.cam.ac.uk † Email: plansa ng@CRM.UMontreal.CA 1 1 In tr o du ction Let ( M , g ) b e a Riemannian four–manifold with an ti–self–dual (ASD) W eyl curv a t ure. The metric g is said to ha v e cohomogeneit y–one if ( M , g ) a dmits an isometry group acting tra nsitiv ely on co dimension-one surfaces in M . W e shall sa y that a conformal structure [ g ] on M has cohomogeneit y–one, if there exists a cohomogeneit y–one metric g ∈ [ g ]. The four-dimensional cohomogeneit y-one metrics can b e classified ac- cording to the Bianc hi ty p e of the three-dimensional real Lie algebra [1] of the isometry group G . Lo cally M = R × G , and the problem of find- ing ASD cohomogeneit y–o ne metrics reduces to solving a system of ODEs, as the group co ordinates do not app ear in the W eyl tensor. Mor eov er, the reduction–in tegrabilit y dogma suggests tha t the resulting ODEs will b e in some sense solv able as the underlying an ti–self–dualit y equations are in te- grable b y tw istor transform [15, 11, 4 ]. This is indeed the case. If G = S U (2) then the ODEs reduce to P ainlev e VI, or (if additio na l assumptions are made ab out the metric) P a inlev e I I I ODEs [17 , 8, 18, 19, 13, 3, 2]. In [19 ] T o d has a nalysed t he general case of conforma l ASD cohomogeneit y–one metrics with a rbitrary group, diagonal in t he basis of left– inv ariant one forms on the group. He has sho wn that cohomogeneity –one diag onal Bianc hi V met- rics are confo r ma lly flat. The diagona lisabilty assumption can only b e made without loss of generality if the underlying metric is Einstein, so the existenc e of non–diagonal Bianc hi V confor mal structures has not b een ruled out in T o d’s w o rk. In part icular he has r a ised a question whether such structures (if they exist) can admit a K¨ ahler metric in a conformal class. In this pap er w e shall use the method of isomono dromic deformations to construct all suc h conformal structures. Let g b e a L ie algebra of Bia nchi ty p e V, and let G b e a corresp onding simply connected Lie group (see App endix). The left–in v arian t v ector fields on G satisfy [ L 0 , L 1 ] = L 1 , [ L 0 , L 2 ] = L 2 , [ L 1 , L 2 ] = 0 . (1.1) A g eneral cohomogeneit y–one metric on M = R × G will b e hav e the form g = dt 2 + h j k ( t ) λ j ⊙ λ k , j, k = 0 , 1 , 2 where λ j are the Maurer–Cartan one–for ms on G suc h that L j λ k = δ j k . W e will say that a group action on a complex manifold is holo mo r phic if Lie deriv ativ es of the complex stucture along any o f the generators v anish. 2 Theorem 1.1 A ny c oh omo geneity–one Bianchi V ASD c onformal structur e which admits a K¨ ahler m etric such that the gr oup acts h olomorphic al ly c an b e lo c al ly r epr esente d by a c ohomo geneity–one metric g = e 1 ⊙ e 1 + e 2 ⊙ e 2 , (1.2) wher e the c omp lex one–forms e 1 , e 2 ar e dual to the ve ctor fi elds E 1 = ∂ t − i t ( r 0 L 0 + r 1 L 1 + r 2 L 2 ) , E 2 = p 0 L 0 + p 1 L 1 + p 2 L 2 and the r e al functions r ( t ) = ( r 0 ( t ) , r 1 ( t ) , r 2 ( t )) and c omplex functions p ( t ) = ( p 0 ( t ) , p 1 ( t ) , p 2 ( t )) s a tisfy the line ar system of ODEs d r dt = tI m ( p 0 p ) , t d p dt = i ( r 0 p − p 0 r ) , dp 0 dt = 0 , dr 0 dt = 0 (1.3) such that det [ r , p , p ] 6 = 0 . W e shall pro v e this theorem in Section 2, where w e shall also sho w that equations (1.3) reduce to the Bessel equation and quadratures. All ASD K¨ ahler metrics hav e v anishing scalar curv ature - w e shall follo w LeBrun [10 ] in calling them scalar–flat K¨ ahler - and con v ersely all scalar– flat K¨ a hler metrics ha v e a n ti–self–dual W eyl tensor (w e choose the nat ura l orien tation, where the K¨ ahler t w o– form is self–dual). In Section 3 w e shall construct the K¨ ahler structure in the conformal class of Theorem 1.1. W e shall demonstrate that although a G –in v ar ia n t conformal factor turning the metric (1.2) into a K¨ ahler metric do es not exist, there is a simple function on G whic h do es the job. W e shall prov e Prop osition 1.2 Any c onformal ly Bianchi – V sc ala r–flat K¨ ahler m etric c an lo c al ly b e p ut i n to a form g K = Ω 2 g , (1.4) wher e g is given by (1.2 ) and Ω : G → R is a function on the gr oup such that L 0 (Ω) = Ω , L 1 (Ω) = 0 , L 2 (Ω) = 0 . (1.5) The holom orphic (1,0) ve ctor fields ar e given by E 1 and E 2 . Mor e o ver g K is Ric ci flat if and only i f it is flat. 3 The righ t–in v ar ia n t v ector fields R j on G generate the conformal transfor- mations of the K¨ ahler metric (1.4 ) L R 0 g K = 2 g K , L R 1 g K = 0 , L R 2 g K = 0 , where L denotes the Lie deriv ative. If w e c ho ose co o rdinates ( ρ, x 1 , x 2 ) on G (see App endix) suc h that L 0 = ρ ∂ ∂ ρ , L 1 = ρ ∂ ∂ x 1 , L 2 = ρ ∂ ∂ x 2 , ρ ∈ R + , ( x 1 , x 2 ) ∈ R 2 then Ω = ρ . Equations (1.3) imply that r 0 and p 0 are constan ts. The remaining equa- tions reduce to the Bessel equation p ossibly with imagina r y order. A partic- ularly simple class of solutions c haracterised by r 0 = 0 is r = (0 , tJ 1 ( t ) , tY 1 ( t )) , p = (1 , iJ 0 ( t ) , iY 0 ( t )) , where J α ( t ) a nd Y α ( t ) a r e Bessel functions of first and second t yp e of order α . This leads to the follow ing example o f a Bianc hi-V scalar fla t K ¨ ahler metric g K = 1 4 ( dρ 2 + ρ 2 dt 2 ) + G AB ( t ) dx A dx B , A, B = 1 , 2 (1.6) where G ( t ) = π 2 t 2 16  Y 2 0 + Y 2 1 − J 0 Y 0 − J 1 Y 1 − J 0 Y 0 − J 1 Y 1 J 2 0 + J 2 1  . All scalar–flat K¨ ahler metrics admitting a Killing v ector which also pre- serv es the K¨ ahler form arise from the L eBrun’s a nsatz [10], and ar e deter- mined b y a solution to the S U ( ∞ )-T o da field equation u xx + u y y + ( e u ) z z = 0 , where u = u ( x, y , z ) , (1.7) together with a solution to its linearisation. K¨ ahler metrics arising from Prop osition 1.2 admit a tw o–dimensional g r oup of symmetries (generated b y the right translations R 1 and R 2 ) preserving the K¨ ahler from. Th us an y linear com bination of R 1 and R 2 should lead to a solution of ( 1 .7). In Section 4 we shall c haracterise the solutions corresp onding to the metric (1.6). They admit a t w o–dimensional non–ab elian group of Lie p oint symmetries and can b e found b y making an ansatz u = u ( y /z ). Ac kno w ledgemen ts. W e are gra teful to Philip Bo a lc h, Andrew Dancer and P a ul T o d for helpful discussions. 4 2 ASD stru ctures and is omono drom y A K¨ ahler structure on a four– dimensional real manifold M consists of a pair ( g , I ) where g is a Riemannian metric and I : T M → T M is a complex structure suc h that, f o r an y v ector fields X, Y , g ( X , Y ) = g ( I X , I Y ) and the t w o–f orm ω defined b y ω ( X , Y ) = g ( I X , Y ) is closed. Giv en an orien tat io n on M the Ho dge op erator ∗ : Λ 2 → Λ 2 satisfies ∗ 2 = Id and giv es a decomp o sition Λ 2 = Λ 2 + ⊕ Λ 2 − of t w o—f orms in to self–dual (SD) and anti–se lf–dual (ASD) comp onen ts. The tw o–form ω induces a nat ur a l orientation on M giv en by the v olume fo rm ω ∧ ω . With resp ect to this orientation ω is self–dual. It is w ell known [1 6, 4] that if the scalar curv ature of a K¨ ahler metric v anishes, then the W eyl tensor of t he underlying conformal structure is ASD. Con v ersely ASD K¨ ahler metrics a re scalar–flat. A conv enien t wa y to express the ASD condition on a conformal structure is summarised in the following theorem. The theorem b elo w is originally due to P enrose [15], but take n in this form from [1 1, 4]. Theorem 2.1 L et E 1 , E 2 b e two c omplex ve ctor fields in T M ⊗ C and let e 1 , e 2 b e the c orr esp onding dual one –forms. The c onfo rmal structur e d e fi ne d by g = e 1 ⊙ e 1 + e 2 ⊙ e 2 is ASD if and only if ther e ex i s ts functions f 0 , f 1 on M × CP 1 holomorphic in λ ∈ CP 1 such that the distribution l = E 1 − λE 2 + f 0 ∂ ∂ λ , m = − E 2 − λE 1 + f 1 ∂ ∂ λ (2.8) is F r ob en ius inte gr a b l e , that is , [ l , m ] = 0 mo dulo l and m. A general ASD conformal structure [ g ] do es not admit a K¨ ahler metric in its conformal class. The existence of suc h metric is characterise d by v anishing of hig her order conformal in v a rian ts o f [ g ] [6]. I n what follows w e shall use a simpler twistor c haracterisation of ASD metrics which are conformal to K¨ ahler. 5 2.1 Twistors and divisors Let us complexify ( M , g ) and regard M as a holomo r phic f o ur–manifold with a holomorphic metric g g =  V 00 ′ ⊙ V 11 ′ − V 01 ′ ⊙ V 10 ′  , (2.9) where V AA ′ , A, A ′ = 0 , 1 is the n ull tetrad of one f o rms written in the tw o– comp onen t spinor notation. The reality conditions can b e imp osed to recov er the Riemannian metric b y setting V 00 ′ = e 1 , V 10 ′ = − e 2 , V 01 ′ = e 2 , V 11 ′ = e 1 . Let V AA ′ b e the corresp onding tetrad of v ector fields so that the in tegrable distribution of Theorem 2.1 is l = V 00 ′ − λV 01 ′ + f 0 ∂ ∂ λ , m = V 10 ′ − λV 11 ′ + f 1 ∂ ∂ λ . (2.10) A t wistor space of ( M , g ) is the space of t w o–dimensional totally n ull surfaces spanned by V 00 ′ − λV 01 ′ , V 10 ′ − λV 11 ′ in M . It is a three dimensional complex manifold Z whic h arises as a quotien t of M × CP 1 b y l, m . T he p oints in M cor r esp o nd to rational curv es (called tw istor curv es) in Z with normal bundle O (1) ⊕ O (1), where O ( n ) is a line bundle ov er CP 1 with Chern class n . The holomorphic canonical line bundle κ of Z restricted to a n y of these curv es is isomorphic to O ( − 4). T o reconstruct a r eal fo ur– manifold M , the t wistor curv es m ust b e in v arian t under an anti–holomorphic in v olution τ on Z whic h restricts to an an tip o dal map on eac h curv e. A theorem o f Pon tecorv o [16] states that an ASD conformal structure ( M , [ g ]) admits a K¨ ahler metric if and only if there exists a section D of the bundle O (2) = κ − 1 / 2 o v er the twistor space Z o f ( M , [ g ]) , with exactly tw o distinct zeros on eac h t wistor line. The section m ust b e τ inv ariant, so its zero es lie on the an tip o dal p o in ts on eac h t wistor curv e. No w consider the group G acting o n an ASD confo r mally K¨ ahler manifold M by ho lomorphic conformal isometries with generically t hree–dimensional orbits. T his g iv es rise to a complexified g r o up action o f G C on the t wistor space Z . Le t e R j , j = 0 , 1 , 2 b e holomorphic vec tor fields on Z corresp ond- ing to the righ t–in v a r ia n t conformal Killing v ectors R j on M . The sub- set of Z where e R j are linearly dependen t is giv en by the zero set of s = v ol Z ( e R 0 , e R 1 , e R 2 ). As the cano nical bundle κ = O ( − 4), the divisor s = 0 6 defines a quartic and v anishes at f our p oints on eac h twistor line. Hitc hin ([8], Prop osition 3) sho w ed 1 that s is not iden t ically zero if g is not Ricci–flat and tha t the divisor when s v anishes is equal to the Pon tecorv o’s divisor D in the case when [ g ] is cohomogeneit y one and con tains a K ¨ ahler class. In the double fibration picture M ← − M × CP 1 − → Z the section s pulls bac k to s = ( dλ ∧ ν )( l, m, e R 0 , e R 1 , e R 2 ) , (2.11) where ν = V 01 ′ ∧ V 10 ′ ∧ V 11 ′ ∧ V 00 ′ is the volume form on M a nd e R j are the lift s of the three generators of G to M × CP 1 suc h that [ l, e R j ] = 0 , [ m, e R j ] = 0 mo dulo l , m, . Th us in the pro o f o f Theorem 1.1 w e will require that this quartic has t w o distinct zeros of o rder t w o. This will gua r an tee the existence of a K¨ ahler metric in the cohomo g eneit y–one confo rmal class. Pro of of Theorem 1.1. W e will work in the complexified category and imp ose the reality conditio ns at the end. Let G C b e a three-complex di- mensional Lie gr o up ( we will ev entually tak e G C to b e a complexification of the Bianchi V group, but the first par t of the pro of do es not dep end on the c ho ice o f G C ). W e assume that the orbits of G C are three–dimensional, and the metric g o n C × G C is in v aria n t under the left tra nslations of G C on itself. One can write the null tetrad V AA ′ in terms of the v ector field ∂ t and three linearly indep enden t v ector fields P , Q, R ta ng en t to G C (in the complexified setting the fields Q and P are indep enden t. Once the r eality conditions are imp osed at the end o f the pro of, w e shall set Q = − P ) whic h are t -dep endent and in v ariant under the left translations, as V 00 ′ = ∂ t + i R t , V 11 ′ = ∂ t − i R t , V 01 ′ = P , V 10 ′ = Q. (2.12) No w, let R 0 , R 1 , R 2 b e the righ t-in v arian t v ector fields on G corresp onding to three generators of the left translations. Since R j are indep enden t o f t, 1 His result was derived for G C = S L (2 , C ) but it r emains v alid for any three-dimensio nal group acting transitively on M , as the corre sp o nding action o f G C preserves the p oints where Pontecorv o’s diviso r v anishes as long a s the group action is holomor phic. Mo reov er we are allow e d to work in a conformal class of g K , as the t wistor equation underlying the Pon tecorvo’s divis or is conformally inv ar iant [16, 6]. 7 one has [ l, R j ] = − R j ( f 0 ) ∂ λ , [ m, R j ] = − R j ( f 1 ) ∂ λ . A direct calculation sho ws t hat there is no lift of R j of the form R j + Q j ∂ λ for some function Q j suc h that [ l , R j + Q j ∂ λ ] = 0 , [ m, R j + Q j ∂ λ ] = 0 mo dulo l , m . Hence , w e conclude that [ l , R j ] , [ m, R j ] , are iden tically zero. This implies that f 0 and f 1 are constant on G, and hence they are functions of λ and t only . W e claim that the quartic s as defined in (2.11) is prop ortional to λf 0 + f 1 , with the prop ortionality factor giv en b y a function h on M . Indeed , using the fact t hat V 01 ′ , V 00 ′ − V 11 ′ , V 10 ′ are linearly indep endent and in v aria nt under the left tra nslations of G C , the right-in v ariant v ector fields R j can b e written in the basis of P = V 01 ′ , Q = V 10 ′ , R = it 2 ( V 11 ′ − V 00 ′ ) . Th us, p erforming all t he con tractions w e are left with s ∝ (det H ) ( λf 0 + f 1 ), where H is the matrix of co efficien ts of R 0 , R 1 , R 2 written in the basis of P , Q, R. As det H do es not dep end on λ, the quartic s has t w o distinct zeros of order tw o if a nd only if λf 0 + f 1 has tw o distinct zeros of order t w o (for the momen t w e rule out the case where f 0 and f 1 are b oth zero whic h corresp onds to a h yp er-K¨ ahler metric [4]). W e shall a ssume that this is the case so that the ASD conformal structure admits a K¨ a hler metric. It is no w p ossible to use M¨ obius transformation to put the tw o zeros at 0 and ∞ . The M¨ obius transformation in λ corresp onds to a c hange of null tetrad by a rig ht rotation V AA ′ → V AA ′ r A ′ B ′ , where r is an S L (2 , C )-v alued function. Since the co efficien ts o f the quartic λf 0 + f 1 are functions of t only , the required r A ′ B ′ will only dep end on t. Thu s, the r o tated tetrad is still G C -in v a rian t. With the zeros a t 0 and ∞ , λf 0 + f 1 is o f the form a ( t ) λ 2 . Let us first assume that a do es not v a nish identically . One still has a M¨ o bius degree of freedom that preserv es (0 , ∞ ) , that is, the m ultiplication of λ b y a function of t. Let us use t his freedom to set a ( t ) = 2 /t. The current tetrad is some righ t rotation of the original one. It is p ossible t o use another freedom: a left rotation V AA ′ → l A B V AA ′ , l ∈ S L (2 , C ) to k eep V 00 ′ − V 11 ′ , V 01 ′ , V 10 ′ tangen t to G C . The r igh t ro tation do es not change the quartic λf 0 + f 1 , and we now ha v e f 0 , f 1 of the form f 0 = b ( t ) λ 2 + c ( t ) λ + d ( t ) , f 1 = − b ( t ) λ 3 +  2 t − c ( t )  λ 2 − d ( t ) λ (2.13) for some functions b ( t ) , c ( t ) , d ( t ) . 8 No w, consider a pair o f linear com binations of l a nd m (2.10) L = λl + m λf 0 + f 1 = ∂ ∂ λ + 2 λiRt − 1 − λ 2 P + Q λf 0 + f 1 (2.14) M = f 1 l − f 0 m λf 0 + f 1 = ∂ ∂ t + ( f 1 − λf 0 ) iRt − 1 − λf 1 P − f 0 Q λf 0 + f 1 . Since the confo rmal class is ASD, Theorem 2.1 means [ l, m ] = ( . . . ) l + ( . . . ) m. This in turn implies that [ L, M ] = 0 , mo dulo L and M . Ho w ev er, one sees that [ L, M ] do es not con tain ∂ λ or ∂ t , th us [ L, M ] must b e identically zero. It turns out that [ L, M ] = 0 implies that b ( t ) = 0 = d ( t ) . The compatibility conditions [ L, M ] = 0 are then giv en b y tP t − i [ R, P ] + ( tc ( t ) − 1) P = 0 , 2 iR t − t [ P , Q ] = 0 , (2.15) tQ t + i [ R, Q ] − ( tc ( t ) − 1) Q = 0 . This show s that a coho mo g eneit y-o ne metric (1.2), in the tetrad (2.1 2) is ASD if the v ector fields P , Q, R satisfy the system (2.15), where c ( t ) is defined in (2.13). No w, let ˆ R = R , ˆ P = h ( t ) P , ˆ Q = h − 1 ( t ) Q, where h ( t ) = e R ( c ( t ) − 1 t ) dt . (2.1 6) The system (2.15) implies that the v ector fields ˆ P , ˆ Q, ˆ R satisfy tP t − i [ R, P ] = 0 , 2 iR t − t [ P , Q ] = 0 , (2.17) tQ t + i [ R, Q ] = 0 , where w e hav e dro pp ed the hat from the r escaled v ector fields. Moreo v er, the tetrad (2 .1 2) constructed f rom a solution ( P , Q, R ) of (2 .17) giv es the same metric (1.2 ) as the tetrad determined from ( h − 1 ( t ) P , h ( t ) Q, R ) whic h satisfy (2.15) with c ( t ) = h t h + 1 t . Neither the metric nor the equations (2.17) dep end on h , so w e can set it equal to 1. The resulting Lax pair (2 .14) is L = ∂ ∂ λ + ( tQ + 2 iλR − λ 2 tP ) 2 λ 2 , M = ∂ ∂ t − ( λQ + λ 3 P ) 2 λ 2 , (2.18) where w e again dropp ed the hat from the rescaled v ector fields. 9 W e conclude that any cohomogeneit y-one metric (1.2), whic h b elongs t o an ASD conformal structure a dmitting a quartic s defined in (2.11) with tw o distinct zeros of order tw o, can b e written in terms of a null tetrad (2.1 2), where the v ector fields P , Q, R satisfy the system (2.1 7). The three vec tor fields P , Q, R can be written in the basis of left-in v arian t v ector fields L 0 , L 1 , L 2 satisfying (1.1) as P = p 0 ( t ) L 0 + p 1 ( t ) L 1 + p 2 ( t ) L 2 Q = q 0 ( t ) L 0 + q 1 ( t ) L 1 + q 2 ( t ) L 2 , (2.19) R = r 0 ( t ) L 0 + r 1 ( t ) L 1 + r 2 ( t ) L 2 , for some functions p j ( t ) , q j ( t ) , r j ( t ). Then using the comm utation relation (1.1) the system (2.17) implies that p 0 , q 0 , r 0 are constan t and that t ( p j ) t = i ( r 0 p j − p 0 r j ) , t ( q j ) t = i ( − r 0 q j + q 0 r j ) , 2 i ( r j ) t = t ( p 0 q j − q 0 p j ) . (2.20) The realit y conditions corresp onding to Riemannian metrics come down to c ho osing R real (so that ( r 0 , r 1 , r 2 ) are all r eal functions) and Q = − P so that q j = − p j ). Equations (2.20) then give the linear system (1.3). Let us no w return to the case a = 0 whic h corresp onds to f 0 and f 1 v anishing in the distribution (2 .1 0). The resulting conformal structure must then b e h yp er-Hermitian [4] a nd (as the v ector fields in (2.10) are v olume pre- serving) it is actually conformal to h yper-K ¨ ahler. The in tegrabilit y [ l , m ] = 0 mo dulo l , m implies the system of Nahm equations P t − i [ R/t, P ] = 0 , 2 i ( R/t ) t − [ P , Q ] = 0 , Q t + i [ R/t, Q ] = 0 . (2.21) Imp osing the reality condition that R is real and Q = − P , (2.21) b ecome d ( r t − 1 ) dt = I m ( p 0 p ) , t d p dt = i ( r 0 p − p 0 r ) , dp 0 dt = 0 , d ( r 0 t − 1 ) dt = 0 . The general solutions for r and p are giv en in terms of trigono metric and exp o nen tia l f unctions dep ending on the constants p 0 and r 0 /t , and the r e- sulting metric is conformally flat (in the pro of of Prop osition 1.2 w e shall find the conformal factor whic h make s it flat). ✷ 10 Our deriv atio n of (2 .1 7) and (2.1 8) did not dep end on the c hoice of t he isometry group. The system (2.17) describes the general isomono dromic deformation equations with t w o double p oles. The cor r espo nding Lax pair (2.18) with P , Q, R given by 2 × 2 matrices w as sho wn b y Jim b o and Miwa [9] to give rise t o the Painlev ´ e I I I equation. The same Lax pair a lso arises as the reduced Lax pair of the ASD YM equation, b y t he Painlev ´ e I I I group. It is sho wn to b e t he isomono dromic L a x pair for the P ainlev ´ e I I I equation when the gauge group of the ASD YM connection is S L (2 , C ) [12], or (with certain algebraic constraints on normal forms) S L (3 , C ) [5]. 2.2 Bessel equation F or generic v alues of the constants ( r 0 , p 0 ) , the solutions to (1.3) ar e deter- mined b y t w o linearly indep enden t solutions of the Bessel equation 2 . If p 0 = 0 then r is a constan t v ector a nd equations fo r p 1 , p 2 can b e easily in tegrated. The resulting metric is conformally flat. If p 0 6 = 0 we rescale the metric b y a constan t | p 0 | 2 and redefine ( r , p 1 , p 2 , t ) to set p 0 = 1. Differen tia ting the first set of equations in (1.3) twice , and using the second set of equations sho ws that the real functions f 1 = 2 t − 1 ( r 1 ) t and f 2 = 2 t − 1 ( r 2 ) t satisfy a pair of Bessel equations t 2 d 2 f k dt 2 + t d f k dt + ( t 2 + r 2 0 ) f k = 0 , k = 1 , 2 . (2.22) If r 0 6 = 0 the general solution of the Bessel equation (2.2 2) is given in terms of Bessel functions of pure imaginary o rder ir 0 f 1 = c 1 J ir 0 ( t ) + c 2 Y ir 0 ( t ) , f 2 = c 3 J ir 0 ( t ) + c 4 Y ir 0 ( t ) . The constan t complex co efficien ts c 1 , c 2 , c 3 , c 4 can b e c hosen so that the func- tions f 1 and f 2 are real, see for example [7]. Given functionally indep enden t f 1 and f 2 w e find r 1 , r 2 , p 1 , p 2 b y in tegrations and a lgebraic manipulations. The case r 0 = 0 is sp ecial. The linear system (1.3) now reduces to a pair of ODEs t 2 d 2 r k dt 2 − t dr k dt + t 2 r k = 0 , k = 1 , 2 2 The r elation with the Bessel equation is already exp ected from the res ult of [14 ], who classified all r eductions of a n ti–self–dual Y ang Mills equatio ns leading (by s witch map) to co ho mogeneity–one ASD conformal str uc tur es without howev e r deter mining a K¨ ahler class. 11 whose solutions are g iven by r 1 = c 1 tJ 1 ( t ) + c 2 tY 1 ( t ) , r 2 = c 3 tJ 1 ( t ) + c 4 tY 1 ( t ) , where J α are Y α are Bessel functions of the first and second kind resp ective ly of order α , a nd c 1 , . . . , c 4 are real constan ts of in tegrations. In this case p 1 , p 2 m ust b e purely imaginar y and the recursion relations ∂ t J 0 = − J 1 , ∂ t Y 0 = − Y 1 , ∂ t ( tJ 1 ) = tJ 0 , ∂ t ( tY 1 ) = tY 0 (2.23) imply that p 1 = i ( c 1 J 0 ( t ) + c 2 Y 0 ( t )) , p 2 = i ( c 3 J 0 ( t ) + c 4 Y 0 ( t )) . (2.24) P erfo rming a linear tr a nsformation of the v ector fields L 1 and L 2 w e can alw a ys set c 1 = 1 , c 2 = 0 , c 3 = 0 , c 4 = 1 whic h yields r = (0 , tJ 1 ( t ) , tY 1 ( t )) , p = (1 , iJ 0 ( t ) , iY 0 ( t )) . T o write down the metric w e in vert the ve ctor fields E 1 , E 2 from Theorem 1.1 and use the fact that 2 t ( Y 0 J 1 − J 0 Y 1 ) = 4 /π . This yields g = 1 4 ( dt 2 + ( λ 0 ) 2 ) + G AB ( t ) λ A λ B , where A, B = 1 , 2 G ( t ) = π 2 t 2 16  Y 2 0 + Y 2 1 − J 0 Y 0 − J 1 Y 1 − J 0 Y 0 − J 1 Y 1 J 2 0 + J 2 1  , where λ 0 , λ 1 , λ 2 are the left–in v arian t one forms on G whic h satisfy (A1). 3 K¨ ahler s tructure Pro of of P rop osition 1.2 . The K¨ ahler structure on M can b e read o ff from the divisor (2 .1 1). In t he pro of of Theorem (1.1) w e hav e mo v ed the double zeros of s to 0 and ∞ , whic h in spinor notation means tha t ω A ′ B ′ = o ( A ′ ι B ′ ) and s ≈ ( ω A ′ B ′ π A ′ π B ′ ) 2 where π A ′ are the homogeneous co or dina t es on CP 1 suc h that λ = − π 0 ′ /π 1 ′ , a nd the spinor basis is o A ′ = (0 , 1) , ι A ′ = ( − 1 , 0). Th us, in the null tetra d (2.12), the K ¨ ahler form is prop ortional to ˆ ω = i 2 ε AB ω A ′ B ′ V AA ′ ∧ V B B ′ = i 2 ( e 1 ∧ e 1 + e 2 ∧ e 2 ) , 12 and the space T 1 , 0 ( M ) of holomorphic v ector fields on M is spanned b y V 11 ′ , V 01 ′ (equiv alently b y the v ectors E 1 and E 2 in Theorem 1.1). The F rob e- nius in tegrabilit y conditions [ T 1 , 0 , T 1 , 0 ] ⊂ T 1 , 0 guaran teeing the v anishing of Nijenh uis torsion follows from the construction, but w e can also ve rify it directly as [ E 1 , E 2 ] =  dp 1 dt − i t r 0 p 1 + i t r 1 p 0  L 1 +  dp 2 dt − i t r 0 p 2 + i t r 2 p 0  L 2 = 0 , where w e ha v e used the constancy of ( r 0 , p 0 ) and equations (1 .3). T o determine the conformal factor w e lo ok for a function Ω : M → R suc h that d ( Ω 2 ˆ ω ) = 0 . Once this has b een fo und the K¨ ahler metric g K and the asso ciated tw o–form ω will b e giv en b y g K = Ω 2 g , ω = i Ω 2 2 ( e 1 ∧ e 1 + e 2 ∧ e 2 ) . It can b e v erified by explicit calculation that there is no G –in v a rian t Ω (i.e. there is no confo rmal factor which dep ends o nly on t ). T o demonstrate that Ω : G → R suc h tha t (1 .5) holds giv es the correct confor mal fa ctor, consider 2Ω d Ω ∧ ˆ ω + Ω 2 d ˆ ω = 0 . (3.25) Since ∂ t Ω = 0 , one can write d Ω in the basis o f left- in v ar ian t one-forms d Ω = L 0 (Ω) λ 0 + L 1 (Ω) λ 1 + L 2 (Ω) λ 2 . The three-form d ˆ ω can b e calculated using the expressions fo r the dual one- forms of E 1 and E 2 in Theorem 1.1 and the Maurer-Cartan’s structure equa- tion (A1). Finally , the system (1.3) is used to simplify the LHS of (3.25 ), and o ne finds that (3.25) is satisfied if a nd only if (1.5) holds. This conformal factor is in fact unique - it could ha v e also b een read off fro m the divisor a s it is prop ortiona l to a p ow er of ω A ′ B ′ ω A ′ B ′ [6]. In the conformally flat case where s in (2.11) is iden tically zero, whic h corresp onds to the system (2.21) , the same conformal factor mak es g flat. Moreo v er we ve rify by explicit calculation that g K giv en b y (1.4 ) where s 6 = 0 is Ricci flat if and only if it is flat, whic h prov es the last part the Propo sition. ✷ 13 The conformal Killing v ectors generating t he group action on g K are giv en b y the righ t–in v aria n t v ector fields on G . If the co o rdinates are c hosen for the group, these ve ctors are giv en b y (A3). Example. The simplest explicit example of the scalar–fla t K ¨ ahler metric corresp onds to r 0 = 0 in Theorem 1.1 and is g iv en by (1.6). The determinan t of the metric (1.6) given by det g K = π 2 ρ 2 t 2 1024 . Since b y definition ρ 6 = 0 , g K ma y b e degenerate only a t t = 0 or t = ∞ . The Ricci scalar R is iden tically zero b ecause the K¨ ahler metric is ASD. The remaining curv ature inv ariants a r e R abcd R abcd = 256 ρ 4 t 2 , W abcd W abcd = 128 ρ 4 t 2 whic h indicates that t = 0 is a singularity . Rescaling the metric b y a confo r - mal facto r ρ − 2 ( tf ( t )) − 1 giv es W abcd W abcd = 128 f 2 , whic h needs to b e regular if the conformal class contains a complete metric, but the regularit y of the conformal factor r equires that tf ( t ) is a lso regular and non-zero. Thus the norm of the W eyl tensor blows up at 0 . The asymptotic b eha viour of g K for large t is g K = 1 4  dρ 2 + ρ 2 dt 2  + π t 8 ( d ( x 1 ) 2 + d ( x 2 ) 2 ) . 4 S U ( ∞ ) T o da e quation LeBrun [10] has sho wn that any K¨ ahler metric g K with symmetry preserving the K¨ a hler form admits a lo cal co ordinate system { τ , x, y , z } suc h that g K = W h + 1 W ( dτ + θ ) 2 , ω = W e u dx ∧ dy + dz ∧ ( dτ + θ ) , (4.26) where h = e u ( dx 2 + dy 2 ) + dz 2 . (4.27) Here τ is a co ordinate along the orbits of the Killing v ector K = ∂ τ , { x, y , z } are co ordinates on the space of orbits, and ( u, W ) and θ are functions and a one–for m on the space o f orbits suc h that u satisfies the S U ( ∞ ) T o da equation (1.7), W satisfies the so–called monop ole equation W xx + W y y + ( W e u ) z z = 0 (4.28) 14 and θ is determined b y W together with the condition dω = 0. The ansatz (4.26) can b e understo o d a s follows. Giv en that K = ∂ τ is a Killing v ector, the metric necessarily takes the fo r m (4.26), where 1 W = g K ( K , K ) a nd h is a metric on the three–dimensional space of orbits. No w, since the K¨ ahler form ω Lie deriv es along K , we hav e K ω = dz (4.29) for some function z on the space of orbits of K . The isothermal co ordinates x, y on the or thogonal complemen t of the space spanned by K and I ( K ) (where I is the complex structure) can b e used tog ether with z to parametrise the space of orbits. The metric then tak es the form (4.26) with h g iv en b y (4.27) for some u = u ( x, y , z ). The in tegrabilit y of the complex structure and the closure of the ω imply (4 .2 8). The scalar- fla t condition gives (1 .7). 4.1 Bessel solutions to S U ( ∞ ) T o da equation The scalar–flat K¨ ahler metric (1 .6) has tw o Killing symmetries ∂ /∂ x 1 and ∂ /∂ x 2 preserving the K¨ ahler form. W e can follo w the algorithm describ ed ab ov e and find the solutio n u of the S U ( ∞ ) T o da equation and the asso ciated monop ole W corresp onding to a linear comb ination K = c 1 ∂ /∂ x 1 + c 2 ∂ /∂ x 2 . W e shall set c 2 = 0 for simplicit y . Set τ = x 1 so that the ve ctor field K = ∂ τ and dx 1 = dτ . Then the metric (1.6) takes the form (4.26), where h = 1 4 W  ρ 2 dt 2 + dρ 2  + t 2 π 2 64 d ( x 2 ) 2 , W = 16 t 2 π 2 ( Y 2 0 + Y 2 1 ) , and θ = − ( J 0 Y 0 + J 1 Y 1 ) Y 2 0 + Y 2 1 d ( x 2 ) . (4.30) Using identities (2 .23) a nd contracting the K¨ ahler form of (1 .6) with ∂ /∂ x 1 , one reco v ers the equations ( 4.29) for z . The other tw o co ordinates are 3 τ = x 1 , x = − π x 2 8 , y = − π ρY 0 8 , z = π ρtY 1 8 . (4.31) 3 F or a K illing vector given by a g eneral linear combination o f ∂ /∂ x 1 and ∂ /∂ x 2 , a linear combination of B e ssel functions app ears in the final formula, and in particular z y = − t  c 2 J 1 − c 1 Y 1 c 2 J 0 − c 1 Y 0  . 15 The solution to S U ( ∞ ) T o da equation is now implicitly giv en b y e u = t 2 . (4.32) F orm ulae (4.31) imply that u = u ( v ), where v = z / y . Thus u is constan t on the plane y v ( u ) − z = 0 (compare [18] where solutions constan t on quadrics w ere constructed) and is in v arian t under a t wo–dimen sional group of Lie p oin t symmetries generated b y v ector fields ∂ /∂ x, x∂ x + y ∂ y + z ∂ z . (4.33) W e shall now sho w that the existence of these symmetries uniquely charac- terises (4.32). An y solution u of (1.7) whic h is in v arian t under symmetries generated by the v ector fields ( 4 .33) is a function u = u ( v ). Then (1.7) b ecomes an ODE ( v 2 + e u ) u vv + 2 v u v + u 2 v e u = 0 . Equiv alently , in t erchanging the dep enden t and indep enden t v ar iables w e hav e ( v 2 + e u ) 2 ∂ u  v u v 2 + e u  = 0 , whic h in tegrates to v u = c ( v 2 + e u ) (4.34) for some constant c . Now we shall arg ue t ha t this constan t c can alwa ys b e set to − 1 / 2 , pro vided that w e consider solutions to the S U ( ∞ ) T o da equation as equiv a len t if they determine the confo rmally equiv alen t metrics (4.27). First, note that a transformation u − → u + 2 β , z − → ± e β z , (4.35) where β is a constan t is a symmetry of (1.7) whic h rescalles the three–metric b y a constant factor. No w, under (4.35), the v a r iable v tra nsforms as v − → ± e β v . This can b e used to set c = − 1 / 2 . The equation (4.34) with c = − 1 / 2 is equiv alent to the Bessel equation of order 0 t 2 Y tt + tY t + t 2 Y = 0 . (4.36) T o see it use t 2 = e u and v = 2 ∂ u ln Y . The four–dimensional metric (4.2 6 ) resulting from the solution u = u ( z /y ) tog ether with the mono p ole (4 .30) admits a conformal actio n of the Bianch i V gro up g enerated by R 0 = τ ∂ τ + x∂ x + y ∂ y + z ∂ z , R 1 = ∂ τ , R 2 = ∂ x . (4.37) 16 W e shall now establish tha t give n a solution u = u ( z /y ) to the S U ( ∞ ) T o da equation, the f unction W given by (4.30) is ( up to ga uge transformation) the only solution to the linearised S U ( ∞ ) T o da equation (4.28 ) suc h that the resulting metric (4.26) in four dimensions defines a cohomog eneit y–one Bianc hi V conformal class. T o sho w it, observ e that the metric h in (4.27) is inv ariant under R 1 , R 2 and conforma lly inv ariant h → c 2 h under the one parameter gr oup of tra nsformations g enerated by R 0 . This implies that t he function W mus t b e inv ariant under the Bianc hi V group G and the one-form θ is in v aria n t under the translations generated by R 1 , R 2 , but transforms a s θ → c θ under the action generated by R 0 . Th us the metric (4.2 6 ) is confor- mally inv arian t under G if a nd only if t he function W and the comp onents of the o ne- f orm θ are functions of t only . Th erefore the PDE (4.28) fo r W b ecomes a second or der ODE W tt v t ( v 2 + t 2 ) + 2 W t ( v t (2 t + v v t ) − v tt ( v 2 + t 2 )) + 2 W ( v t − tv tt ) = 0 . The in v aria nce prop erties of θ lead to one f ur t her constrain t W t ( t 2 + v 2 ) + 2 tW = 0 . (4.38) The relation (4.31) yields v = − t Y 1 Y 0 and using (2.23) w e find the general solution W of (4.38) W = k t 2 ( Y 2 0 + Y 2 1 ) , k = const . (4.39) Changing the prop o rtionality constan t k in W amounts to c hanging the co- ordinate τ − → τ /k in (4.26) and rescaling the metric by the constan t k . Hence w e can alw a ys set k = 64 /π 2 , whic h giv es the monop ole W in (4.31). Example: ASD E instein metric with symmetry . Here w e shall presen t an example of a metric (4.26) whic h is obtained from a monop ole W differen t from (4.3 9). It w as sho wn in [20 ] that any ASD Einstein metric with symmetry a nd non-zero scalar curv ature can b e written a s g E = W z 2  e u ( dx 2 + dy 2 ) + dz 2  + 1 W z 2 ( dτ + θ ) 2 , (4.40) where W = const ( z u z − 2), and u is a solutio n to the S U ( ∞ ) T o da equation. T a k e u giv en b y (4.32) and the Einste in monop ole W with const= 1 / 2. Then W = Y 0 Y 1 t ( Y 2 0 + Y 2 1 ) − 1 , θ =  1 2  Y 2 0 − Y 2 1 Y 2 0 + Y 2 1  + ln  π ρ 8   dx. 17 The resulting Einstein metric (4.40) has negativ e scalar curv a ture and is non-conformally flat. Note that the metric is not conformal to a Bianc hi V metric. It only admits a t w o–dimensional group of symmetries . App endix The real three–dimensional Lie a lg ebra of Bianc hi ty p e V is defined by com- m utation relations [ X 0 , X 1 ] = X 1 , [ X 0 , X 2 ] = X 2 , [ X 1 , X 2 ] = 0 . W e can c ho ose its represen t a tion b y 3 × 3 matrices X 0 =   1 0 0 0 0 0 0 0 0   , X 1 =   0 1 0 0 0 0 0 0 0   , X 2 =   0 0 1 0 0 0 0 0 0   . The corresp onding Lie group G is the mu ltiplicativ e gro up of real matrices of the form g =   ρ x 1 x 2 0 1 0 0 0 1   , where ρ ∈ R + , ( x 1 , x 2 ) ∈ R 2 . The left- inv ariant one-fo rms { λ j , j = 0 , 1 , 2 } corresp o nding to a basis { X j } of a Lie algebra are giv en b y g − 1 d g = λ j X j , where g ∈ G. Hence λ 0 = ρ − 1 dρ, λ 1 = ρ − 1 dx 1 , λ 2 = ρ − 1 dx 2 , and dλ 0 = 0 , dλ 1 = λ 1 ∧ λ 0 , dλ 2 = λ 2 ∧ λ 0 . (A1) The left in v arian t v ector fields defined b y L j λ k = δ j k are found to b e L 0 = ρ ∂ ∂ ρ , L 1 = ρ ∂ ∂ x 1 , L 2 = ρ ∂ ∂ x 2 . 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