The Erwin Schrodinger International Boltzmanngasse 9. Institute for Mathematical Physics A-1090 Wien, Austria

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1 ESI The Erwin Schrodinger International Boltzmanngasse 9 Institute for athematical Physics A-9 Wien, Austria Hyper{Kahler Hierarchies and their Twistor Theory aciej Dunajski Lionel J. ason Vienna, Preprint ESI 8 () January 4, Supported by Federal inistry of Science and Transport, Austria Available via

2 Hyper-Kahler Hierarchies and their twistor theory aciej Dunajski, Lionel J. ason The athematical Institute, 4-9 St Giles, Oxford OX 3LB, UK Abstract A twistor construction of the hierarchy associated with the hyper-kahler equations on a metric (the anti-self-dual Einstein vacuum equations, ASDVE, in four dimensions) is given. The recursion operator R is constructed and used to build an innite-dimensional symmetry algebra and in particular higher ows for the hyper-kahler equations. It is shown that R acts on the twistor data by multiplication with a rational function. The structures are illustrated by the example of the Sparling-Tod (Eguchi-Hansen) solution. An extended space-time N is constructed whose extra dimensions correspond to higher ows of the hierarchy. It is shown that N is a moduli space of rational curves with normal bundle O(n) O(n) in twistor space and is canonically equipped with a Lax distribution for ASDVE hierarchies. The space N is shown to be foliated by four dimensional hyper-kahler slices. The Lagrangian, Hamiltonian and bi-hamiltonian formulations of the ASDVE in the form of the heavenly equations are given. The symplectic form on the moduli space of solutions to heavenly equations is derived, and is shown to be compatible with the recursion operator. Introduction Roger Penrose's twistor theory gives rise to correspondences between solutions to dierential equations on the one hand and unconstrained holomorphic geometry on the other. The two most prominent systems of nonlinear equations which admit such correspondences are the anti-self-dual vacuum Einstein equations (ASDVE) [3] which in Euclidean signature determine hyper-kahler metrics, and the anti-self-dual Yang{ ills equations (ASDY) [3]. Richard Ward [3] observed that many lower-dimensional integrable systems are symmetry reductions of ASDY. This has led to an overview of the theory of integrable systems [], which provides a classication of those lower-dimensional integrable systems that arise as reductions of the ASDY equations and a unication of the theory of such integrable equations as symmetry reduced versions of the corresponding theory of the ASDY equations. In [], Lagrangian and Hamiltonian frameworks for ASDY were described together with a recursion operator. This leads to the corresponding structures for symmetry reductions of the ASDY equations. In this paper we investigate these structures for the second important system of equations the ASDVE or hyper-kahler equation (this system also admits known integrable systems as symmetry reductions []). We shall give a twistor-geometric construction of the hierarchies associated to the ASDVE in the `heavenly' forms due to Plebanski [5]. In this context it is more natural to work with complex (holomorphic) metrics on complexied space-times and so we use the term ASDVE equations rather than hyper-kahler equations. Our considerations will generally be local in space-time which will be understood to be a region in C 4. In Section we summarise the twistor correspondences for at and curved spaces. We establish a spinor notation (which will not be essential for the subsequent sections) and recall basic facts about the ASD conformal condition and the geometry of the spin bundle. In Section 3 the recursion operator R for the ASDVE is constructed as an integro- dierential operator mapping solutions to the linearised heavenly equations to other solutions. We then use this to give an alternate development of the twistor correspondence by using R to build a family of foliations by twistor surfaces. We show that R corresponds to multiplication of the twistor data by a given twistor function. We then analyse the hidden symmetry algebra of the ASDVE, and use the recursion operator to construct Killing spinors. We illustrate the ideas using the example of the Sparling{Tod solution and show how R can be used to construct rational curves with normal bundle O() O() in the associated twistor space. In Section 4 we give the twistor construction for the ASDVE hierarchies. The higher commuting ows can be thought of as coordinates on an extended space-time. This extended space-time has a twistor dunajski@maths.ox.ac.uk

3 correspondence: it is the moduli space of rational curves with normal bundle O(n) O(n) in a twistor space. This moduli space is canonically equipped with the Lax distribution for ASDVE hierarchies, and conversely that truncated hierarchies admit a Lax distribution that gives rise to such a twistor space. The Lax distribution can be interpreted as a connecting map in a long exact sequence of sheaves. In Section 5 we investigate the Lagrangian and Hamiltonian formulations of heavenly equations. The symplectic form on the moduli space of solutions to heavenly equations will be derived, and is shown to be compatible with the recursion operator. We end this introduction with some bibliographical remarks. Signicant progress towards understanding the symmetry structure of the heavenly equations was achieved by Boyer and Plebanski [3, 4] who obtained an innite number of conservation laws for the ASDVE equations and established some connections with the nonlinear graviton construction. Their results were later extended in papers of Strachan [6] and Takasaki [8, 9]. The present work is an extended version of [8, 9, ]. Preliminaries. Spinor notation We work in the holomorphic category with complexied space-times: thus space-time is a complex four-manifold equipped with a holomorphic metric g and compatible volume form. In four complex dimensions orthogonal transformations decompose into products of ASD and SD rotations SO(4;C ) = (SL(;C) f SL(;C))= : (.) The spinor calculus in four dimensions is based on this isomorphism. We use the conventions of Penrose and Rindler [4]. Indices will generally be assumed to be concrete unless stated otherwise: a; b; : : :, a = ; : : :3 are four-dimensional space-time indices and A; B; : : :; A ; B ; : : :, A = ; etc. are twodimensional spinor indices. The tangent space at each point of is isomorphic to a tensor product of the two spin spaces T a = S A S A : (.) The complex Lorentz transformation V a?! a bv b, a b c dg ac = g bd, is equivalent to the composition of the SD and the ASD rotation V AA?! A BV BB A B ; where A B and A B are elements of SL(;C) and SL(;C). f Spin dyads (o A ; A ) and (o A ; A ) span S A and S A respectively. The spin spaces S A and S A are equipped with symplectic forms " AB and " A B such that " = " =. These anti-symmetric objects are used to raise and lower the spinor indices. We shall use normalised spin frames so that o B C? B o C = " BC ; o B C? B o C = " B C : Let e AA be a null tetrad of -forms on and let r AA be the frame of dual vector elds. The orientation is given by xing the volume form = e ^ e ^ e ^ e : Apart from orientability, must satisfy some other topological restrictions for the global spinor elds to exist. We shall not take them into account as we work locally in. The local basis AB and A B of spaces of ASD and SD two-forms are dened by The rst Cartan structure equations are e AA ^ e BB = " AB A B + " A B AB : (.3) de AA = e BA ^? A B + e AB ^? A B ; where? AB and? A B their indices, and are the SL(;C) and f SL(;C) spin connection one-forms. They are symmetric in? AB =? CC ABe CC ;? A B =? CC A B ecc ;? CC A B = o A r CC B? A r CC o B : The curvature of the spin connection R A B = d? A B +? A C ^? C B

4 decomposes as R A B = C A BCD CD + (=)R A B + A BC D C D ; and similarly for R A B. Here R is the Ricci scalar, ABA B is the trace-free part of the Ricci tensor R ab, and C ABCD is the ASD part of the Weyl tensor. The at twistor correspondence C abcd = " A B " C D C ABCD + " AB " CD C A B C D : The at twistor correspondence is a correspondence between points in complexied inkowski space, C 4 (or its conformal compactication) and holomorphic lines in CP 3. The at twistor correspondence has an invariant formulation in terms of spinors. A point in C 4 has position vector with coordinates (w; z; x; y). The isomorphism (.) is realised by y w x AA := ; so that g = "?x z AB " A B dxaa dx BB : A two-plane in C 4 is null if g(x; Y ) = for every pair (X; Y ) of vectors tangent to it. The null planes can be self-dual (SD) or anti self-dual (ASD), depending on whether the tangent bi-vector X ^Y is SD or ASD. The SD null planes are called -planes. The -planes passing through a point in C 4 are parametrised by = = CP. Tangents to -planes are spanned by two vectors which form the kernel of A B A B L A A AA The set of all -planes is called a projective twistor space and denoted PT. For C 4 it is a three-dimensional complex manifold bi-holomorphic to CP 3?CP. The ve complex dimensional correspondence space F := C 4 CP bres over C 4 by (x AA ; )! x AA and over PT with bres spanned by L A. Twistor functions (functions on PT ) pull back to functions on F which are constant on -planes, or equivalently satisfy L A f =. Twistor space can be covered by two coordinate patches U and e U, where U is a complement of = and e U is a compliment of =. If ( ; ; ) are coordinates on U and (~ ; ~ ; ~ ) are coordinates on e U then on the overlap ~ = =; ~ = =; ~ = =: The local coordinates ( ; ; ) on PT pulled back to F are We can introduce homogeneous coordinates on the twistor space = w + y; = z? x; : (.5) (! A ; A ) = (! ;! ; ; ) := ( ; ; ; ): The point x AA C 4 lies on the -plane corresponding to the twistor (! A ; A ) PT i! A = x AA A : (.6) For A 6= and (! A ; A ) xed, The solution to (.6) is a complex two plane with tangent vectors of the form A A for all A. Alternatively, if we x x AA, then (.6) denes a rational curve, CP, in PT with normal bundle O() O(). Kodaira theory guarantees that the family of such rational curves in PT is four complex dimensional. There is a canonical (quadratic) conformal structure ds on C 4 : the points p and q are null separated with respect to ds in C 4 i the corresponding rational curves l p and l q intersect in PT at one point..3 Curved twistor spaces and the geometry of the primed spin bundle. Given a complex four-dimensional manifold with curved metric g, a twistor in is an -surface, i.e. a null two-dimensional surface whose tangent space at each point is an plane. There are Frobenius integrability conditions for the existence of such -surfaces through each -plane element at each point and these are equivalent, after some calculation, to the vanishing of the self-dual part of the Weyl curvature, C A B C D. Thus, given C A B C D =, we can dene a twistor space PT to be the three complex dimensional manifold of -surfaces in. If g is also Ricci at then PT has further structures which are listed in the Nonlinear Graviton Theorem: Here O(n) denotes the line bundle over CP with transition functions?n from the set 6= to 6= (i.e. Chern class n). 3

5 Theorem. (Penrose [3]) There is a - correspondence between complex ASD vacuum metrics on complex four-manifolds and three dimensional complex manifolds PT such that There exists a holomorphic projection : PT?! CP PT is equipped with a four complex parameter family of sections of each with a normal bundle O() O(), (this will follow from the existence of one such curve by Kodaira theory), Each bre of has a symplectic structure?( (? ()) O()); where CP. To obtain real metrics on a real 4-manifold, we can require further that the twistor space admit an anti-holomorphic involution. The correspondence space F = CP is coordinatized by (x; ), where x denotes the coordinates on and is the coordinate on CP that parametrises the -surfaces through x in. We represent F as the quotient of the primed-spin bundle S A with bre coordinates A by the Euler vector eld = A =@ A. We relate the bre coordinates to by = =. A form with values in the line bundle O(n) on F can be represented by a homogeneous form on the non-projective spin bundle satisfying = ; L = n: The space F possesses a natural two dimensional distribution called the twistor distribution, or Lax pair, to emphasise the analogy with integrable systems. The Lax pair on F arises as the image under the projection T S A?! TF of the distribution spanned by AA +? AA B C C on T S A where AA are a null tetrad for the metric on, and? AA B C are the components of spin connection in the associated spin frame (@ AA +? AA B C B is the horizontal distribution C S A ). We can also represent the Lax pair on the projective spin bundle by L A = (? AA + f ); where f A = (? )? AA B C A B C : (.8) The integrability of the twistor distribution is equivalent to C A B C D =, the vanishing of the self-dual Weyl spinor. When the Ricci tensor vanishes also, a covariant constant primed spin frame can be found so that? AA B C =. We assume this from now on. The projective twistor space PT arises as a quotient of F by the twistor distribution. With the Ricci at condition, the coordinate descends to twistor space and A descends to the non-projective twistor space. It can be covered by two sets, U = fjj < + g and U ~ = fjj >? g. On the non-projective space we can introduce extra coordinates! A of homogeneity degree one so that (! A ; A ); A 6= A are homogeneous coordinates on U and similarly (~! A ; A ); A 6= o A ) on U. ~ The twistor space PT is then determined by the transition function ~! B = ~! B (! A ; A ) on U \ U. ~ The correspondence space has the alternate denition F = PT j lx = CP where l x is the line in PT that corresponds to x and PT lies on l x. This leads to a double bration p? F?! q PT : (.9) The existence of L A can also be deduced directly from the correspondence. From [3], points in correspond to rational curves in PT with normal bundle O A () := O()O(). The normal bundle to l x consists of vectors tangent to x (horizontally lifted to T (x;) F) modulo the twistor distribution. Therefore we have a sequence of sheaves over CP?! D?! C 4?! O A ()?! : The map C 4?! O A () is given by V AA?! V AA A. Its kernel consists of vectors of the form A A with A varying. The twistor distribution is therefore D = O(?) S A and so there is a canonical L A?(D O() S A ), as given in (.8). Various powers of in formulae like (.8) guarantee the correct homogeneity. We usually shall omit them when working on the projective spin bundle. In a projection S A?! F we shall use the replacement formula This is because (on functions @ A?! : (.7) = oa? A = A : 4

6 .4 Some formulations of the ASD vacuum condition The ASD vacuum conditions C A B C D =, ABA B = = R imply the existence of a normalised, covariantly constant frame (o A ; A ) of S A, so that? AA B C =. One can further choose an unprimed spin frame so that the Lax pair (.8) consists of volume-preserving vector elds on : Proposition. (ason & Newman [8].) Let b raa = (b r ; b r b r b r ) be four independent holomorphic vector elds on a four-dimensional complex manifold and let be a nonzero holomorphic four-form. Put L = b r? b r ; L = b r? b r : (.) Suppose that for every CP [L ; L ] = ; L LA = : (.) Here L V denotes the Lie derivative. AA = f? b raa ; where f := (b r ; b r b r b r ); is a null-tetrad for an ASD vacuum metric. Every such metric locally arises in this way. In [7] the last proposition is generalised to the hyper-hermitian case. A choice of unprimed spin frame with f = is always possible and we shall assume this here-on so that r AA = b raa. For easy reference we rewrite the eld equations (.) in full Let A B [r A ; r B ] = ; (.) [r A ; r B ] + [r A ; r B ] = ; (.3) [r A ; r B ] = : (.4) be the usual basis of SD two-forms. On the correspondence space, dene The formulation of the ASDVE condition dual to (.) is: () := A B A B : (.5) Proposition.3 (Plebanski [5], Gindikin []) If a two-form of the form on the correspondence space satises () := A B A B d h () = ; () ^ () = (.6) where d h is the exterior derivative holding A constant, then there exist one-forms e AA related to A B by equation (.3) which give an ASD vacuum tetrad. Note that the simplicity condition in (.6) arises from the condition that A B comes from a tetrad. To construct Gindikin's two-form starting from the twistor space, one can pull back the brewise complex symplectic structure on PT?! CP to the projective spin bundle and x the ambiguity by requiring that it annihilates vectors tangent to the bres. The resulting two-form is O() valued. (To obtain Gindikin's two-form one should divide it by a constant section of O().) Put =?~; =!; =. The second equation in (.6) becomes! ^! = ^ ~ :=?; ^! = ~ ^! = ^ = ~ ^ ~ = : Equations (.6) can be seen to arise from (.) by observing that () can be dened by " AB () = (L A ; L B ; :::; :::): Note also that L A spans a two-dimensional distribution annihilating (). The two one-forms e A := A e AA by denition annihilate the twistor distribution. Dene (; ) A B := r eab AA so that where (@ e A L A = B B A + ) If the eld equations are satised then the Euclidean slice of is equipped with three integrable complex structures given by J i := ); ~ (@ )g and three symplectic structures! i = f(i(? ~); i!; ( + ~)g compatible with the J i. It is therefore a hyper-kahler manifold. 5

7 .5 The ASD condition and heavenly equations Part of the residual gauge freedom in (.) is xed by selecting one of Plebanski's null coordinate systems.. Equations (.3) and (.4) imply the existence of a coordinate system and a complex-valued function such that (w; z; ~w; ~z) =: (w A ; ~w A ) AA = ~z? w = z ~z? z Equation (.) yields the rst heavenly equation The dual A : (.7) w~z z ~w? w ~w z~z @w ~w ~w B = : (.8) e A = dw A ; e ~w B d ~w B (.9) with the at solution = w A ~w A. The only nontrivial part of A B is so that is a Kahler scalar. The Lax pair for the rst heavenly equation is L : = w ~z? w ; L : = z ~z? z : (.) Equations L = L = have solutions provided that satises the rst heavenly equation (.8). Here is a function on F.. Alternatively equations (.) and (.3) imply the existence of a complex-valued function and coordinate system (w; z; x; y) =: (w A ; x A ), w A as above, @ AA = w + x? y =?@ z? x + y A As a consequence of (.4) satises second heavenly equation xw + yz + xx yy? xy = or The dual frame is given by e A = dx : B @x A = : @x B dw B ; e A = dw A A with = dening the at metric. The Lax pair corresponding to (.) is L y? (@ w? y + x ); L x + (@ z + y? x ): (.4) Both heavenly equations were originally derived by Plebanski [5] from the formulation (.6). The closure condition is used, via Darboux's theorem, to introduce! A, canonical coordinates on the spin bundle, holomorphic around = such that the two-form (.5) is () = d h! A ^ d h! A. The various forms of the heavenly equations can be obtained by adapting dierent coordinates and gauges to these forms. 3 The recursion operator In xx3. the recursion operator R for the anti-self-dual Einstein vacuum equations is constructed. In xx3. then show that the generating function for R i is automatically a twistor function, and is in fact a Cech representative for. It is shown that R acts on such a twistor function by multiplication. A similar application to the coordinates used in the heavenly equations yields the coordinate description of the twistor space starting. In xx3.3 we show how that the action of the recursion operator on space-time corresponds to multiplication of the corresponding twistor functions by. In xx3.4 the algebra of hidden symmetries of the second heavenly equation is constructed by applying the recursion operator to the explicit symmetries. In xx3.5, R is used to build a higher valence Killing spinors corresponding to hidden symmetries. In the last subsections examples of the use of the recursion operator are given. 6

8 3. The recursion relations The recursion operator R is a map from the space of linearised perturbations of the ASDVE equations to itself. This can be used to construct the ASDVE hierarchy whose higher ows are generated by acting on one of the coordinate ows with the recursion operator R. We will identify the space of linearised perturbations to the ASDVE equations with solutions to the background coupled wave equations in two ways as follows. Lemma 3. Let and denote wave operators on the ASD background determined by and respectively. Linearised solutions to (.8) and (.) satisfy = ; = : (3.5) Proof. In both cases g = r A r A since g = p a (g ab p g@b ) = g b + (@ a g ab )@ b a g ab = for both heavenly coordinate systems. For the rst equation (@ + )) = implies = (@ = d(@ ^ (@? = d d: Here is the Hodge star operator corresponding to g. For the second equation we make use of the tetrad (.) and perform coordinate calculations. From now on we identify tangent spaces to the spaces of solutions to (.8) and (.) with the space of solutions to the curved background wave equation, W g. We will dene the recursion operator on the space W g. The above lemma shows that we can consider a linearised perturbation as an element of W g in two ways. These two will be related by the square of the recursion operator. The linearised vacuum metrics corresponding to and are h I AA BB = (A o B )r (A r B) ; h II AA BB = o A o B r A r B : where o A = (; ) and A = (; ) are the constant spin frame associated to the null tetrads given above. Given W g we use the rst of these equations to nd h I. If we put the perturbation obtained in this way on the LHS of the second equation and add an appropriate gauge term we obtain - the new element of W g that provides the which gives rise to h II ab = h I ab + r (a V b) : (3.6) To extract the recursion relations we must nd V such that h I AA BB? r (AA V BB ) = o A o B AB : Take V BB = o B r B, which gives r (AA V BB ) =? (A o B )r (A r B) + o A o B r A r B : This reduces (3:6) to r A r B = r A r B : (3.7) Denition 3. Dene the recursion operator R : W g?! W g by A r AA = o A r AA R; (3.8) so formally R = (r A )? r A (no summation over the index A). Remarks: From (3.8) and from (.) it follows that if belongs to W g then so does R. If R = then and correspond to the same variation in the metric up to gauge. The operator 7! r A is over-determined, and its consistency follows from the wave equation on. This denition is formal in that in order to invert the operator 7! r A we need to specify boundary conditions. 7

9 To summarize: Proposition 3.3 Let W g be the space of solutions of the wave equation on the curved ASD background given by g. (i) Elements of W g can be identied with linearised perturbations of the heavenly equations. (ii) There exists a (formal) map R : W g?! W g given by (3.8). The recursion operator can be generalised to act on solutions to the higher helicity ero Rest-ass equations on the ASD vacuum backgrounds [] by using Herz potentials. We restrict ourselves to the gauge invariant case of left-handed neutrino eld A on a heavenly background. First note that any solution of r AA A = must be of the form r A where W g. Dene the recursion relations R A := r A R : (3.9) It is easy to see that R maps solutions into solutions, although again the denition is formal in that boundary conditions are required to eliminate the ambiguities. A conjugate recursion operator R will play a role in the Hamiltonian formulation in Section The recursion operator and twistor functions A twistor function f can be pulled back to the correspondence space F. A function f on F descends to twistor space i L A f =. Given W g, dene, for i, a hierarchy of linear elds, i P R i. Put =? i i and observe that the recursion equations are equivalent to L A =. Thus is a function on the twistor space PT. Conversely every solution of L A = dened on a neighbourhood of jj = can be expanded in a Laurent series in with the coecients forming a series of elements of W g related by the recursion operator. The function, when multiplied by =( ), is a Cech representative of the element of H (PT ; O(?)) that corresponds to the solution of the wave equation under the Penrose transform (i.e. by integration around jj = ). The ambiguity in the inversion of r A means that there are many such functions that can be obtained from a given. However, they are all equivalent as cohomology classes. It is clear that a series corresponding to R is the function?. As noted before, R is not completely well dened when acting on W g because of the ambiguity in the inversion of r A. However, the denition R = = is well dened as a twistor function on PT, but the problem resurfaces when one attempts to treat () as a representative of a cohomology class since pure gauge elements of the rst sheaf cohomology group H (PT ; O(?)) are mapped to functions dening a non-trivial element of the cohomology. Note, however, that with the denition R = =, the action of R is well dened on twistor functions and can be iterated without ambiguity. We can in this way build coordinate charts on twistor space from those on space-time arising from the choices in the Plebanski reductions. Put! A = w A = (w; z); the surfaces of constant! A are twistor surfaces. We have that r A!B = so that in particular r A r A!B = and if we dene! i A = R i! A then we can choose! i A = for negative i. We dene! A = X i=! A i i : (3.3) We can similarly dene ~! A by ~! A = ~w A and choose ~! A i = for i >. Note that! A and ~! A are solutions of L A holomorphic around = and = respectively and they can be chosen so that they extend to a neighbourhood of the unit disc and a neighbourhood of the complement of the unit disc and can therefore be used to provide a patching description of the twistor space. 3.3 The Penrose transform of linearised deformations and the recursion operator The recursion operator acts on linearised perturbations of the ASDVE equations. Under the twistor correspondence, these correspond to linearised holomorphic deformations of (part of) PT. Cover PT by two sets, U and U ~ with jj < + on U and jj >? on U ~ with (! A ; ) coordinates on U and (~! A ;? ) on U. ~ The twistor space PT is then determined by the transition function ~! B = ~! B (! A ; A ) on U \ U ~ which preserves the brewise -form, d! A ^ d! A j =const: = d~! A ^ d~! A j =const:. 8

10 Innitesimal deformations are given by elements of H (PT ; ), where denotes a sheaf of germs of holomorphic vector elds. Let Y = f A (! B ; A dened on the overlap U \ ~ U and dene a class in H (PT ; ) that preserves the bration PT 7! CP. The corresponding innitesimal deformation is given by ~! A (! A ; A ; t) = ( + ty )(~! A ) + O(t ): (3.3) From the globality of () = d! A ^d! A it follows that Y is a Hamiltonian vector eld with a Hamiltonian f H (PT ; O()) with respect to the symplectic structure. A nite deformation is given by integrating from t = to. Innitesimally we can put d~! B dt = ~! A : ~! ~! A : (3.3) If the ASD metric is determined by and then " B, (or more simply f) is a linearised deformation corresponding to W g. The recursion operator acts on linearised deformations as follows Proposition 3.4 Let R be the recursion operator dened by (3.8). Its twistor counterpart is the multiplication operator R f = f =? f: (3.33) [Note that R acts on f without ambiguity; the ambiguity in boundary condition for the denition of R on space-time is absorbed into the choice of explicit representative for the cohomology class determined by f.] Proof. Pull back f to the primed spin bundle on which it is a coboundary so that f( A ; x a ) = h( A ; x a )? ~ h( A ; x a ) (3.34) where h and ~ h are holomorphic on U and ~ U respectively (here we abuse notation and denote by U and ~U the open sets on the spin bundle that are the preimage of U and ~ U on twistor space). A choice for the splitting (3.34) is given by h = ~h = I i? i I ~? ( A o A ) 3 ( C C )( B o B ) 3 f( E ) D dd ; (3.35) ( A o A ) 3 ( C C )( B o B ) 3 f( E ) D dd : Here A are homogeneous coordinates of CP pulled back to the spin bundle. The contours? and ~? are homologous to the equator of CP in U \ ~ U and are such that?? ~? surrounds the point A = A. The functions h and ~ h are homogeneous of degree in A and do not descend to PT, whereas their dierence does so that A r AA h = A r AA ~ h = A B C AA B C (3.36) where the rst equality shows that the LHS is global with homogeneity degree and implies the second equality for some AA B C which will be the third potential for a linearised ASD Weyl spinor. AA B C is in general dened modulo terms of the form r A(A B C ) but this gauge freedom is partially xed by choosing the integral representation above; h vanishes to third order at A = o A and direct dierentiation, using r AA f = A f A for some f A, gives AA B C = o A o B o C r A where = i I? f ( B o B ) 4 D dd : (3.37) This is consistent with the Plebanski gauge choices (there is also a gauge freedom in arising from cohomology freedom in f which we shall describe in the next subsection.) The condition r A(D A A B C ) = follows from equation (3.36) which, with the Plebanski gauge choice, implies W g. Thus we obtain a twistor integral formula for the linearisation of the second heavenly equation. 9

11 Now recall formula (3.8) dening R. Let Rf be the twistor function corresponding to R by (3.37). The recursion relations yield I so Rf =? f.? Rf A ( B o B ) 3 D dd = I? f A ( B o B ) ( B B ) D dd Let be the linearisation of the rst heavenly potential. From R = it follows that = i 3.4 Hidden symmetry algebra I? f ( A o A ) ( B B ) C dc : The ASDVE equations in the Plebanski forms have a residual coordinate symmetry. This consists of area preserving dieomorphisms in the w A coordinates together with some extra transformations that depend on whether one is reducing to the rst or second form. By regarding the innitesimal forms of these transformations as linearised perturbations and acting on them using the recursion operator, the coordinate (passive) symmetries can be extended to give `hidden' (active) symmetries of the heavenly equations. Formulae (3.37) and (3.33) can be used to recover the known relations (see for example [8]) of the hidden symmetry algebra of the heavenly equations. We deal with the second equation as the case of the rst equation was investigated by other methods []. Let be a volume preserving vector eld on. Dene r AA := [; r AA ]. This is a pure gauge transformation corresponding to addition of L g to the space-time metric and preserves the eld equations. Note that [; N]r AA := [;N]r AA : Once a Plebanski coordinate system and reduced equations have been obtained, the reduced equation will not be invariant under all the SDi() transformations. The second form will be preserved if we restrict ourselves to transformations which preserve the SD two-forms = dw A ^dw A and = dx A^dw A. The conditions L = L = imply that is given A +? x h A where h = h(w A ) and g = g(w A ). The space-time is now viewed as a cotangent bundle = T N with w A being coordinates on a two-dimensional complex manifold N. The full SDi() symmetry breaks down to the semi-direct product of SDi(N ), which acts on by a Lie lift, with?(n ; O) which acts on by translations of the zero section by the exterior derivatives of functions on N. Let correspond to r AA by The `pure gauge' elements are r @x B B = F + x A G + x A x B + 3 h x A x B x @w A A (3.38) where F; G A ; g; h are functions of w B only. The above symmetries can be seen to arise from symmetries on twistor space as follows. Since we have the symplectic form = d! A ^ d! A on the bres of : PT?! CP, a symmetry is a holomorphic dieomorphism of the set U that restricts to a canonical transformation on each bre. Let H = H(x a ; ) = P i= h i i be the Hamiltonian for an innitesimal such transformation pulled back to the projective spin bundle. The functions h i depend on space time coordinates only. In particular h and h give h and g from the previous construction (3.38). This can be seen by calculating how transforms if! A = w A + x A A + :::?! ^! A. Now is treated as an object on the rst jet bundle of a xed bre of PT and it determines the structure of the second jet. These symmetries take a solution to an equivalent solution. The recursion operator can be used to dene an algebra of `hidden symmetries' that take one solution to a dierent one as follows. Let be an expression of the form (3.38) which also satises g =. We set i := R i W g :

12 Proposition 3.5 Generators of the hidden symmetry algebra of the second heavenly equation satisfy the relation [ i ; N j ] = [;N] i+j : (3.39) Proof. This can be proved directly by showing that the ambiguities in R can be chosen so that R = R. It is perhaps more informative to prove it by its action on twistor functions. Let i f be the twistor function corresponding to i (by (3.37)) treated as an element of?(u \ ~U; O()) rather than H (PT ; O()). Dene [ i ; j N ] by [; i j N ] := i I f i f; j N fg ( ) 4 A d A where the Poisson bracket is calculated with respect to a canonical Poisson structure on PT. Proposition (3.33) it follows that as required. [ i ; j N ] = i I?i?j f f; N fg ( ) 4 A d A = R i+j [;N] From 3.5 Recursion procedure for Killing spinors Let (; g) be an ASD vacuum space. We say that L A :::A n is a Killing spinor of type (; n) if r A (A L B :::B n ) = : (3.4) Killing spinors of type (; n) give rise to Killing spinors of type (; n? ) by r A A L B :::B n = " A (B KA B :::B n ) : In an ASD vacuum, K BB :::B n is also a Killing spinor Put (for i = ; :::; n) r (A (A KB) B :::B n ) = : L i := B ::: B io B i+ :::o B nl B :::B n ; and contract (3.4) with B ::: Bi o B i+ :::o B n+ to obtain We make use of the recursion relations (3.8): ir A L i? =?(n? i + )r A L i ; i = ; :::; n? :?i n +? i R(L i?) = L i : This leads to a general formula for Killing spinors (with r A L = )? L i = (?) i n X n R i i (L ); L B = o B :::B n (B :::o B i B i+ ::: B n )L i (3.4) i= and equation (3.4) is then satised i R? L = RL n =. 3.6 Example Let us demonstrate how to use the recursion procedure to nd metrics with hidden symmetries. tn := n be a linearisation of the rst heavenly equation. We have R : z?! w t. Look for solutions to (.8) with an additional t =. The recursion relations (3.8) imply wz = ww =, therefore (w; z; ~w; ~z) = wq( ~w; ~z) + P (z; ~w; ~z): The heavenly equation yields dq ^ dp ^ dz = d~z ^ d ~w ^ dz. With the z P = p the metric is ds = dwdq + dzdp + fdz ; where f =?P zz. We adopt (w; z; q; p) as a new coordinate system. Heavenly equations imply that f = f(q; z) is an arbitrary function of two variables. These are the null ASD plane wave solutions.

13 3.7 Example Now we shall illustrate the Propositions 3.3 and 3.4 with the example of the Sparling{Tod solution [7]. The coordinate formulae for the pull back of twistor functions are: Consider = w + y? x + 3 z + ::: ; = z? x? y? 3 w + ::: : (3.4) = wx + zy ; (3.43) where = const. It satises both the linear and the nonlinear part of (.). The at case: First we shall treat (3.43), with =, as a solution to the wave equation on the at background. The recursion relations are (R ) x = y (wx + zy) ; (R ) y =?x (wx + zy) : They have a solution := R = (?y=w). ore generally we nd that n := R n y n =? w wx + zy : (3.44) The last formula can be also found using twistor methods. The twistor function corresponding to is =( ), where = w + y and = z? x. By Proposition 3.33 the twistor function corresponding to n is?n =( ). This can be seen by applying the formula (3.37) and computing the residue at the pole =?w=y. It is interesting to ask whether any n (apart from ) is a solution to the heavenly equation. Inserting = n to (.) yields n = or n =. We parenthetically mention that yields (by formula (.3)) a metric of type D which is conformal to the Eguchi-Hanson solution. The curved case. Now let given by (3.43) determine the curved metric The recursion relations are where satises ds = dwdx + dzdy + 4(wx + zy)?3 (wdz? zdw) : y (R) = (@ w? y + x );?@ x (R) = (@ z + y? x )?@ x (R ) = (@ z + w(wx + zy)?3 (w@ x? z@ y )) y (R ) = (@ w + z(wx + zy)?3 (w@ x? z@ y )) ; = (@ w z + (wx + zy)?3 x + y? wz@ y )) = : (3.46) One solution to the last equation is = (wx + zy)?. We apply the recursion relations to nd the sequence of linearised solutions y =? w wx + zy ; 3 =? 3 (wx + zy) + y? 3 w wx + zy ; :::; nx n = A k y k(wx (n)? + zy) k?n : w k= To nd A k note that the recursion relations imply (n) R = y k(wx? + zy) j = w This yields a recursive formula y y??? w w?(wx + zy)? k j + y k(wx? + zy) j : w A k = (n+) Ak?? k + (n) n? k + Ak+; (n) A = ; () A () = ; A? = ; k = :::n; (3.47) (n)

14 which determines the algebraic (as opposed to the dierential) recursion relations between n and n+. It can be checked that functions n indeed satisfy (3.46). Notice that if = (at background) then we recover (3.44). We can also nd the inhomogeneous twistor coordinates pulled back to F X X n = w + y + n+ B k w y k(wx (n)? + zy) k?n? ; w n= k= where = z? x + X n= n+ n X k= B k (n) z x z k(wx + zy) k?n? : B k = (n+) Bk?? k + (n) n? k + Bk+; (n) B = ; () B () = ; B? = ; k = :::n: (n) The polynomials A solve L A ( B ) =, where now L =?@ w? z (wz + x + ( + wz(wz + zy)?3 )@ y ; L z + (? wz(wz + zy)?3 )@ x + w (wz + zy)?3 )@ y : 4 Hierarchies for the ASD vacuum equations The hidden symmetries corresponding to higher ows associated to translations along the coordinate vector elds give `higher ows' of a hierarchy. This yields a hierarchy of ows of the anti-self-dual Einstein vacuum equations. We rst give this for the equations in their second heavenly form but then give the equations in the form of consistency conditions for a Lax system of vector elds generalizing equations.. The nonlinear graviton construction generalizes to give a construction for the corresponding system of equations and is presented in xx4.. In xx4.3 the geometric structure of solutions to the truncated hierarchy are explored in further detail. Finally in xx4.4 innitesimal deformations are studied. 4. Hierarchies for the heavenly equations The generators of higher ows are rst obtained by applying powers of the recursion operator to the linearised perturbations corresponding to the evolution along coordinate vector elds. This embeds the second heavenly equation into an innite system of over-determined, but consistent, PDEs (which we will truncate at some arbitrary but nite level). These equations in turn can be naturally embedded into a system of equations that are the consistency conditions for an associated linear system that extends (.). We shall discuss here the hierarchy for the second Plebanski form; that for the rst arises from a dierent coordinate and gauge choice. Introduce the coordinates x Ai, where for i = ; ; x Ai = x AA are the original coordinates on, and for < i n; x Ai are the parameters for the new ows (with n? dimensional parameter space X). The propagation of along these parameters is determined by the recursion y (@ Bi+ ) = (@ w? y + x )@ Bi ;?@ x (@ Bi+ ) = (@ z + y? x )@ Bi ; A (@ Bi+ ) = (@ A C )@ Bi : (4.48) However, we will take the hierarchy to be the system (containing the above when j = Ai? + f@ Ai? Bj? g yx = ; i; j = :::n: (4.49) Here f:::; :::g yx is the Poisson bracket with respect to the Poisson A A x y. Lemma 4. The linear system for equations (4.49) is where L Ai s = (?D Ai+ + Ai )s = ; i = ; :::; n? ; (4.5). s := s(x Ai ; ) is a function on a spin bundle (a CP -bundle) over N = X,. D Ai+ Ai+ + [@ Ai ; V ], (V = " B ) and Ai Ai are 4n vector elds on N. 3

15 Proof. This follows by direct calculation. The compatibility conditions for (4.5) are: [D Ai+ ; D Bj+ ] = ; (4.5) [ Ai ; Bj ] = ; (4.5) [D Ai+ ; Bj ]? [D Bj+ ; Ai ] = : (4.53) It is straightforward to see that equations (4.5) and (4.53) hold identically with the above denitions and (4.5) is equivalent to (4.49). As a converse to this lemma, we will see in xx4. using the twistor correspondence, that given the Lax system above, in which the vector elds D Ai and Aj are volume preserving vector elds, then coordinate and gauge choices can be made so that the Lax system takes on the above form. 4.. Spinor notation The above can also be represented in a spinorial formulation that will be useful later. We introduce the spinor indexed coordinates x AA :::A n = x A(A :::A n ) on N which correspond to the x Ai by x Ai n = x AA A :::A no A i :::o A i A i+ ::: A n (?) n?i : The vector elds D Ai+ and Ai are then represented by the 4n vector elds on N, D AA (A :::A n ) where D AA i = A ::: A io A i+ :::o A nd AA (A :::A n ) ; D A i = D Ai+ ; D A i = Ai and L A(A :::A n ) = A DAA (A :::A n ) ; L Ai = A DAA i. In the adopted gauge D A A :::A n A A :::A n ; D A A :::A n A A :::A n + [@ A A :::A n ; V ]: In what follows we will often be interested in r A(A A :::A n ), the symmetric part of D AA A :::A n. Put D A ::: A. The n + vector elds span T N. r Ai = D A(A A :::A n ) A ::: A io A i+ :::o A n (4.54) = n (id A i? + (n? i)d A i) Ai + i n [@ Ai?; V ]: (4.55) r AA :::A n = f@ A; r A A :::A n? ; D Ang 4. The twistor space for the hierarchy The twistor space PT for a solution to the hierarchy associated to the Lax system on N as above is obtained by factoring the spin bundle N CP by the twistor distribution (Lax system) L Ai. This clearly has a projection q : N CP 7! PT and we have a double bration N p. N CP & q PT Since the twistor distribution is tangent to the bres of N CP 7! CP, twistor space inherits the projection : PT 7! CP. The twistor space for the hierarchy is three-dimensional as for the ordinary hyper-kahler equations, but has a dierent topology. We have Lemma 4. The holomorphic curves q(cp x) where CP x = p? x, x N, have normal bundle N = O(n) O(n). 4

16 Proof. To see this, note that N can be identied with the quotient p (T x N )=fspanl Ai g, i = ; : : :; n. In their homogeneous form the operators L Ai have weight, so the distribution spanned by them is isomorphic to the bundle C n O(?). The denition of the normal bundle as a quotient gives 7! C n O(?) 7! C n+ 7! N 7! and we see, by taking determinants that the image is O(n + a) O(n? a) for some a. We see that a = as the last map, in the spinor notation introduced at the end of the last section, is given explicitly by V AA :::A n 7! V AA :::A n A : : : A n clearly projecting onto O(n) O(n). A nal structure that PT possesses is a skew form taking values in O(n) on the bres of the projection. This arises from the fact that the vector elds of the distribution preserve the coordinate volume form on N in the given coordinates system. Furthermore, the Lax system commutes exactly [L ai ; L Bj ] = so that = (; ; L ; : : :; L n ; L ; : : :; L n ) descends to the bres of PT 7! CP and clearly has weight n as each of the L Ai has weight one. Thus we see that, given a solution to the hyperkahler hierarchy in the form of a commuting Lax system, we can produce a twistor space with the above structures. Now we shall prove the main result of this section and demonstrate that, given PT, with the above structures, we can construct N (as the moduli space of rational curves in PT ) which is naturally equipped with a function satisfying (4.49) and with the Lax distribution (4.5). Proposition 4.3 Let PT be a 3 dimensional complex manifold with the following structures ) a projection : PT?! CP, ) a section s : CP 7! PT of with normal bundle O(n) O(n), 3) a non-degenerate -form on the bres of, with values in the pullback from CP of O(n). Let N be the moduli space of sections that are deformations of the section s given in (). n + dimensional and Then N is a) There exists coordinates, x Ai, A = ;, and i = ; : : :; n and a function : N?! C on N such that equation (4.49) is satised. b) The moduli space N of sections is equipped with { a factorisation of the tangent bundle TN = S A n S A, { a n-dimensional distribution on the `spin bundle' D T (N CP ) that is tangent to the bres of r over CP and, as a bundle on N CP has an identication with O(?) S AA :::A so n? that the linear system can be written as in equation (4.5). This correspondence is stable under small perturbations of the complex structure on PT preserving () and (3). Proof: The rst claim, that N has dimension n + follows from Kodaira theory as dimh (CP ; N) = n + and dimh (CP ; N) = dimh (CP ; EndN) =. Proof of (a): we rst start by dening homogeneous coordinates on PT. These are coordinates on T, be homogeneous the total space of the pullback from CP of the tautological line bundle O(?). Let A coordinates on CP pulled back to T and let! A be local coordinates on T chosen on a neighbourhood of? f = g that are homogeneous of degree n and canonical so that = " AB d! A ^ d! B. We also use = = as an ane coordinate on CP. Let L p be the line in PT that corresponds to p N and let PT lie on L p. We denote by F the correspondence space PT Nj Lp = N CP. (See gure for the double bration picture.) Pull back the twistor coordinates to F and dene (n + ) coordinates on N by x A(A A :::A n ) := where the derivative is along the bres of F over n! A :::@ A n A =o A ; This can alternatively be expressed in ane coordinates on CP by expanding the coordinates! A pulled back to F in powers of = = :! A = ( ) n n X i= x Ai n?i + n+ X i= s A i i! ; (4.56) 5

17 N λ Figure : Double bration. dim F = n+3 L p q dim PT= 3 p r L p p dim N = n+ λ CP where the s A i are functions of x AA :::A n and will be useful later. The symplectic -form on the bres of, when pulled back to the spin bundle, has expansion in powers of that truncates at order n + by globality and homogeneity, so that = d h! A ^ d h! A = A ::: A n B ::: B n A :::A nb :::B n for some symmetric spinor indexed -form A :::A n B :::B n. We have () ^ () = ; d h () = : (4.57) where in the exterior derivative d h, is understood to be held constant. If we express the forms in terms of the x Ai and the s A i, the closure condition is satised identically, whereas the truncation condition will give rise to equations on the s A i allowing one to express them in terms of a function (x AA :::A n ) and to eld equations on as follows. To deduce the existence of (x AA :::A n ) observe that the vanishing of the coecient of n+ in d! A ^ d! A gives nx nx ds Ai ^ dx Ai = d s Ai dx Ai = =) s Ai : i= i= The equations of the hierarchy arise from the vanishing of the coecient of n+ nx i= dx Ai ^ ds i+ A + dsa ^ ds A = : This leads to the equations (4.49) on for i; Ai + and further equations that determine s @x Bj = Proof of b). The isomorphism TN = S A n S A follows simply from the structure of the normal bundle. From Kodaira theory, since the appropriate obstruction groups vanish, we have T x N =?(CP x; N x ) = S A x n S A (4.58) where N x is the normal bundle to the rational curve CP x in PT corresponding to the point x N. The bundle S A on space-time is the Ward transform of O(?n) T V PT where the subscript V denotes the 6

18 sub-bundle of the tangent bundle consisting of vectors up the bres of, the projection to CP, so that Sx A =?(CP x ; O(?n) T V PT ). The bundle S A =?(CP ; O()) is canonically trivial. Let r AA A = r n A(A be the indexed vector eld that establishes the isomorphism (4.58) and A n ) let e AA A n = e A(A A n ) S A n S A be the dual (inverse) map. We now wish to derive the form of the linear system, equations (4.5). For each xed A = (; ) CP we have a copy of a space-time N. The horizontal (i.e. holding constant) subspace of T (x;) (N CP ) is spanned by r A(A :::A n ). An element of the normal bundle to the corresponding line CP x consists of a a horizontal tangent vector at (x; ) modulo the twistor distribution. Therefore we have the sequence of sheaves over CP?! D x?! T x N?! ea S A O(n)?! ; where D x is the twistor distribution at x and the map T x N?! S A O(n) is given by the contraction of elements of T x N with e A := e AA :::A n A ::: A n since e A annihilates all L Bi s in D. Consider the dual sequence tensored with O(?) to obtain?! O A (?n? )?! T xn(?)?! D x(?)?! : (4.59) From here we would like to extract the Lax distribution L AA :::A n = A DAA A :::A n S AA :::A n O() D: This can be achieved by globalising (4.59) in A groups yields. The corresponding long exact sequence of cohomology?!?(o A (?n? ))?!?(T N (?))?!?(D (?))?! H (T N (?))?! :::?! H (O A (?n? )) which (because T N is a trivial bundle so that O(?) T N has no sections or cohomology) reduces to?!?(d (?))?! H (O A (?n? ))?! : From Serre duality we conclude, since D has rank n, that the connecting map is an isomorphism :?(D (?))?! S AA :::A n. Therefore is a canonically dened object annihilating! A given by (4.56). In index notation we can put?(d O() S AA :::A n ) (4.6) = L AA :::A n = A DAA A :::A n ; where L AA :::A n = L A(A :::A n ), the second identity follows from the globality of L AA :::A n are vector elds on N lifted to N CP using the product structure. It follows from L AA :::A!B = that if A = o A then D n A A :::A xbn = so n D AA A :::A n and the D A A :::A n = ABB :::B A A :::A B B :::B n for some matrix A BB :::Bn A A This matrix must be invertible by dimension counting. By multiplying n. by the inverse :::A of this matrix, we nd we can put L AA :::A n A BB :::B n A A :::A n = " B A" B A :::" B n A : n Therefore we can take L AA :::A n A A? D :::A n A A. Equating the (n? i + )th and (n + )th :::A n powers of in L Ai! B = to zero yields D A A :::A n A A :::A n + [@ A A :::A n ; V ] ; where V = " B. So nally L AA :::A n is of the form L Ai Ai? (@ Ai+ + [@ Ai ; V ]). 7