In citeB3 Bridgeland defined a geometric structure named a Joyce structure conjectured to exist on the space M of stability conditions of a CY3 triangulated category. Given a nondegeneracy assumption a feature of this structure is a complex hyperKhler metric with homothetic symmetry on the total space X  TM of the holomorphic tangent bundle. par Generalising the isomonodromy calculation which leads to the A2 Joyce structure in citeBM we obtain an explicit expression for a hyperKhler metric with homothetic symmetry via construction of the isomonodromic flows of a Schrdinger equation with deformed polynomial oscillator potential of odd degree 2n1. The metric is defined on a total space X of complex dimension 4n and fibres over a 2ndimensional manifold M which can be identified with the unfolding of the A2nsingularity. The hyperKhler structure is shown to be compatible with the natural symplectic structure on M in the sense of admitting an textitaffine symplectic fibration as defined in citeBS. par Separately using the additional conditions imposed by a Joyce structure we consider reductions of Plebaskis heavenly equations that govern the hyperKhler condition. We introduce the notion of a textitprojectable hyperLagrangian foliation and show that in dimension four such a foliation of X leads to a linearisation of the heavenly equation. The hyperKhler metrics constructed here are shown to admit such a foliation.
['Mathematics - Differential Geometry', 'General Relativity and Quantum Cosmology', 'High Energy Physics - Theory', 'Mathematics - Algebraic Geometry', 'Nonlinear Sciences - Exactly Solvable and Integrable Systems']
Heavenly metrics hyperLagrangians and Joyce structures
{'MACIEJ DUNAJSKI AND TIMOTHY MOY': "Abstract. In [11], Bridgeland defined a geometric structure, named a Joyce structure, conjectured to exist on the space M of stability conditions of a CY 3 triangulated category. Given a non-degeneracy assumption, a feature of this structure is a complex hyperKahler metric with homothetic symmetry on the total space X = TM of the holomorphic tangent bundle.  \nGeneralising the isomonodromy calculation which leads to the A 2 Joyce structure in [13], we obtain an explicit expression for a hyper-Kahler metric with homothetic symmetry via construction of the isomonodromic flows of a Schrodinger equation with deformed polynomial oscillator potential of odd degree 2 n + 1. The metric is defined on a total space X of complex dimension 4 n and fibres over a 2 n -dimensional manifold M which can be identified with the unfolding of the A 2 n -singularity. The hyper-Kahler structure is shown to be compatible with the natural symplectic structure on M in the sense of admitting an affine symplectic fibration as defined in [14].  \nSeparately, using the additional conditions imposed by a Joyce structure, we consider reductions of Pleba'nski's heavenly equations that govern the hyper-Kahler condition. We introduce the notion of a projectable hyper-Lagrangian foliation and show that in dimension four such a foliation of X leads to a linearisation of the heavenly equation. The hyper-Kahler metrics constructed here are shown to admit such a foliation.", '1. Introduction': "In [11] it was proposed that the Donaldson-Thomas invariants of a three-dimensional Calabi-Yau triangulated ( CY 3 ) category can be encoded in a geometric structure on its space M of stability conditions. This structure, named a Joyce structure may be outlined as follows: The complex manifold M carries a closed holomorphic 2-form taking rational values on a system of lattices in the holomorphic tangent bundle X = TM . Assuming the 2-form is symplectic (which we will do in the sequel) there is a compatible complex hyper-Kahler structure on the total space X . In [14] it was shown how this hyper-Kahler structure is encoded in a solution to an over-determined system of nonlinear PDEs for a single function Θ. Additional homogeneity and lattice invariance conditions must be imposed on Θ so that it defines a Joyce structure.  \nIn the lowest-dimensional case of interest, where X is a complex four-manifold, the system reduces to a single PDE which is Pleba'nski's second heavenly equation for antiself-dual, Ricci-flat complex metrics [35]. In this case the homogeneity condition is a conformal symmetry reduction in a class studied in [23].  \nIn [15] Bridgeland and Smith realise spaces of stability conditions of CY 3 triangulated categories associated to certain quivers as spaces of quadratic differentials on Riemann surfaces. This motivates constructions of Joyce structures starting from the data of a space of quadratic differentials. The base of the Joyce structure should be M = Quad( γ, µ ), the moduli space of Riemann surfaces of genus γ equipped with meromorphic quadratic differential with l poles of orders µ = { m 1 , ..., m l } and simple zeroes.  \nThere are such constructions for holomorphic quadratic differentials ( µ = ∅ ) in [8] and in the genus zero case, for µ = { 7 } in [13].  \nThe latter construction of [13] proceeds by calculating the isomonodromic flows for a family of second-order linear ODE. The ODE can be thought of as deformations of Schrodinger's equation with a cubic oscillator potential and are specified by points in the manifold X = TM . Interpreting the flows as a Lax pair on X × CP 1 gives rise to a complex hyper-Kahler metric on X .  \nA primary aim of this article is to generalise this construction. We construct the isomonodromic flows corresponding to a family of ODEs with deformed polynomial potential of general odd degree 2 n + 1. This allows us to explicitly construct a complex hyper-Kahler metric with homothety on a complex manifold X . Each point in X specifies an ODE in the family, and X fibres over M = Quad(0 , { 2 n +5 } ), the moduli space of quadratic differentials on CP 1 with a pole of order 2 n + 5 and simple zeroes. The hyper-Kahler metric is compatible with the symplectic form ω on M induced [8] by the cohomology intersection forms on H 1 (Σ p , C ) where we associate to each point p ∈ M a hyper-elliptic curve Σ p . If n = 1, then M identified with the universal unfolding of the A 2 singularity carries a Frobenus structure in the sense of Dubrovin [18] which can be constructed directly from the complex hyper-Kahler structure.  \nThe structure of the article is as follows: § 2 is a review of complex hyper-Kahler structures: 4 n -complex dimensional manifolds X with a holomorphic Riemannian metric g , and a triple of holomorphic, Hermitian, endomorphisms I, J, K of TX satisfying the quaternion relations and parallel for the Levi-Civita connection of g . We do this, taking the spinorial approach of Bailey and Eastwood to paraconformal structures , introduced in [7]. We formulate the hyper-Kahler condition as two equivalent systems of PDEs of a single function which generalise Pleba'nski's first and second heavenly equations [35]. We show that the second system can be interpreted as an infinitesimal version of the first. We explain how a Lax distribution on X × CP 1 defines a paraconformal structure with an associated family of metrics and give sufficient conditions that this family contains a hyper-Kahler metric.  \nIn § 3 -§ 5 we consider complex hyper-Kahler metrics with extra symmetry. The discussion is independent of the construction of the complex hyper-Kahler metrics presented in § 6 -§ 8. In § 3 we introduce the notion of a projectable hyper-Lagrangian foliation .  \nDefinition 1.1 (Projectable hyper-Lagrangian foliation) . Let ( X,g ) be a complex hyperKahler structure of dimension 4 n for complex structures I, J, K . An integrable rank2 n distribution B ⊆ TX is called a hyper-Lagrangian foliation if B is Lagrangian with respect to the holomorphic symplectic forms associated to endomorphisms I, J, K .  \nThe hyper-Lagrangian foliation is called projectable if, writing π : X → M , for the natural fibration over M = X/ ker N where N = ( J -iK ) / 2 there exists a rankn foliation L of M with dπ ( B ) = L .  \nWe show such foliations have a characterisation in terms of the existence of adapted local coordinates as follows:  \nTheorem 1.2. The following conditions are equivalent for a complex hyper-Kahler manifold ( X,g )  \n- (1) X is foliated by projectable hyper-Lagrangian submanifolds.\n- (2) At at each point of M there exists a local coordinate system ( x i , y i ) , such that the second Pleba'nski potential corresponding to the metric g is at most quadratic in half of the fibre coordinates in the fibration π : X → M .  \nIf n = 1 then the conditions (1) and (2) are equivalent to the existence of a two-parameter family of anti-self-dual null surfaces in X which push down to curves in M .  \nWe recall the definition and relevant properties of Joyce structures in § 4. In § 5 we present a number of new reductions and solutions of the heavenly equations in four dimensions with relevance to Joyce structures. We consider the case of a projectable hyper-Lagrangian foliation. We obtain:  \nTheorem 1.3. Suppose a complex hyper-Kahler manifold ( X,g ) of complex dimension four is foliated by projectable hyper-Lagrangian submanifolds. Then in the adapted coordinates of Theorem 1.2 the heavenly equation for the Pleba'nski potential reduces to a system which is linearisable by a contact transformation.  \nWe also consider the stronger condition of the foliation pushing down to all twistorfibres. In this case we can solve the first heavenly equation and give the general form of the metric in terms of two unconstrained holomorphic functions. We consider the case where the vector field generating the homothetic symmetry of a Joyce structure is null, in which case we show the metric coincides with a generalisation of the H -space of Sparling and Tod [37].  \nIn § 6 we review the isomonodromy problem for the deformed cubic oscillator and explain how it is related to one studied by Okamoto [31] and is associated with the Painlev'e I equation. We use the Lax formulation of the hyper-Kahler condition to construct the explicit form of the hyper-Kahler metric in this case. In § 7 we shall consider a deformed polynomial quantum oscillator governed by the Schrodinger equation  \n(1.1) /planckover2pi1 2 d 2 y dx 2 = Q ( x ) y, where Q ( x ) := Q 0 ( x ) + /planckover2pi1 Q 1 ( x ) + /planckover2pi1 2 Q 2 ( x ) .  \nHere  \nQ 0 ( x ) = x 2 n +1 + c 2 n -1 x 2 n -1 + · · · + c 0  \ncorresponds to a quadratic differential Q 0 ( x ) dx 2 on CP 1 with a pole of order 2 n +5 at ∞ paramerised by a point with coordinates c 0 , . . . , c 2 n -1 in the 2 n -dimensional symplectic manifold M = Quad(0 , 2 n +5). The first deformation term is  \nQ 1 ( x ) = n ∑ i =1 p i x -q i + R ( x )  \nwhere R ( x ) is the general polynomial of degree at most n -1. Finally the trailing term Q 2 ( x ) is specified so that the ODE (1.1) has 'square-root-like' monodromy at the regular singularity q i .  \nWe shall compute the 2 n linearly-independent isomonodromic flows of (1.1), and show that dependence of the isomonodromic flows on a spectral parameter λ = /planckover2pi1 defines a (2 n, 2)-paraconformal structure on the 4 n -dimensional manifold X of potentials Q ( x ) in (1.1). To every such paraconformal structure is associated a family of metrics. In § 8 we will distinguish a complex hyper-Kahler metric in this family by insisting that it is compatible with the natural symplectic structure on M .  \nTheorem 1.4. Let M = Quad(0 , 2 n +5) be the 2 n -complex-dimensional manifold of quadratic differentials differentials on CP 1 with a single pole of order 2 n + 5 and simple zeros, equipped with its canonical symplectic form ω , and let X be the 4 n -complexdimensional holomorphic manifold parametrising the choices of Q 0 ( x ) and Q 1 ( x ) in the Schrodinger equation (1.1). The family of metrics associated to the paraconformal structure on X defined by the isomonodromic flows of (1.1) contains a complex hyper-Kahler  \nmetric g such that π ∗ ω = g (( J + iK ) / 2 , · ) for some compatible complex structures J, K on M . This metric admits a homothetic Killing vector field, and a projectable hyperLagrangian foliation.  \nAcknowledgements. The authors would like to thank Tom Bridgeland for very helpful suggestions and also the anonymous referees for their detailed comments which have lead to improvements in the manuscript. Timothy Moy is supported by Cambridge Australia Scholarships.", '2. Paraconformal structures and complex hyper-Kahler metrics': "- 2.1. Paraconformal structures. Let X be a complex manifold of complex dimension 4 n . Denote by TX the holomorphic tangent bundle. A complex hyper-Kahler structure is a triple of holomorphic endomorphisms I, J, K of TX satisfying the quaternion relation  \nI 2 = J 2 = K 2 = IJK = -Id TX  \nwhich are Hermitian for a holomorphic metric g on X and parallel for its Levi-Civita connection.  \nComplex hyper-Kahler metrics fit into the framework of paraconformal structures [7] which we now define  \nDefinition 2.1 (Paraconformal structure) . A (2 n, 2) -paraconformal structure on X is an isomorphism  \nTX ∼ = S ⊗ S ' (2.1)  \nwhere S and S ' are holomorphic vector bundles of rank 2 n and rank 2 respectively.  \nAssociated to a paraconformal structure is the family of holomorphic compatible metrics of the form  \ng = e ⊗ /epsilon1 (2.2)  \nwhere e, /epsilon1 are non-degenerate skew-forms on S and S ' respectively. Since /epsilon1 is determined up to scale, we may fix this scale and a choice of compatible metric becomes a choice of non-degenerate skew-form e on S .  \nDefinition 2.2 (Null structure) . A null structure on an 4 n -complex dimensional manifold X with holomorphic metric g is an endomorphism N : TX → TX such that  \nN 2 = 0 , g ( N · , · ) = -g ( · , N · ) , rank(N) = 2 n.  \nThe null structure is called null-Kahler [19] if N is parallel for the Levi-Civita connection of g .  \nA section o ∈ Γ( S ' ) defines a null structure by  \nN := Id S ⊗ /epsilon1 ( o, · ) ⊗ o. (2.3)  \nGiven an additional section ι ∈ Γ( S ' ) for which /epsilon1 ( o, ι ) = 1, we may define endomorphisms of TX  \nI = -Id S ⊗ i/epsilon1 ( o, · ) ⊗ ι -Id S ⊗ i/epsilon1 ( ι, · ) ⊗ o (2.4) J = Id S ⊗ /epsilon1 ( o, · ) ⊗ o +Id S ⊗ /epsilon1 ( ι, · ) ⊗ ι K = Id S ⊗ i/epsilon1 ( o, · ) ⊗ o -Id S ⊗ i/epsilon1 ( ι, · ) ⊗ ι  \nwhich are by construction Hermitian for g . We define an associated non-degenerate 2form Ω I := g ( I · , · ) with Ω J and Ω K defined similarly. Note that  \nN = 1 2 ( J -iK ) .  \nSuppose that there exists connections on S and S ' which preserve e, /epsilon1 and for which the induced connection on TX is the Levi-Civita connection for g . Suppose also that the section o is parallel. Then N together with g is the general null-Kahler structure as defined in [19]. Next, if ι is also parallel, so the connection on S ' is flat, then I, J, K together with g define a complex hyper-Kahler structure. Every complex hyper-Kahler structure (locally) arises this way [14, 19].  \n2.2. Notation. Unless otherwise specified we will employ the Einstein summation convention. Unprimed lower case indices will take values in 1 , ..., 2 n whereas primed indices take values in 0 , 1. We define skew-symmetric matrices with components η ij and /epsilon1 i ' j ' by  \nη ij = 0 , i, j ≤ n η ij = 0 , i, j > n η i ( j + n ) = δ ij , i, j ≤ n, equivalently η = [ 0 Id n -Id n 0 ]  \nand  \n/epsilon1 0 ' 1 ' = 1 .  \nWe use the convention of [33], in that the components η ij , /epsilon1 i ' j ' of the inverses are defined by the relations  \n(2.5) η ik η jk = δ i j , /epsilon1 i ' k ' /epsilon1 j ' k ' = δ i ' j '  \n.  \nWe define the symmetric tensor product according to: a /circledot b ≡ 1 2 ( a ⊗ b + b ⊗ a ).  \n2.3. Pleba'nski's heavenly equations. We now explain how to encode the complex hyper-Kahler condition in two equivalent system of PDEs. The following proposition goes back to Pleba'nski [35] if dim( X ) = 4, and [14, 19, 2] if dim( X ) > 4. We give a self-contained proof of it, which clarifies some of the issues with a freedom to modify the scalar potentials and reabsorb redundant functions if dim( X ) > 4.  \nProposition 2.3. Let ( X,g ) be a complex hyper-Kahler manifold of dimension 4 n . There exist local coordinate systems ( z j , ˜ z j ) and ( x j , y j ) , and functions U = U ( z, ˜ z ) , the first Pleba'nski potential, and Θ = Θ( x, y ) , the second Pleba'nski potential, such that the metric g takes the form  \n(2.6) g = ∂U ∂z j ∂ ˜ z k dz j /circledot d ˜ z k , where  \n∂ 2 U ∂z k ∂ ˜ z i ∂ 2 U ∂z l ∂ ˜ z j η kl = η ij , (2.7)  \nand  \ng = η ij dy i /circledot dx j + ∂ 2 Θ ∂y i ∂y j dx i /circledot dx j , where (2.8)  \n∂ 2 Θ ∂y i ∂x j -∂ 2 Θ ∂y j ∂x i -η kl ∂ 2 Θ ∂y i ∂y k ∂ 2 Θ ∂y j ∂y l = 0 . (2.9)  \nProof. Define the following 2-forms  \nΩ + := e ⊗ /epsilon1 ( ι, · ) ⊗ /epsilon1 ( ι, · ) (2.10)  \nΩ -:= e ⊗ /epsilon1 ( o, · ) ⊗ /epsilon1 ( o, · ) = g ( N · , · ) . (2.11)  \nwhich are closed in the hyper-Kahler case. The (integrable) kernels of these 2-forms are the ∓ i -eigenspaces of I respectively and the closure implies they descend to symplectic forms on the space of leaves of their respective kernels. The i -eigenspace coincides with the kernel of N . Darboux's theorem implies there exists coordinates ( z i , ˜ z i ), i = 1 , ..., 2 n for X such that  \nΩ + = 1 2 η ij d ˜ z i ∧ d ˜ z j Ω -= 1 2 η ij dz i ∧ dz j  \nwhere η ij is the standard symplectic matrix given by (2.5).  \nWe also have the symplectic form  \nΩ I = g ( I · , · ) = -2 ie ⊗ /epsilon1 ( o, · ) /circledot /epsilon1 ( ι, · ) = -i 2 U ij dz i ∧ d ˜ z j  \nwhere the U ij are some holomorphic functions to be determined. Ω I is locally exact and so we may write  \nU ij dz i ∧ d ˜ z j = d ( -φ i dz i + ψ i d ˜ z i )  \nand it follows  \nφ i = ∂φ ∂z i , ψ i = ∂ψ ∂ ˜ z i , U ij = ∂ 2 U ∂z i ∂ ˜ z j  \nwhere φ, ψ are holomorphic functions on X and U = φ + ψ . The metric is now given by (2.6). Similarly we may define symplectic forms Ω J := g ( J · , · ), Ω K := g ( K · , · ). In fact  \nΩ J = Ω + + Ω -= 1 2 η ij ( dz i ∧ dz j + d ˜ z i ∧ d ˜ z j ) .  \nThen J 2 = -Id TX is to say that (2.7) holds. This is a system of PDEs generalising Pleba'nski's first heavenly equation [35] for Ricci-flat anti-self-dual complex spacetimes in complex dimension four. In the same article Pleba'nski obtained another equivalent non-linear PDE of a single function which can also be generalised to a system. To obtain it, note that (2.7) implies that ( x i , y i ) give local coordinates on X , where x i := z i and y i is defined by  \nη ji y j = ∂U ∂z i .  \nWe have  \nη ji dy j = ∂ 2 U ∂z i ∂z k dz k + ∂ 2 U ∂z i ∂ ˜ z k d ˜ z k . (2.12)  \nDefine  \nΘ ij := -∂U ∂z i ∂z j  \nso that from (2.12) we see the metric is  \ng = η ij dy i /circledot dx j +Θ ij dx i /circledot dx j .  \nAlso from (2.12) one may calculate  \nΩ + = 1 2 η ij dy i ∧ dy j +Θ ij dx i ∧ dy j + 1 2 η kl Θ ik Θ jl dx i ∧ dx j .  \nFrom the closure of Ω + we obtain  \n∂ Θ ij ∂y k -∂ Θ ik ∂y j = 0  \nwhich combined with the symmetry of Θ ij implies there exists a holomorphic function ˜ Θ such that  \n∂ 2 ˜ Θ ∂y i ∂y j = Θ ij .  \nAlso from the closure  \n∂ ∂y m ( ∂ 2 ˜ Θ ∂y i ∂x j -∂ 2 ˜ Θ ∂y j ∂x i -η kl ∂ 2 ˜ Θ ∂y i ∂y k ∂ 2 ˜ Θ ∂y j ∂y l ) = 0 (2.13)  \n∂ ∂x m ( η kl ∂ 2 ˜ Θ ∂y i ∂y k ∂ 2 ˜ Θ ∂y j ∂y l ) dx m ∧ dx i ∧ dx j = 0 . (2.14)  \nThese two equations imply that  \n∂ 2 ˜ Θ ∂y i ∂x j -∂ 2 ˜ Θ ∂y j ∂x i -η kl ∂ 2 ˜ Θ ∂y i ∂y k ∂ 2 ˜ Θ ∂y j ∂y l = ∂ρ i ∂x j -∂ρ j ∂x i  \nfor some holomorphic functions ρ i ( x ) so that, defining Θ := ˜ Θ -y i ρ i ( x ) we deduce (2.9), together with the metric (2.8). /square  \nThe system (2.9) has been studied in, for example [38, 14, 19] and generalises Pleba'nski's second heavenly equation. An application of the Cartan-Kahler theorem shows that the general solution to (2.9) depends on two arbitrary functions of (2 n +1) variables. There is also some coordinate freedom in the definition of Θ, but this is measured by functions of 2 n variables.  \nThe null-Kahler structure N takes the form  \nN = dx i ⊗ ∂ ∂y i (2.15)  \nand so the kernel of N is Frobenius integrable. We may consider the leaf space  \n(2.16) M := X/ ker N  \nand associated submersion π : X → M . The coordinates x i descend to coordinates on M and Ω -descends to a symplectic form we will label ω . Locally we may identify X ∼ = TM by decreeing the y i to be the fibre (tangent) coordinates for TM associated to the x i . This has the following convenience: coordinate transformations preserving the form of the metric (2.8) with the second Pleba'nski potential Θ satisfying (2.9) are symplectic coordinate changes on M lifted to fibre coordinates on TM the usual way, together with Γ( TM ) acting by translations of the fibre coordinates. Given such a change, the second Pleba'nski potential Θ needs to be modified by a function cubic in the fibre coordinates. See [13, 19] for a calculation of the modification.  \n- 2.4. The second heavenly system as an infinitesimal limit. In the thesis of the first author ([21] page 38) it is claimed that, with n = 1, the second Pleba'nski equation (2.9) is the infinitesimal version of the first equation (2.7). The argument below makes this precise, and works for the general dimension dim( X ) = 4 n .  \nConsider a one-parameter family of flat metrics, and corresponding volume forms  \ng ξ = ξ -1 η ij dz i /circledot d ˜ z j ,  \nThe function U ξ = ξ -1 η ij z i ˜ z j satisfies the first Pleba'nski system (2.7) with the righthand-side replaced by ξ -2 η ij . Any solution to such a rescaled equation still corresponds to a complex hyper-Kahler metric, as scalling a metric by a non-zero constant does not change its hyper-Kahler property. Now set  \nU := U ξ -ξ 2 Θ( x i , y i ) , where x i = z i , y i = z i -˜ z i ξ .  \nWe find  \nand  \n∂ 2 U ∂z i ∂ ˜ z j = 1 ξ η ij + ∂ 2 Θ ∂y i ∂y j + ξ ∂ 2 Θ ∂z i ∂y j , dz i /circledot d ˜ z j = dz i /circledot dz j + ξdz i /circledot dy j  \ng = ∂ 2 U ∂z i ∂ ˜ z j dz i /circledot d ˜ z j = η ij dz i /circledot dy j + ∂ 2 Θ ∂y i ∂y j dz i /circledot dz j + o ( ξ ) .  \nIn the limit ξ → 0 the first heavenly system  \n∂ 2 U ∂z k ∂ ˜ z i ∂ 2 U ∂z l ∂ ˜ z j η kl = η ij ξ 2 (2.17)  \nreduces to the second Pleba'nski system (2.9) to leading order in ξ . In the twistor interpretation U is the generating function for the symplectomorphism between M and X/ ker Ω + , and Θ is the infinitesimal generating function.  \n- 2.5. Twistor fibrations for paraconformal structures. The family of compatible metrics associated to a (2 , 2)-paraconformal structure is a conformal structure in the usual sense. The condition of anti-self-duality of the Weyl tensor leads to Penrose's holomorphic twistor correspondence [32].  \nTo summarise the twistor correspondence for (2 n, 2)-paraconformal structures, first note that given a compatible metric (2.2) and x ∈ X there is a distinguished cone of null vectors in T x X of the form  \n(2.18) V = p ⊗ q, where p ∈ S x , q ∈ S ' x .  \nFixing q while varying p the vectors (2.18) span a 2 n -dimensional totally-null subspace of T x X . We shall call this an α -plane. Similarly, for each fixed p ∈ Γ( S ) there is a 2-dimensional totally-null β -plane obtained by varying q in (2.18). α -planes and β -planes generalise the classification of null planes into self-dual or anti-self-dual planes for a holomorphic metric in four dimensions; if n = 1 all null vectors take the form (2.18).  \nA 2 n -dimensional submanifold is called an α -surface if its tangent planes are α -planes. In the terminology of [7], a paraconformal structure is called right-flat if there is an α -surface tangent to each α -plane. This is equivalent to the anti-self-duality condition if n = 1. For general n it is equivalent to integrability of the rank-2 n twistor distribution D α on P ( S ' ) that has leaves that then push down to α -surfaces. The α -surface tangent to an α -plane is unique. In the right-flat case the we get CP 1 = P ( S ' x ) worth of α -surfaces through each point of X . This gives rise to a double-fibration picture  \n(2.19) X ←-P ( S ' ) -→ Z  \n,  \nwhere Z is the twistor space of α -surfaces in X . We compute its dimension to be  \ndim( Z ) = dim( X ) + dim( CP 1 ) -rank( D α ) = 2 n +1 .  \nIf there is an integrable β -surface tangent to the β -plane corresponding to each p ∈ S x , then there exists another double-fibration picture  \nX ←-P ( S ) -→ W ,  \nwhere W is the dual twistor space arising as the quotient of P ( S ) by the rank-2 β -surface distribution D β . If W exists, then its dimension is  \ndim( W ) = dim( X ) + dim( CP 2 n -1 ) -rank( D β ) = 4 n +(2 n -1) -2 = 6 n -3 .  \nIf both Z and W exist, then the paraconformal structure is locally modelled on the Grassmanian of 2-planes in C 2 n +2 and there is a compatible metric with vanishing Riemann tensor. Any compatible metric which is hyper-Kahler will be flat. Note that if n = 1 and X is a four-manifold, then both Z and W are three-dimensional. If n > 1 then dim( W ) > dim( X ) which makes W of limited interest as far as solving equations on X is concerned 1 .  \n- 2.6. Paraconformal structures from Lax distributions. Let S be the vector bundle over X of rank 2 n trivialised by σ i , i = 1 , ..., 2 n and S ' rank-2 and trivialised by o, ι . A Lax distribution is a a sub-bundle of T P ( S ' ) ∼ = T ( X × CP 1 ) which takes the form  \nL ( λ ) = span { L i = E i 0 ' + λE i 1 ' + f i ∂ ∂λ } 2 n i =1 . (2.20)  \nHere E ij ' are vector fields on X such that TX = span { E i 0 ' , E i 1 ' } 2 n i =1 , the f i are holomorphic functions on P ( S ' ) and λ = u 1 ' u 0 ' where ( u 0 ' , u 1 ' ) are coordinates for the fibres of S ' induced by the trivialisation o, ι . We call the fibre coordinate λ the spectral parameter .  \nTo such a Lax distribution we may associate a (2 n, 2)-paraconformal structure (2.1) given by  \nσ i ⊗ o ↦→ E i 0 ' , σ i ⊗ ι ↦→ E i 1 ' .  \nIf L ( λ ) is a Frobenius integrable distribution for each value of λ then this paraconformal structure is right-flat.  \nFixing /epsilon1 by /epsilon1 ( o, ι ) = 1 and letting { E i 0 ' , E i 1 ' } be the dual sections trivialising T ∗ X defined by E ik ' ( E jl ' ) = δ i j δ k ' l ' the compatible metrics (2.2) are of the form  \n(2.21) g = e ij /epsilon1 i ' j ' E ii ' /circledot E jj '  \nfor some non-degenerate skew-matrix of holomorphic functions e ij on X .  \nIn the case f i = 0 for i = 1 , ..., 2 n it is natural to define endomorphisms I, J, K of TX by the formulae (2.4) for which the metric g is Hermitian. Explicitly the endomorphisms are determined by  \nJ ( E j 0 ' ) = -E j 1 ' K ( E j 0 ' ) = iE j 1 ' (2.22)  \nand the quaternion relations. Note that the family of metrics and quaternionic structure does not depend on the choice of basis L i for the Lax distribution. It is also simple to verify that in this setting, Frobenius integrability of the Lax distribution L ( λ ) for  \nall λ implies that the ± i -eigenspaces of the endomorphisms I, J, K will themselves be Frobenius integrable. Given a choice of metric, the associated 2-forms (2.10) are given by  \n1  \nΩ + = e ij E i 0 ' ∧ E j 0 ' (2.23)  \n1 .  \n2 2  \nΩ -= e ij E i 1 ' ∧ E j 1 ' (2.24)  \n2.7. Hyper-Kahler structures from Lax distributions. We state conditions under which the family of metrics (2.21) contains a complex hyper-Kahler metric.  \nIn the n = 1 case, if f 1 = f 2 = 0 and the flows of E ij ' preserve a holomorphic volume form vol on X and also  \n[ L 1 , L 2 ] = 0  \nthen the metric  \n(2.25)  \ng = ν 2 ( E 10 ' /circledot E 21 ' -E 11 ' /circledot E 20 ' ) , where ν 2 = vol( E 10 ' , E 11 ' , E 20 ' , E 21 ' )  \nis complex hyper-Kahler [29].  \nFor n ≥ 1 a condition is given in [8]. It can be reformulated as follows: Suppose that f i = 0 for i = 1 , ..., 2 n and also  \n[ E i 0 ' , E j 0 ' ] = 0  \nfor i, j = 1 , ..., 2 n . Suppose there exists a symplectic form ω on M := X/ span { E i 0 ' } . Its pull-back to X necessarily takes the form  \nπ ∗ ω = 1 2 e ij E i 1 ' ∧ E j 1 ' (2.26)  \nfor some non-degenerate skew-matrix of holomorphic functions e ij on M . Then, the corresponding metric (2.21) is complex hyper-Kahler if  \n( L E i 1 ' Ω + ) | ker π = 0 , i = 1 , ..., 2 n (2.27)  \nwhere π : X → M is the natural projection.", '3. Hyper-Lagrangians': "3.1. Hyper-Lagrangian foliations and projectability. In this section we shall impose an additional condition on complex hyper-Kahler structures which imposes a differential constraint on the associated second Pleba'nski potential Θ. If dim( X ) = 4, then this constraint is sufficient to linearise the heavenly equation.  \nDefinition 3.1 (Hyper-Lagrangian foliation) . Let ( X,g ) , I, J, K be a complex hyperKahler structure. An integrable rank2 n distribution B ⊆ TX is called a hyper-Lagrangian foliation if B is Lagrangian with respect to the holomorphic symplectic forms associated to endomorphisms I, J, K .  \nA hyper-Lagrangian foliation is equivalent to a sub-bundle B ⊂ S maximally isotropic for e such that the 2 n -dimensional sub-bundle B = B ⊗ S ' of TX is Frobenius integrable  \n[ B , B ] ⊂ B .  \nSuch a foliation is preserved by the action of I, J, K .  \nDefinition 3.2 (Projectable hyper-Lagrangian foliation) . A hyper-Lagrangian foliation is called projectable if there exists a rankn foliation L of M = X/ ker N such that dπ ( B ) = L . In this case L is Lagrangian in M with respect to the symplectic form ω .  \nThe projectability condition imposes additional local restrictions on the complex hyperKahler manifold: That B pushes down to M is equivalent to  \n(3.1) [ B , ker N ] ⊂ B +ker N.  \nAs both ker N and B are Frobenius integrable, this is equivalent to Frobenius integrability of the distribution B + ker N . The intersection B ∩ ker N is a distribution of rank n , spanned by vectors of the form p ⊗ ι , where p ⊂ Γ( B ), and o ⊂ Γ( S ' ) defines the nullKahler structure. Therefore the foliation B + ker N is of rank 2 n + 2 n -n = 3 n . The leaves are inverse images (with respect to the projection π ) of the leaves of the Lagrangian foliation L of M .  \n- 3.2. Adapted coordinates. To make the hyper-Lagrangian notion more amenable to calculations take the coordinates ( x i , y i ) so the metric takes the form (2.8) and introduce a basis of 4 n vector fields E ij ' , i = 1 , . . . , 2 n, j ' = 0 , 1 on X given by  \nE i 0 ' = ∂ ∂y i , E i 1 ' = ∂ ∂x i + η jk ∂ 2 Θ ∂y i ∂y j ∂ ∂y k (3.2)  \nso that  \ng = η ij /epsilon1 i ' j ' E ii ' /circledot E jj ' .  \nGiven a projectable hyper-Lagrangian foliation we have the coordinate freedom to choose the x i to be Darboux coordinates adapted to the Lagrangian foliation dπ ( B ) = L of M in the sense  \nL = span { ∂ ∂x i } n i =1 .  \nThe hyper-Lagrangian condition then implies  \nB = span { E i 0 ' , E i 1 ' } n i =1 . (3.3) Next, we consider the differential consequences of [ B , B ] ⊆ B . Firstly, [ E i 1 ' , E j 1 ' ] = 0 is already implied by the heavenly equations (2.9) so the only non-trivial condition is  \n[ E i 0 ' , E j 1 ' ] = η kl ∂ 3 Θ ∂y i ∂y j ∂y k ∂ ∂y l ∈ B , i, j ≤ n. (3.4)  \nwhich implies  \n∂ 3 Θ ∂y i ∂y j ∂y k = 0 , i, j, k ≤ n. (3.5)  \nIf n = 1, the hyper-Lagrangian foliation is the same as a foliation by β -planes 2 . Altogether this proves  \nTheorem 1.2. The following conditions are equivalent for a complex hyper-Kahler manifold ( X,g )  \n- (1) X is foliated by projectable hyper-Lagrangian submanifolds.\n- (2) At at each point of M there exists a local coordinate system ( x i , y i ) , such that the second Pleba'nski potential corresponding to the metric g is at most quadratic in half of the fibre coordinates in the fibration π : X → M .  \nIf n = 1 then the conditions (1) and (2) are equivalent to the existence of a two-parameter family of anti-self-dual null surfaces in X which push down to curves in M .  \n3.3. Discussion. In [9] Bridgeland defines a good Lagrangian submanifold of M . A good Lagrangian B ⊂ M is a submanifold such that the Lax distribution L ( λ ) ⊆ TX descends to a distribution on the total space of the normal bundle NB when pushed forward by the map X B → NB , where X B := π -1 ( B ) ⊂ X . An equivalent condition is that in Pleba'nski coordinates ( x i , y i ) coming from Darboux coordinates adapted to B in the sense B corresponds to holding x n +1 , ..., x 2 n constant, the equation (3.5) holds on X B . It follows that a projectable hyper-Lagrangian foliation is equivalent to a foliation of M by good Lagrangian submanifolds.  \nIn [9] it is shown that for the Joyce structures on moduli spaces M of Riemann surfaces of fixed genus equipped with holomorphic quadratic differential, M admits a foliation by good Lagrangian submanifolds and hence a projectable hyper-Lagrangian foliation of X = TM . The leaves in M are spaces of quadratic differentials holding the Riemann surface fixed. The hyper-Kahler metrics we construct in § 8 (which includes as a special case the metric of [13]) will also be shown to admit projectable hyper-Lagrangian foliations.", '4. Joyce structures': "Joyce structures were introduced in [11], with a refined definition (which we shall follow below) given in [14]. Let ( M,ω ) be a complex manifold of complex dimension 2 n with holomorphic symplectic form ω . A lattice G p at a point is a discrete subgroup of T p M such that complex multiplication G p ⊗ Z C → TM is an isomorphism. A holomorphically varying bundle of lattices G ↪ → TM defines a linear connection ∇ on the bundle TM by picking out a 2 n -dimensional vector space of parallel sections. Specifically, these are taken to be the sections which take values in the lattice.  \nThe first ingredient of a Joyce structure on M is such a lattice on which the symplectic form takes rational values, and a compatible C ∗ -action, or equivalently a holomorphic vector field W 0 such that ∇ W 0 is the identity endomorphism. It follows that there are canonical complex coordinates ζ i on any suitably small neighbourhood of M in which  \n(4.1) W 0 = ζ i ∂ ∂ζ i  \nand  \n(4.2) ω = 1 2 η ij dζ i ∧ dζ j .  \nWe will call these coordinates period coordinates for M .  \nNext, the non-linear data of a Joyce structure is a compatible complex hyper-Kahler structure on the complex manifold X := TM , along with some symmetry conditions.  \nIn the notation introduced in the previous section the compatibility just means we can take the metric g to take the second heavenly form (2.8) in the period coordinates ζ i and the associated fibre coordinates θ i . Then Ω -= π ∗ ω . We will also call ( ζ i , θ i ) period coordinates (for X ).  \nThe symmetry conditions are that the second Pleba'nski potential in the period coordinates (which we will call Φ) can be taken to satisfy:  \n- (1) L W g = g where W is the ∇ -horizontal lift of W 0 .\n- (2) Φ is odd under simultaneously multiplying all the fibre coordinates by -1.\n- (3) Φ is invariant under translations of the fibres by the lattice.  \nCondition (1) is equivalent to the function Φ being homogeneity -1 under simultaneous rescalings of the ζ i since W = ζ i ∂ ∂ζ i , is the coordinate expression for W in the period coordinates ( ζ i , θ i ).", '5.1. Projectability condition.': "Theorem 1.3. Suppose a complex hyper-Kahler manifold ( X,g ) of complex dimension four is foliated by projectable hyper-Lagrangian submanifolds. Then in the adapted coordinates of Theorem 1.2 the heavenly equation for the Pleba'nski potential reduces to a system which is linearisable by a contact transformation.  \nProof. Given a hyper-Kahler metric on a complex four-manifold X take coordinates ( z, w, x, y ) adapted to Pleba'nski's second heavenly equation so that  \nΘ wx -Θ zy -Θ xx Θ yy +Θ 2 xy = 0 , and (5.1) g = dwdx -dzdy +Θ xx dz 2 +2Θ xy dzdw +Θ yy dw 2 .  \nFrom Theorem 1.2 given X admits a projectable hyper-Lagrangian foliation we may take Θ xxx = 0, or  \n(5.2) Θ = Ax 2 +2 Bx + C,  \nwhere ( A, B, C ) are functions of ( z, w, y ) which satisfy  \n2 AA yy -4 A y 2 + A yz = 0 , (5.3)  \n2 AB yy -4 B y A y + B yz -A w = 0 , (5.4)  \n2 AC yy -4 B y 2 + C yz -2 B w = 0 . (5.5)  \nGiven a general solution A ( z, w, y ) the remaining two equations are linear. We shall show how to linearise the first equation by a Legendre transformation, assuming A yy is non-vanishing on an open region. Rewrite this equation as a differential ideal  \nT ≡ dA -A dy -A dz -A dw = 0 ,  \nWe use p = A as a coordinate and set F ( z, w, p ) = A -py so that T = 0 becomes  \n1 y z w T 2 ≡ (2 Adz ∧ dA y -4 A y 2 dz ∧ dy + dA y ∧ dy ) ∧ dw = 0 . y 1  \ndF = A z dz + A w dw -ydp  \n= F z dz + F w dw + F p dp  \nwhich yields A z = F z , A w = F w , y = -F p . Substituting this into T 2 = 0 gives a linear PDE  \n(5.6) 2( F -pF p ) + 4 p 2 F pp + F pz = 0 .  \n/square  \nRemark 5.1. In the case ( X,g ) is the hyper-Kahler structure for a Joyce structure on M then A and C must be odd functions of y , while B even.  \nFor a Joyce structure with projectable hyper-Lagrangian foliation, there are two different types of useful Pleba'nski coordinates: those adapted to the foliation, and the period coordinates. Seeing the hyper-Lagrangian foliation in the period coordinates and vice versa is not so easy. This motivates the following:  \nProposition 5.2. Suppose a complex hyper-Kahler manifold ( X,g ) of complex dimension four corresponds to a Joyce structure on M . Given Pleba'nski coordinates ( z, w, x, y ) such that the decomposition (5.2) holds, A y is an even elliptic function of y .  \nProof. The coordinates ( x 1 , x 2 , y 1 , y 2 ) := ( z, w, x, y ) and the period coordinates ( ζ 1 , ζ 2 , θ 1 , θ 2 ) are related by the lift of the canonical transformation between ( x 1 , x 2 ) and ( ζ 1 , ζ 2 ) to X . The potential will be modified by some function cubic in the fibre coordinates. Write  \nΦ( ζ 1 , ζ 2 , θ 1 , θ 2 ) = Θ( x 1 , x 2 , y 1 , y 2 ) + D ijk y i y j y k (5.7) = Ay 1 y 1 +2 By 1 + C + D ijk y i y j y k  \nwhere Φ is the Pleba'nski potential corresponding to the period coordinates and D ijk are functions on M . Differentiate (5.7) by y 1 twice and y 2 once. Since the two sets of fibre coordinates are related by linear transformations depending on the base the result is  \n∂ζ i ∂x 1 ∂ζ j ∂x 1 ∂ζ k ∂x 2 ∂ 3 Φ ∂θ i ∂θ j ∂θ k = 2 A y 2 +2 D 112 .  \nThe partial derivatives of Φ taken with respect to the period coordinates will be invariant under lattice transformations, while the Jacobian matrices and D 112 are functions on M . Accordingly, A y 2 will be invariant under lattice transformations  \nζ i ↦→ ζ i + L j i v j  \nwhere v i ∈ Z and L j i defines a non-degenerate lattice. This corresponds to  \ny i ↦→ y i + ∂x i ∂ζ j L k j v k . (5.8)  \nSo A y 2 is doubly periodic in y 2 with periods  \nω 1 := ∂x 2 ∂ζ i L 1 i , ω 2 := ∂x 2 ∂ζ i L 2 i  \nwhich are functions on M .  \nFix the point on M . If the periods are linearly independent over R then A y 2 is an elliptic function of y 2 . Otherwise, in the degenerate case where these are linearly dependent over R then, we can find a sequence of transformations (5.8) defined by k i ( m ) ∈ Z such that ω j k j ( m ) → 0 and ∣ ∣ ∂x 1 ∂ζ i L j i k j ( m ) ∣ ∣ → ∞ . That (5.7) is invariant under this sequence of transformations implies the right hand side has no polynomial dependence on y 1 . This implies A = -D 112 y 2 so A y 2 = -D 112 is constant (so elliptic) with respect to y 2 .  \nSince  \nA  \nis odd,  \nA  \ny  \n2  \nwill be an even function of  \ny  \n2  \n.  \n/square  \n5.2. Strengthened projectability condition. The projectability condition on the hyperLagrangian foliation B is that B pushes down to the quotient by the α -plane foliation given by the null structure (2.3). The hyper-Lagrangian foliation is preserved by the action of I, J, K . In the n = 1 case, if we insist that B pushes down to the quotients corresponding to the whole family of α -plane foliations then we can solve the (first) heavenly equation completely. This condition in fact makes sense for any n and is the condition for B to define a rank-2 n foliation of the twistor space Z .  \nProposition 5.3. Given a complex hyper-Kahler metric on a complex four-manifold X , if X admits a hyper-Lagrangian foliation such that for any α -plane foliation D α we have  \n[ B , D α ] ⊆ B + D α  \nthen there exist coordinates ( u 1 , u 2 , ˜ u 1 , ˜ u 2 ) such that  \n(5.9) g = Adu 1 d ˜ u 2 + 1 A du 2 d ˜ u 1 + ( u 1 ∂A ∂u 2 + ˜ u 1 A -2 ∂A ∂ ˜ u 2 + D ) du 2 d ˜ u 2  \nfor arbitrary holomorphic functions A ( u 2 , ˜ u 2 ) , D ( u 2 , ˜ u 2 ) .  \nProof. As before take Pleba'nski coordinates ( x 1 , x 2 , y 1 , y 2 ) adapted to the Lagrangian foliation of M so that in the trivialisation (3.2)  \nB = span { E 10 ' , E 11 ' } .  \nRecalling the relationship between the coordinates ( x 1 , x 2 , y 1 , y 2 ) and ( z 1 , z 2 , ˜ z 1 , ˜ z 2 ) adapted to the first and second Pleba'nski equations respectively, we have  \n∂ ∂y i = η jk ∂ 2 U ∂z i ∂ ˜ z j ∂ ∂ ˜ z k  \nand so  \n(5.10) ∂ 3 Θ ∂y i ∂y j ∂y k = ∂ 2 U ∂z i ∂ ˜ z l η lp ∂ 3 U ∂ ˜ z p ∂z j ∂z k .  \nThe strengthened projectability condition implies [ B , D ] ⊆ B + D where D is the α -foliation D = span { E 11 ' , E 21 ' } . Then  \n[ E 10 ' , E i 1 ' ] = η kl ∂ 3 Θ ∂y i ∂y 1 ∂y k ∂ ∂y l ∈ B + D , i = 1 , 2  \nwhich implies  \n∂ 3 Θ ∂y i ∂y 1 ∂y 1 = 0 , i = 1 , 2 .  \nSo (5.10) then implies  \n∂ 3 U ∂ ˜ z i ∂z 1 ∂z 1 = 0 , i = 1 , 2  \nso that U = A ( z 2 , ˜ z 1 , ˜ z 2 ) z 1 + B ( z 2 , ˜ z 1 , ˜ z 2 ) + C ( z 1 , z 2 ). Substituting this ansatz into the first heavenly equation yields two equations  \n∂A ∂ ˜ z 1 ∂ 2 A ∂z 2 ∂ ˜ z 2 -∂A ∂ ˜ z 2 ∂ 2 A ∂z 2 ∂ ˜ z 1 = 0 (5.11)  \n∂A ∂ ˜ z 1 ∂ 2 B ∂z 2 ∂ ˜ z 2 -∂A ∂ ˜ z 2 ∂ 2 B ∂z 2 ∂ ˜ z 1 = 1 . (5.12)  \nInterchanging ˜ z 1 and ˜ z 2 if necessary, the first equation is equivalent to the existence of function F (˜ z 1 , ˜ z 2 ) such that  \n(5.13) ∂A ∂z 1 = F (˜ z 1 , ˜ z 2 ) ∂A ∂ ˜ z 2 .  \nNow introduce coordinates ˜ u 1 (˜ z 1 , ˜ z 2 ) , ˜ u 2 (˜ z 1 , ˜ z 2 ) , u 1 = z 1 , u 2 = z 2 arranged so that  \n∂ ∂ ˜ u 1 = ∂ ∂ ˜ z 1 -F (˜ z 1 , ˜ z 2 ) ∂ ∂ ˜ z 2 .  \nThe form of the metric (2.6) is preserved. The system of PDEs (5.12, 5.13) becomes  \n∂A  \n∂B ∂ ˜ u 1 ∂u 2 = ( ∂A ∂ ˜ u 2 ) .  \n∂ ˜ u 1 = 0 -1  \nThe general solution for U is then  \n(5.14) U = A (˜ u 2 , u 2 ) u 1 + ∫ ˜ u 1 ∂A ∂ ˜ u 2 du 2 + C ( u 1 , u 2 ) + D (˜ u 2 , u 2 ) + E (˜ u 1 , ˜ u 2 ) .  \nDifferentiating gives the metric (5.9).  \n/square  \n5.3. The null-case. For a Joyce structures with n = 1, the condition that the homothety is a null-vector implies the metric takes an explicit form generalising that of Sparling and Tod [37].  \nProposition 5.4. Given a hyper-Kahler metric on a complex four-manifold X in the form (2.8) with homothety W = x i ∂ ∂x i satisfying L W g = g and g ( W,W ) = 0 , the metric is given by  \n(5.15) g = η ij dy i /circledot dx j + η kp η lq F ( x 1 x · y , x 2 x · y ) ( x · y ) 3 x k x l dx p /circledot dx q  \nwhere F is an arbitrary holomorphic function of two variables and x · y = η ij x i y j .  \nProof. We work locally away from the hypersurface where x · y = 0. Then, we may use the fact that x i x j , x ( i y j ) and y i y j span the space of symmetric rank-two spinors. g ( W,W ) does not vanish unless we have the following decomposition with respect to this basis:  \n(5.16) ∂ 2 Θ ∂y i ∂y j = η ki η lj ( Gx ( k y l ) + Hx k x l )  \nfor some holomorphic functions G and H on X . The homothety condition is that the above expression has homogeneity -1 with respect to rescalings by the x i coordinates. Accordingly  \n(5.17) W ( G ) = -2 G, W ( H ) = -3 H.  \nDifferentiating (5.16) by y k and enforcing that the result is symmetric on all indices implies  \n(5.18) y i i +2 x i i = -3 G, x i i = 0 .  \n∂G ∂y ∂H ∂y ∂G ∂y  \nThe second heavenly equation yields  \nη ij ∂ 2 Θ ∂y i ∂x j = 1 4 G 2 ( x · y ) 2  \nwhich we differentiate by the fibre coordinates to get  \n1 2 ∂G ∂x j x j y i + 1 2 ∂G ∂x j y j x i + 3 2 Gy i + ∂H ∂x j x j x i +3 Hx i -1 2 Gη ij ∂G ∂y j ( x · y ) 2 -1 2 G 2 x i ( x · y ) = 0  \nContracting the above with η ik x k and using (5.17) and (5.18) implies G = 0. We thus have  \n(5.19) x i ∂H ∂y i = 0 .  \nSince H is homogeneity -3 in the x i coordinates we can write:  \n(5.20) H = F ( x 1 x · y , x 2 x · y , y 1 ) ( x · y ) 3 .  \nfor some function F . Write F 3 to denote the derivative of the function F by its third argument. Differentiating (5.20) using the chain rule yields, after a short calculation  \nx i ∂ ∂y i F ( x 1 x · y , x 2 x · y , y 1 ) = x 1 F 3 ( x 1 x · y , x 2 x · y , y 1 ) ,  \nbut the left hand side vanishes due to (5.19) so F does not depend on its third argument. Substituting H back into (5.16) gives the metric components. /square  \nAny metric of the form (5.15) is complex hyper-Kahler and it admits a Killing vector  \n(5.21) T := x i ∂ ∂y i .  \nThe metric of [37] takes F to be constant. In this case it admits a projectable hyperLagrangian foliation span { x i E i 0 ' , x i E i 1 ' } .  \nRemark 5.5. It is possible via a similar calculation to show if n = 1 that the metric (5.15) is the general complex hyper-Kahler metric in the form (2.8) with both sides of the second heavenly equation (2.9) vanishing separately and with homothety W = x i ∂ ∂x i .", "6. The deformed cubic oscillator, and Painlev'e I": 'In [13] a complex hyper-Kahler metric was constructed via a Lax pair of isomonodromic flows for an isomonodromy problem for a Schrodinger equation with deformed cubic oscillator potential. This calculation leads to the A 2 Joyce structure on the space M = Quad(0 , { 7 } ). The reason for this name is that the moduli space realises the space of stability conditions of a CY 3 triangulated category associated to the A 2 quiver. Furthermore the base M can be identified with unfolding of the A 2 singularity y 2 = x 3 in the ADE classification.  \nIn this section we shall explicitly construct the complex hyper-Kahler metric, and analyse its singularities. Our tool will be the Lax pair characterisation of the hyperKahler metric (2.25).', '6.1. Deformed cubic oscillator. Consider the differential equation': "(6.1) y '' = Q ( x ) y  \nwhere  \n(6.2) Q ( x ) := Q 0 ( x ) λ 2 + Q 1 ( x ) λ + Q 2 ( x )  \nis a deformation of the polynomial cubic oscillator potential given by  \n3  \nQ 0 ( x ) = x + ax + b Q 1 ( x ) = p x -q + r Q 2 ( x ) = 3 4( x -q ) 2 + r 2 p ( x -q ) + r 2 4 p 2  \n.  \nThe corresponding isomonodromy problem was the one studied by Bridgeland and Masoero in [13] and is a modification to the isomonodromy problem corresponding to the PI Hamiltonian studied by Okamoto [31].  \nLet the four-manifold X be defined as a submanifold of C 5 with coordinates ( a, b, q, p, r ) given by  \n(6.3) p 2 = q 3 + aq + b.  \nPoints in the space X specify potentials of the form (6.2) and there is a fibration π : X → M over the complex manifold M = Quad(0 , { 7 } ) with coordinates ( a, b ) corresponding to quadratic differentials Q 0 ( x ) dx 2 . The period data of the Joyce structure is constructed via an identification (the Abelian holonomy map ) of the space X with a discrete quotient of the holomorphic tangent bundle TM . We focus on the metric and homothety here.  \nThe generators of the isomonodromic deformations of (6.1) were calculated in [13]. We will revisit this calculation later in § 7, but for now, using ( a, b, q, r ) as local coordinates, all we need to know is that they are given by vector fields  \nl 1 = -∂ ∂r + λ ( ∂ ∂b + r 2 p 2 ∂ ∂r ) (6.4) l 2 = -2 p ∂ ∂q + λ ( ∂ ∂a -q ∂ ∂b -r p ∂ ∂q -r 2 (3 q 2 + a ) 2 p 3 ∂ ∂r ) .  \nIn particular, the distribution takes the form (2.20) where we recognise λ as the spectral parameter and f 1 = f 2 = 0. These flows satisfy [ l 1 , l 2 ] = 0, and moreover finding the integral curves of l 2 reduces to solving the PI. Parametrise the flow so ˙ a = -λ then  \n˙ q = 2 p 2 + λr p ˙ r = λr 2 (3 q 2 + a ) 2 p 3  \nwhile differentiating the constraint and using the above yields  \n(6.5) ˙ p = 6 p 2 q 2 +3 λq 2 r +2 ap 2 + λar 2 p 2 .  \nDifferentiating the first of these ODEs, and eliminating ( ˙ p, ˙ r ) using the remaining two ODEs yields a rescaled form of Painlev'e I.  \n(6.6) q = 6 q 2 +2 a.  \nThis calculation generalises that of [13], where it was shown how Painlev'e I arises from (6.4) in the case where r = 0.  \n- 6.2. The metric. For the purposes of writing down the hyper-Kahler metric it is convenient to work with the basis L 1 = l 1 , L 2 = l 2 + ql 1 so  \nL 1 = -∂ ∂r + λ ( ∂ ∂b + r 2 p 2 ∂ ∂r ) L 2 = -q ∂ ∂r -2 p ∂ ∂q + λ ( ∂ ∂a -r p ∂ ∂q -r (3 q 2 r + ar -qp ) 2 p 3 ∂ ∂r ) .  \nThese flows commute and preserve the volume form  \nvol := 1 2 p db ∧ da ∧ dq ∧ dr. (6.7)  \nand so the conformal class of metrics distinguished by the Lax distribution contains a complex hyper-Kahler metric. Reading off from L 1 , L 2 above the tetrad in (2.20) we can  \nuse the formula (2.25) for the hyper-Kahler metric. We have ν 2 = 1, so calculating the dual frame of one-forms E ij ' , the complex hyper-Kahler metric is  \n(6.8) g = ( r (3 q 2 r + ar -2 qp ) 2 p 3 da -r 2 p 2 db -q 2 p dq + dr ) /circledot da -( r 2 p 2 da + 1 2 p dq ) /circledot db.  \nThe vector field  \n(6.9) W = 4 a 5 ∂ ∂a + 6 b 5 ∂ ∂b + 2 q 5 ∂ ∂q + r 5 ∂ ∂r  \nsatisfies L W g = g and g does not admit any other Killing vectors, or conformal Killing vectors.  \nTo investigate the singular nature of the hyper-surface p = 0 in X consider a local diffeomorphism of X , and use ( a, p, q, r ) as coordinates (so that b is eliminated using (6.3)). This puts the metric in the form  \ng = ( r (3 q 2 r + ar -qp ) 2 p 3 da -r p dp + 3 q 2 r + ar -pq 2 p 2 dq + dr ) /circledot da (6.10) -( r 2 p 2 da + 1 2 p dq ) /circledot (2 pdp -qda -( a +3 q 2 ) dq ) .  \nComputing the determinant of the quadratic form given by g gives an expression which is regular at p = 0. We therefore turn to the curvature invariants. The simplest of these is the squared norm of the Weyl tensor of g . A cumbersome computation gives  \n| Weyl | 2 = 96 (3 q 2 + a ) 2 p 6 .  \nThis implies the singularity at p = 0 is not removable.  \nLetting ω = db ∧ da define the symplectic structure on M , the metric has the property that Ω -= π ∗ ω where span { ∂ r , ∂ b } is a projectable hyper-Lagrangian foliation 3 . Quadratic coordinates of Theorem 1.2 will be obtained as Pleba'nski coordinates for X corresponding to the Darboux coordinates ( a, b ) for M . They are obtained in [13] as follows: With each point of M associate an elliptic curve  \nΣ ( a,b ) = { ( µ, ζ ) ∈ C 2 , µ 2 = ζ 3 + aζ + b } .  \nThen define ( z, w, x, y ) on X by  \n(6.11) w = a, z = b, x = -r -∫ ( q,p ) ( q, -p ) ζdζ µ , y = -1 4 ∫ ( q,p ) ( q, -p ) dζ µ .  \nwhere µ = √ ζ 3 + aζ + b , and the path of integration between the two points ( q, p ) and ( q, -p ) in Σ ( a,b ) is invariant under the involution ( ζ, µ ) → ( ζ, -µ ).  \nFrom a calculation with the MAPLE DifferentialGeometry package we obtain the metric components in this frame and hence the second derivatives of Θ. We omit the full expressions which are long and in terms of elliptic and related functions but by integrating we can indeed take Θ such that  \n2  \n(6.12) Θ = Ax +2 Bx + C.  \nIn light of Proposition 5.2, we note A y has a simple expression in terms of Weierstrass elliptic functions  \nA y ( z, w, y ) = -1 ℘ ' ( y ; -4 z, -4 w ) 2 . (6.13)  \nThe linear systems governing the remaining five Painlev'e equations [31] can be similarly modified so that (conformal classes of) metrics can be read off from the isomonodromic flows. In these cases the constraint (6.3) needs to be replaced by a constraint coming from the relevant Painlev'e Hamiltonian from the Table (H) in [31] (Note that in our case the variable b is just the Hamiltonian H I ). This work has been carried out in the case of PII and PIII [12].  \n6.3. Series expansions. In this subsection we shall independently show how to construct the metric (6.8) starting directly from the fact the heavenly potential must take the form (6.12) in the Pleba'nski coordinates (6.11), along with the additional homothety condition. In this coordinate system the homothety is given by 4  \nW = 4 w 5 ∂ ∂w + 6 z 5 ∂ ∂z + x 5 ∂ ∂x -y 5 ∂ ∂y , and W (Θ) = -Θ .  \nThe quadraticity, homogeneity, and the parity imply  \nW ( A ) = -7 5 A, W ( B ) = -6 5 B, W ( C ) = -C,  \nwhere A, B, C are functions of ( w, z, y ). Therefore  \nwhere  \nA = ∞ ∑ k =1 a 2 k +1 y 2 k +1 , B = ∞ ∑ k =2 b 2 k y 2 k , C = ∞ ∑ k =2 c 2 k +1 y 2 k +1  \na 3 = w -1 α 3 ( t ) , a 5 = wz -1 α 5 ( t ) , a 7 = α 7 ( t ) , a 9 = w -1 zα 9 ( t ) , a 11 = wα 11 ( t ) , . . . b 4 = wz -1 β 4 ( t ) , b 6 = β 6 ( t ) , b 8 = w -1 zβ 8 ( t ) , b 10 = wβ 10 ( t ) , . . . c 5 = γ 5 ( t ) , c 7 = w -1 zγ 7 ( t ) , c 9 = wγ 9 ( t ) , c 11 = zγ 11 ( t ) , . . . .  \nThe invariant coordinate t = w 3 z -2 satisfies W ( t ) = 0, and the coefficients ( α, β, γ ) can be found by solving a recursive set of ODEs resulting from (5.3), (5.4), and (5.5). The particular solution corresponding to the metric (6.8) is  \nA = -1 28 y 7 + 1 110 wy 11 + 1 91 zy 13 -1 300 w 2 y 15 + . . . , (6.14) B = 1 30 y 6 -1 150 wy 10 -23 2310 zy 12 + 127 47775 w 2 y 14 + . . . C = -1 30 y 5 + 17 3780 wy 9 + 52 5775 zy 11 -323 150150 w 2 y 13 + . . . .  \nW = c 1 ( w∂ w -1 3 x∂ x + 2 3 y∂ y ) + c 2 ( z∂ z + 2 3 x∂ x -1 3 y∂ y ) + 1 3 c 3 ( x∂ x + y∂ y ) .  \nHere c 3 is another constant, and Θ is constrained by the condition W (Θ) = c 3 Θ. The three constants ( c 1 , c 2 , c 3 ) are defined up to an overall constant scaling.", '7. Isomonodromy of the deformed polynomial oscillator': "Motivated by the A 2 case, in this section we construct the isomonodromic flows for the equation (6.1) taking the potential (6.2) to be a deformed polynomial oscillator potential of odd-degree.  \nWe will suppress the summation convention in this section.  \n- 7.1. Isomonodromic flows. The solutions of linear ODE (6.1) with meromorphic potential may have branching behaviour near a set of poles x 0 , ..., x M . The fundamental group of the punctured space CP 1 \\ { x 0 , ..., x M } then has a linear representation on the space of solutions called the monodromy .  \nAs in § 6 we will consider a complex manifold X parametrising a family of ODEs with deformed polynomial oscillator potential. By isomonodromic flow we will mean a vector field on X generating a one-parameter group of deformations which preserves the monodromy. In our case, the ODE (6.1) will have an irregular (non-Fuchsian) singular point and so strictly speaking, by monodromy we mean the partial monodromy data consisting of the connection matrices and Stokes' multipliers as defined in [28]. The only part of the theory we will really utilise is the sufficient and necessary condition for a flow to be isomonodromic. This is a generalisation [39, 27] of Schlesinger's result for ODE with regular singular points.  \nIn our case the space X will be of complex dimension 4 n . The important conclusion will be that the isomonodromic flows span a rank-2 n distribution of the form (2.20), and thus, as explained in § 2.6 distinguish a family of holomorphic metrics on X .  \n7.2. Deformed polynomial oscillator potential. Take an integer n ≥ 1. We take the first term in the potential (6.2)  \nQ 0 ( x ) := x 2 n +1 + a n x 2 n -1 + ... + a 1 x n + b n x n -1 + ... + b 1  \nto correspond to a meromorphic quadratic differential Q 0 ( x ) dx 2 on CP 1 with a single pole of order 2 n +5 at x = ∞ . Next define  \nQ 1 ( x ) := n ∑ i =1 p i x -q i + R ( x )  \nwhere R ( x ) is the general polynomial of degree at most n -1. The parameters q i specify the locations of n singularities with residues p 2 i := Q 0 ( q i ). We then define the trailing term Q 2 ( x ) as the unique meromorphic function with the condition that the Laurent expansion of Q ( x ) at x = q i is of the form  \n(7.1) 3 4( x -q i ) 2 + u i x -q i + u 2 i + O ( x -q i )  \nfor some u i for each i = 1 , ..., n . The condition ensures that the ODE (6.1) has 'squareroot-like' monodromy at the singularity q i (analytic continuation once around the singularity corresponds to multiplication by 1 or -1). Such a singularity is referred to as an apparent singularity in the literature [28]. The only monodromy data that the parameters can affect is then at the irregular singularity at x = ∞ .  \nIn order to simplify calculation of the isomonodromic flows it is helpful to introduce parameters ( v 1 , ..., v n ) defined so that  \n/negationslash  \n/negationslash  \nR ( x ) = n ∑ i =1 ( 2 p i v i -n ∑ k = i p k q i -q k ) n ∏ j = i ( x -q j ) ( q i -q j ) .  \nIf n = 1 we have v = r/ 2 p . Explicitly  \nwhere  \nQ 2 ( x ) = n ∑ i =1 3 4( x -q i ) 2 + n ∑ i =1 v i x -q i + S ( x )  \n/negationslash  \n/negationslash  \n/negationslash  \nLocally, the equation (6.1) depends on 4 n parameters ( a, b, q, v ) where a := ( a 1 , ..., a n ), b = ( b 1 , ..., b n ) specify a quadratic differential Q 0 ( x ) dx 2 and together with q := ( q 1 , ..., q n ), v := ( v 1 , ..., v n ) specify the terms of lower degree in λ . Globally there is some sign ambiguity in p = ( p 1 , ..., p n ) so we should think of the potential as corresponding to a point in a 4 n -complex-dimensional manifold X with a fibration π : X → M given in local coordinates by ( a, b, q, v ) ↦→ ( a, b ) with half dimensional fibres isomorphic to Σ 0 ( a,b ) × C where Σ 0 ( a,b ) ⊂ C 2 is the affine part of the curve defined by y 2 = Q 0 ( x ). We also require that Q 0 ( x ) has simple zeroes and so we take M ⊆ C 2 n to be the open region on which this holds. If n = 1 this is just the condition that 4 a 3 +27 b 2 = 0. The 2 n -dimensional complex manifold M is identified as the unfolding of the A 2 n singularity minus the discriminant locus [18].  \nS ( x ) = n ∑ i =1 ( v 2 i -n ∑ k = i 3 4( q i -q k ) 2 -n ∑ k = i v k q i -q k ) n ∏ j = i ( x -q j ) ( q i -q j ) .  \n/negationslash  \n- 7.3. Calculation of the isomonodromic flows. We will now construct the isomonodromic flows in a similar fashion to the calculation in [13]. A flow ( a ( t ) , b ( t ) , q ( t ) , v ( t )) of the parameters in equation (6.1) depending on an auxiliary parameter t is isomonodromic if there exists a matrix B ( x, t ) of meromorphic functions such that the connection  \n∇ = d -[ 0 1 Q ( x, t ) 0 ] dx -B ( x, t ) dt  \nis flat (this condition is derived in a more general setting in [39], [27]). The general form of B ( x, t ) such that ∇ is flat is  \nB ( x, t ) = [ -1 2 A ' A AQ -1 2 A '' 1 2 A ' ]  \nfor some function A ( x, t ), where an apostrophe denotes a derivative with respect to x , and A must satisfy  \n(7.2) -2 ∂Q ∂t = ∂ 3 A ∂x 3 -4 Q ∂A ∂x -2 ∂Q ∂x A.  \nSubstituting Q and the Laurent expansion for A at x = q i into (7.2), a short calculation yields that A has at most a simple pole at q i . Next, taking the Laurent expansion at x = ∞ of both sides of (7 . 2), we see that A is constrained to take the form  \nn  \n(7.3) A = ∑ i =1 ζ i x -q i  \nwhere the ζ i are functions on X .  \nProposition 7.1 ( A 2 n isomonodromic flows) . There are 2 n linearly independent isomonodromic flows for the equation (6.1) . They are given by, for i = 1 , ..., n ,  \nU i := U i 0 ' + λU i 1 ' V i := V i 0 ' + λV i 1 '  \nwhere  \nU i 0 ' := -n ∑ j =1 q i -1 j 2 p j ∂ ∂v j (7.4)  \nU i 1 ' := ∂ ∂b i (7.5)  \nand  \n/negationslash  \nV i 0 ' := -2 p i ∂ ∂q i -n ∑ j = i ( ∑ n k =1 T k ( q i ) q k + n -1 j 2 p j + Q ' 0 ( q j ) 2 p j ( q i -q j ) + p j ( q i -q j ) 2 ) ∂ ∂v j (7.6) -( ∑ n k =1 T k ( q i ) q k + n -1 i 2 p i -Q ' 0 ( q i ) v i p i + R ' ( q i ) -n ∑ j = i p j ( q i -q j ) 2 ) ∂ ∂v i V i 1 ' := n ∑ j =1 T j ( q i ) ∂ ∂a j -2 v i ∂ ∂q i + n ∑ j = i 1 q i -q j ∂ ∂q j (7.7) -n ∑ j = i ( 3 2( q i -q j ) 3 + v j ( q i -q j ) 2 ) ∂ ∂v j + ( n ∑ j = i 3 2( q i -q j ) 3 + n ∑ j = i v j ( q i -q j ) 2 -S ' ( q i ) ) ∂ ∂v i  \nwhere  \n(7.8) T j ( x ) := (2 j -1) x n -j + n -j -1 ∑ k =1 (2 j + k ) a n -k +1 x n -j -1 -k .  \nProof. First we find n flows U i that satisfy (7.2) with A = 0 and so preserve the potential in the sense Q ( x ) is constant along the flows. We call these isopotential flows. Clearly q i must be constant along the flow since differentiation by q i leads to cubic terms which cannot be cancelled. Similarly a i must be constant since there is nothing to cancel the x i term resulting from differentiation by this parameter. If we think of the flows as being defined by a vector field U on X then U must annihilate the terms in the Laurent expansion at x = q i . By (7.1) a necessary condition is  \n(7.9) U ( u i ) = 0 ,  \nfor i = 1 , ..., n . Conversely, if this is the case then U ( Q ( x )) has no poles and vanishes at q 1 , ..., q n , but since a i does not flow we see that U ( Q ( x )) is a polynomial of degree at most n -1 and therefore must vanish since it has n roots. So it is in fact sufficient for U to annihilate the u i in order to annihilate Q ( x ). As a corollary we see that there are n linearly independent isopotential flows since we are searching for flows in 2 n variables b 1 , ..., b n , v 1 , ..., v n which annihilate n independent complex functions u i which we may calculate as  \n(7.10) u i = p i λ + v i .  \nIn particular, for these flows ˙ p i = -λ ˙ v i , where we are denoting the derivative along the flow parameter t by a dot. Using the relation  \n2 p i ˙ p i = ˙ Q 0 ( q i ) + Q ' 0 ( ˙ q i ) (7.11)  \n/negationslash  \n/negationslash  \n/negationslash  \n/negationslash  \n/negationslash  \nwe see that for these flows 2 p i ˙ p i = ˙ b n q n -1 i + ... + ˙ b 1 and after substituting to eliminate ˙ p i we deduce the linearly independent flows are given by (7.4, 7.5) above.  \nNext we will find the flows which solve (7.2) with A = ( x -q i ) -1 for each i . By linearity, and the constraint (7.3) this will give all the isomonodromic flows. For the ensuing calculation we need the Laurent expansion of Q ( x ) at x = q i to the next order. For this write:  \nQ ' ( x ) = P i ( x ) + O (( x -q i ) -1 ) , i = 1 , ..., n  \nwhere P i ( x ) is holomorphic at x = q i and the omitted terms comprise the principal part of the Laurent expansion of Q ' ( x ) at q i . Then  \nQ ( x ) = 3 4( x -q i ) 2 + u i x -q i + u 2 i + P i ( q i ) · ( x -q i ) + O ( ( x -q i ) 2 ) .  \nWe will need the explicit expression for P i ( x ) later but it is best to leave it suppressed for the moment.  \n/negationslash  \nWe now take (7.2) with A = 1 x -q i and insist that the ( x -q k ) l terms, in the Laurent expansion of both sides of (7.2), l ≤ 0, agree for k = 1 , ..., n . We need to handle the k = i and k = i terms separately. For k = i use the fact that the constant term in the Laurent expansion of  \n4 F ( x ) ( x -q i ) 2 -2 F ' ( x ) ( x -q i )  \nvanishes at x = q i for meromorphic F ( x ). The conditions are  \n( x -q i ) -3 : ( x -q i ) -2 : ( x -q i ) -1 : ( x -q i ) 0 : ˙ q i = -2 u i -2 ˙ q i u i = 4 u 2 i -2 ˙ u i = 2 P i ( q i ) -4 u i ˙ u i +2˙ q i P i ( q i ) = 0 .  \n/negationslash  \nA priori we have too many equations to specify ˙ q i and ˙ u i but the ( x -q i ) 0 and ( x -q i ) -2 equations are implied by the other two. For k = i , a similar, although somewhat more complicated computation taking the product of Laurent expansions gives  \n( x -q k ) -3 : ( x -q k ) -2 : ( x -q k ) -1 : ( x -q k ) 0 : ˙ q k = -( q k -q i ) -1 -2 ˙ q k u k = 2 u k ( q k -q i ) -1 -2 ˙ u k = 2 u k ( q k -q i ) -2 -3( q k -q i ) -3 -4 u k ˙ u k +2 ˙ q k P k ( q k ) = -2 P k ( q k ) q k -q i -6 u k ( q k -q i ) 3 + 4 u 2 k ( q k -q i ) 2  \nand similarly the ( x -q k ) 0 and ( x -q k ) -2 equations are implied by the other two.  \nGiven all the above are satisfied, (7.2) is a polynomial in x with roots x = q k , k = 1 , ..., n . The conditions give equations for ˙ q k and ˙ u k for k = 1 , ..., n . Now considering (7.10) and (7.11) we can solve for ˙ v k given we know the ˙ a k and ˙ b k for k = 1 , ..., n . Since the isopotential flows preserve the potential we can insist ˙ b 1 , ..., ˙ b n vanish. Lastly, ˙ a 1 , ..., ˙ a n are determined by the condition that the x 2 n -1 , ..., x n terms in the Laurent expansion at x = ∞ of both sides of (7.2) agree. It would then follow that (7.2) is a polynomial of degree at most n -1 with n roots, and therefore vanishes. The relevant terms of (7.2) are  \n-2 ∂Q 0 ∂t = -4 Q 0 ∂A ∂x -2 ∂Q 0 ∂x A + O ( x n -1 ) (7.12)  \nwhere the trailing terms are of degree n -1 or lower in the expansion, so can be ignored for this part of the calculation. Equating terms in the Laurent expansion yields  \n-2˙ a j = -2 T j ( q i )  \nfor j = 1 , ..., n .  \nWriting everything out in full using  \n/negationslash  \n/negationslash  \n/negationslash  \nP i ( x ) = Q ' 0 ( x ) λ 2 + λ -1 ( R ' ( x ) -n ∑ j = i p j ( x -q j ) 2 ) -n ∑ j = i 3 2( x -q j ) 3 -n ∑ j = i v j ( x -q j ) 2 + S ' ( x ) /square  \ngives (7.6, 7.7).  \nRemark 7.2. Frobenius integrability of the span of the isomonodromic flows follows from general theory: The map which sends the parameters ( a, b, q, v ) to the Stokes' data of the equation (6.1) is holomorphic, and the isomonodromy distribution is the kernel of the differential. Alternatively, this can be shown by calculation from the equation (7.2).", '7.4. Quasi-homogeneity properties. In what follows we will consider the vector field': "(7.13) W = n ∑ i =1 ( (2 n -2 i +4) a i 2 n +3 ∂ ∂a i + (4 n -2 i +4) b i 2 n +3 ∂ ∂b i + 2 q i 2 n +3 ∂ ∂q i -2 v i 2 n +3 ∂ ∂v i ) .  \nThe explanation for the various weights in the rescaling of coordinates defined by W is the following: these are the precise weights under which, together with rescaling x with weight 2 / (2 n +3) and λ with weight 1 the Schrodinger equation (6.1) is preserved.  \nIt is useful now to write down various quasi-homogeneity properties with respect to the rescaling defined by W . From  \nW ( p j ) = 2 n +1 2 n +3 p j  \nwe deduce  \n[ W,U i 0 ' ] = 2 i 2 n 1 U i 0 ' (7.14)  \n--2 n +3  \n4 n +4 -2 i  \n[ W,U i 1 ' ] = -2 n +3 U i 1 ' (7.15)  \nfor i = 1 , ..., n . From  \nW ( T j ( q i )) = 4 n -2 j 2 n +3 T j ( q i )  \nwhere j = 1 , ..., n we obtain  \n2 n -1  \n2 n +3 [ 4  \n[ W,V i 0 ' ] = V i 0 ' (7.16)  \n(7.17)  \nfor i = 1 , ..., n .", '8. The A 2 n hyper-Kahler metric': "In this section we will consider the family of metrics associated to the isomonodromic flows of Proposition 7.1 and identify a hyper-Kahler representative. We will reinstate the summation convention with lower case primed indices taking values in 0 ' , 1 ' as before, but with lower case unprimed indices taking values in 1 , ..., n .  \nW,V i 1 ' ] = -2 n +3 V i 1 '  \nConsider the basis { U i , V i } n i =1 of isomonodromic flows from (7.1). We will see the vector fields { U i 0 ' , U i 1 ' , V i 0 ' , V i 1 ' } n i =1 trivialise TX on the dense open subset of X where the q i are distinct and the p i are non-zero. On this open subset the flows define a Lax distribution of the form (2.20), taking U ij ' = E ij ' and V ij ' = E ( i + n ) j ' for i = 1 , ...n . Given the dual basis of one-forms { U i 0 ' , U i 1 ' , V i 0 ' , V i 1 ' } n i =1 , compatible metrics (2.21) for the associated (2 n, 2)-paraconformal structure are given by  \n(8.1) g = 2 /epsilon1 i ' j ' ( µ kl U ki ' /circledot U lj ' +2 ν kl U ki ' /circledot V lj ' + ξ kl V ki ' /circledot V lj ' ) specified by matrices µ kj , ν kj , ξ kj for which µ kj , ξ kj are skew and in the notation of (2.21)  \ne = 2 [ µ ν -ν ξ ] .  \nIn this trivialisation the quaternionic structures (2.4) are determined by  \nJ ( U j 0 ' ) = -U j 1 ' K ( U j 0 ' ) = iU j 1 ' (8.2) J ( V j 0 ' ) = -V j 1 ' K ( V j 0 ' ) = iV j 1 '  \nand the associated 2-forms are given by  \nΩ + = µ kl U k 0 ' ∧ U l 0 ' +2 ν kl U k 0 ' ∧ V l 0 ' + ξ kl V k 0 ' ∧ V l 0 ' (8.3)  \nand  \nΩ -= µ kl U k 1 ' ∧ U l 1 ' +2 ν kl U k 1 ' ∧ V l 1 ' + ξ kl V k 1 ' ∧ V l 1 ' . (8.4)  \nSince the span of the isomonodromic flows is integrable for each value of λ , the hyperKahler condition for g , that J, K (and hence I ) are parallel for the Levi-Civita, is equivalent to the closure of Ω J , Ω K .  \nNow Ω J = Ω + +Ω -and Ω K = Ω + -Ω -. This means to find a hyper-Kahler metric, we seek µ kj , ν kj , ξ kj for which Ω + , Ω -are closed. It will turn out the choice is quite natural to the geometry at hand.  \nAs we will see, the base M carries a canonical symplectic form ω induced by the cohomology intersection pairing on each hyper-elliptic curve defined by y 2 = Q 0 ( x ).  \nIn the next few subsections we will show the following:  \nProposition 8.1 ( A 2 n hyper-Kahler metric) . The unique metric g ω in the class (8.1) such that the associated 2 -form Ω + given by (8.3) satisfies  \n(8.5) Ω + | ker π = dv i ∧ dq i  \nis equal to the unique metric in the class (8.1) for which the associated 2 -form Ω -given by (8.4) satisfies Ω -= π ∗ ω and this metric is complex hyper-Kahler.  \nRemark 8.2. In the case n = 1, this metric agrees (up to a constant factor) with (6.8).  \nRemark 8.3. In the langauge of [14], the hyper-Kahler structure on X defines an affine symplectic fibration for the symplectic form ω .  \n- 8.1. Hyper-elliptic curves and the Gauss-Manin connection. We first explain the definition of ω . A point ( a, b ) ∈ M specifies non-singular genusn hyper-elliptic curve Σ ( a,b ) defined by the equation y 2 = Q 0 ( x ). Take the holomorphic vector bundle E → M with fibres  \n(8.6) E ( a,b ) := H 1 (Σ ( a,b ) , C ) .  \nE ( a,b ) carries the usual intersection form 〈· , ·〉 : H 1 (Σ ( a,b ) , C ) × H 1 (Σ ( a,b ) , C ) → C given by  \n(8.7) 〈 [ α ] , [ β ] 〉 = 1 2 ∫ Σ ( a,b ) α ∧ β  \nfor smooth representative one-forms α, β . Letting ( a, b ) vary we get a holomorphic skewform on E we will also denote by 〈· , ·〉 .  \nChoose a basis { µ 1 , ...µ 2 n } dual to fundamental cycles { γ 1 , ..., γ 2 n } . The Gauss-Manin connection ∇ GM on E is the flat connection constructed by decreeing sections taking values in { µ 1 , ...µ 2 n } at each point are covariantly constant [17].  \nUsing x as a local holomorphic coordinate on Σ ( a,b ) the differential form  \nZ ( a,b ) = ydx  \nis clearly holomorphic away from the branch points but in fact extends to a meromorphic 1-form on Σ ( a,b ) with a single pole at infinity. To see this use y as a holomorphic coordinate in a neighbourhood of a (finite) branch point and note  \nydx = 2 Q 0 ( x ) Q ' 0 ( x ) dy.  \nThe branch point will be a root of Q 0 ( x ) but recalling the assumption that the roots of Q 0 ( x ) were distinct at points of M , the denominator will not vanish. A similar argument shows that Z ( a,b ) has no residue at the branch point x = ∞ . Since integrals of this oneform around cycles (away from infinity) are well defined Z is a well defined section of E . We will see that the map µ : TM → E defined by  \nµ ( X ) = ∇ GM X Z  \nis an isomorphism of holomorphic vector bundles.  \nIt is a well-known fact that the one-forms η i with local representation  \nη i = ∂y dx = x i -1 dx, i = 1 , ..., n (8.8)  \n∂b i 2 y  \nextend to holomorphic 1-forms on Σ ( a,b ) , as can be easily seen by looking at the orders of the numerator and denominator at x = ∞ . We will also need the one-forms ω i with local representation  \nω i = ∂y ∂a i dx = x n + i -1 2 y dx, i = 1 , ..., n  \nwhich are globally defined and meromorphic on Σ ( a,b ) with a pole of order 2 i with zero residue at the branch point at x = ∞ [6].  \nIt is important that η i and ω i have no residues for i = 1 , ..., n (they may traditionally be referred to as differentials of the first and second kind respectively). This implies, given a tangent vector U ∈ TM , the integral of U ( y ) dx around any trivial cycle vanishes and this meromorphic 1-form U ( y ) dx represents a cohomology class. Furthermore, there exists a meromorphic function φ U defined in a neighbourhood of infinity such that  \ndφ U = U ( y ) dx.  \nPulling 〈· , ·〉 back by the map µ and normalising by a constant factor for convenience we obtain a 2-form  \nω := 1 2 πi µ ∗ 〈· , ·〉  \non M .  \nProposition 8.4 (Hyper-elliptic intersection form) . The 2 -form ω is symplectic and given by the following formula  \nω ( U, V ) = 1 2 πi ∫ C U ( y ) φ V dx. (8.9)  \nwhere C is a contour in CP 1 with the standard orientation, containing all the poles of the integrand except one at x = ∞ .  \nProof. We have  \nµ ( U ) = 2 n ∑ L =1 U ( ∫ γ L ydx ) µ L = 2 n ∑ L =1 ( ∫ γ L U ( y ) dx ) µ L . (8.10)  \nTake a smooth function ρ which takes the constant value 1 on a neighbourhood D of x = ∞ and vanishes on the complement of a neighbourhood ˜ D of D . We may represent the cohomology class of U ( y ) dx by a smooth, globally-defined 1-form as follows:  \nand similarly for V . We therefore have  \nµ ( U ) = [ U ( y ) dx -d ( ρφ U ) ]  \nω ( U, V ) = 1 4 πi ∫ ˜ D d ( ρφ U ) ∧ d ( ρφ V ) -U ( y ) dx ∧ d ( ρφ V ) -d ( ρφ U ) ∧ V ( y ) dx = 1 4 πi ∫ ˜ D d (2 ρφ V U ( y ) dx -ρφ V d ( ρφ U ))  \nwhere we have used d ( ρφ V U ( y ) dx ) = -d ( ρφ U V ( y ) dx ). Next we use Stokes' theorem, taking the boundary contour ∂D so that we can set ρ = 1 in the integrand. The 1-form U ( y ) φ V dx being integrated is pulled-back from CP 1 so we may write the integral as a contour integral in CP 1 . We take the contour C in CP 1 with preimage ∂D under the covering map to have a winding number of 1 anti-clockwise around the origin, which means we need to multiply the integrand by a factor of 2 as ∂D traverses both sheets.  \nω ( U, V ) = 1 4 πi ∮ ∂D U ( y ) φ V dx = 1 2 πi ∮ C U ( y ) φ V dx,  \nwhich is the desired formula.  \nTo see that ω is non-degenerate it suffices to show that ω is represented by a triangular matrix with non-vanishing diagonal entries. Use the fact the wedge product of holomorphic 1-forms vanishes to see  \nω ( ∂ ∂b j , ∂ ∂b k ) = 0 .  \nNext by checking the order of the pole at which the residue is extracted we see that  \nω ( ∂ ∂a j , ∂ ∂b k ) = res x = ∞ ( ∂y ∂a j d -1 ( ∂y ∂b k dx ) dx )  \nvanishes for j + k ≤ n +1 and  \n/negationslash  \nω ( ∂ ∂a j , ∂ ∂b n +1 -j ) = res x = ∞ ( ∂y ∂a j d -1 ( ∂y ∂b n +1 -j dx ) dx ) = 1 2 -4 j = 0 .  \nThe closure of ω follows from the Schwarz rule and integration by parts.  \n/square  \nSince ω is non-degenerate it follows µ : TM → E is an isomorphism. For the first few values of n we have  \n1  \n8.2. Distinguished A 2 n metric. It will be convenient to introduce matrices A ij , ˜ B ij , ˆ B ij , ˜ C ij , ˆ C ij , T ij to compactly write the flows from (7.1) as follows:  \n(8.11)  \nA 2 : ω = 2 db ∧ da A 4 : ω = 1 6 db 1 ∧ da 2 + 1 2 db 2 ∧ da 1 + a 2 6 da 1 ∧ da 2 A 6 : ω = 1 10 db 1 ∧ da 3 + 1 6 db 2 ∧ da 2 + 1 2 db 3 ∧ da 1 -3 a 3 10 db 3 ∧ da 3 + a 3 6 da 1 ∧ da 2 + a 2 5 da 1 ∧ da 3 + ( a 2 3 10 + a 1 30 ) da 2 ∧ da 3 .  \nU i 0 ' = A ij ∂ ∂v j U i 1 ' = ∂ ∂b i V i 0 ' = ˜ C ij ∂ ∂q j + ˆ C ij ∂ ∂v j V i 1 ' = T ij ∂ ∂a j + ˜ B ij ∂ ∂q j + ˆ B ij ∂ ∂v j  \nfor i = 1 , ..., n . Where the q i are distinct and p i are non-zero, the matrices A ij , T ij and ˜ C ij are invertible. To see this note that the matrices A ij and T ij can each be written as the product of the Vandermonde matrix q i -1 j and a triangular matrix with non-zero diagonal entries, while ˜ C ij is diagonal with the p i as its entries. This implies that { U i 0 ' , U i 1 ' , V i 0 ' , V i 1 ' } n i =1 trivialise TX on this open region. We then calculate the dual trivialisation as  \n(8.12)  \nU i 0 ' = A -1 ji dv j -A -1 ki ˆ C lk ˜ C -1 jl dq j -A -1 ki ˆ B lk T -1 jl da j + A -1 ki ˆ C lk ˜ C -1 ml ˜ B rm T -1 jr da j U i 1 ' = db i V i 0 ' = ˜ C -1 ji dq j -˜ C -1 ki ˜ B lk T -1 jl da j V i 1 ' = T -1 ji da j  \nwhere the components ( A -1 ) ij of the matrix inverses are denoted by A -1 ij and satisfy A ij A -1 jk = δ ik and so on.  \nSuppose that we would like to choose the the metric g in the family (8.1) so that the 2form Ω + takes Darboux form when restricted to the vertical bundle as in Proposition8.1. Specifically  \nΩ + | ker π = dv i ∧ dq i .  \nSubstituting the expressions for the dual trivialisation (8.12) into (8.3) we see that µ ij = 0 since otherwise Ω + | ker π would contain terms of the form µ ij A -1 ki A -1 lj dv k ∧ dv l . Given this simplification  \nΩ + | ker π = 2 ν ij A -1 ki ˜ C -1 lj dv k ∧ dq l + ( ξ ij ˜ C -1 mi ˜ C -1 rj -2 ν ij A -1 ki ˆ C lk ˜ C -1 ml ˜ C -1 rj ) dq m ∧ dq r .  \nSo we see the following:  \nLemma 8.5. There is a unique metric g ω in the class (8.1) such that the associated 2 -form Ω + given by (8.3) satisfies (8.5). It is obtained by setting  \n(8.13)  \nA calculation using the explicit form of A ij , ˜ C ij , ˆ C ij then gives give the non-zero components of Ω -with respect to the trivialisation { U i 0 ' , U i 1 ' , V i 0 ' , V i 1 ' } n i =1 :  \nµ ij = 0 ν ij = 1 2 A ik ˜ C jk ξ ij = 1 2 ( ˆ C ik ˜ C jk -ˆ C jk ˜ C ik ) .  \nΩ -( U j 1 ' , U k 1 ' ) = 0  \nΩ -( U j 1 ' , V k 1 ' ) = -A jl ˜ C kl 2 = q j -1 k 2 Ω -( V j 1 ' , V k 1 ' ) = 1 2 ( ˜ C jl ˆ C kl -˜ C kl ˆ C kl ) = 1 2 ( F j ( q k ) -F k ( q j ) )  \nwhere  \nF i ( x ) := V i 1 ' ( Q 0 ( x )) -Q ' 0 ( x ) x -q i + 2 Q 0 ( x ) ( x -q i ) 2 . (8.14)  \nLemma 8.6. The 2 -form Ω -given by (8.4) associated to g ω satisfies Ω -= π ∗ ω .  \nProof. First rewrite (8.14) in terms of 1-forms on Σ ( a,b ) as follows:  \nF j ( x ) 2 y dx = V j 1 ' ( y ) dx -Q ' 0 ( x ) 2 y ( x -q j ) dx + 2 Q 0 ( x ) 2 y ( x -q j ) 2 dx.  \nThe first term on the right hand side has a meromorphic antiderivative on a neighbourhood of x = ∞ . The last two terms together are a total derivative of y/ ( x -q j ). Write I j ( x ) for the (local) antiderivative of the left hand side.  \nApplying the formula (8.9) then yields  \n(8.15) π ∗ ω ( V j 1 ' , V k 1 ' ) = 1 2 πi ∮ C ( F j ( x ) 2 y + ∂y ∂x ( x -q j ) -y ( x -q j ) 2 ) · ( I k ( x )+ y ( x -q k ) ) dx.  \nLet ˜ x = 1 /x . In the proof of (7.1) the vector field V i 1 ' was chosen precisely so that the Laurent expansion at infinity of the right hand side of (8.14) has a pole of order at most n -1 at infinity (see (7.12)). That is  \nF j ( x ) = c ˜ x -n +1 + O (˜ x -n +2 )  \nfor some constant c . The contour integral (8.15) is proportional to the coefficient of the ˜ x term in the Laurent expansion of the integrand at infinity. Meanwhile we have the Puiseux series  \ny -1 = ˜ x n + 1 2 -1 2 a n ˜ x n +2 + O (˜ x n +3 ) .  \nTherefore, the 1-form F j ( x ) I k ( x ) / 2 y dx does not contribute to the contour integral since it is holomorphic. We may evaluate other terms by integration by parts which yields, after skew-symmetrising  \nπ ∗ ω ( V j 1 ' , V k 1 ' ) = 1 4 πi ∮ C F j ( x ) x -q k -F k ( x ) x -q j + Q 0 ( x ) ( x -q k ) 2 ( x -q j ) -Q 0 ( x ) ( x -q j ) 2 ( x -q k ) dx.  \nEvaluating this integral by taking residues at the poles x = q j , q k (note that F j ( x ) has a simple pole at x = q j ) then yields, after some cancellation, the desired equality  \nπ ∗ ω ( V j 1 ' , V k 1 ' ) = 1 2 ( F j ( q k ) -F k ( q j ) ) .  \nSimilarly, noting that x j -1 I k ( x ) / 2 y dx is holomorphic gives  \nπ ∗ ω ( U j 1 ' , V k 1 ' ) = 1 2 πi ∮ C x j -1 2 y · ( I k ( x ) + y ( x -q k ) ) dx = 1 2 πi ∮ C x j -1 2( x -q k ) dx = q j -1 k 2 .  \nLastly, that the wedge product of holomorphic 1-forms vanishes implies  \nπ ∗ ω ( U j 1 ' , U k 1 ' ) = 0 .  \nAltogether we have shown π ∗ ω and Ω -agree when evaluated on the elements of the trivialisation { U i 0 ' , U i 1 ' , V i 0 ' , V i 1 ' } n i =1 which is sufficient to show the desired equality. /square  \nThe closure of Ω -follows from the closure of ω . The final thing to check to show g ω is hyper-Kahler is therefore:", 'Lemma 8.7. Ω + associated to g ω is closed.': "Proof. Let U, V, W be tangent vectors. The standard formula for the exterior derivative is  \nd Ω + ( U, V, W ) = U (Ω + ( V, W )) + V (Ω + ( W,U )) + W (Ω + ( U, V )) (8.16) -Ω + ([ U, V ] , W ) -Ω + ([ V, W ] , U ) -Ω + ([ W,U ] , V ) .  \nNow TX = span { U i 0 ' , V i 0 ' } ⊕ span { U i 1 ' , V i 1 ' } . The first summand is the vertical bundle while the second summand is the kernel of Ω + . Clearly (8.16) vanishes when U, V, W all belong to the second summand. Next, Frobenius integrability of the second summand implies (8.16) vanishes when U is vertical and V, W belong to the second summand. Substituting (8.12) together with (8.13) into (8.3) we obtain (after some cancellation):  \nΩ + = dv i ∧ dq i -˜ B li T -1 kl dv i ∧ da k -ˆ B ki T -1 lk da l ∧ dq i + ... (8.17)  \nwhere the omitted terms vanish when one input is vertical. It follows that (8.16) vanishes when U, V, W are all vertical. It remains to show (8.16) vanishes when U, V are vertical and W belongs second summand. For this it is sufficient to check that the exterior derivative of the terms written out in (8.17) vanish when two inputs are vertical. Because the T ij are independent of the v i this reduces to  \n∂ ˜ B ki ∂v j -∂ ˜ B kj ∂v i = 0 (8.18)  \n∂ ∂q i ( T -1 kl ˆ B lj ) -∂ ∂q j ( T -1 kl ˆ B li ) = 0 (8.19)  \n∂ ˆ B ki ∂v j + ∂ ˜ B kj ∂q i + T kl ∂ T -1 lm ∂q i ˜ B mj = 0 (8.20)  \nfor k = 1 , ..., n . Checking (8.18) is straightforward. The other two equalities can be shown by long (although elementary) calculations. It helps to use  \nT jk ∂T -1 kl ∂q i = -∂T jk ∂q i T -1 kl  \nand the fact we can decompose T ij = F ik Q kj where F ik is independent of the q l and Q kj := q j k . Then  \n∂T jk ∂q i T -1 lk = ∂Q j k ∂q i Q -1 kl . (8.21)  \nThe right hand side of (8.21) vanishes unless i = j in which case  \n/negationslash  \n/negationslash  \nn ∑ k =1 ∂Q ik ∂q i Q -1 kl = ∏ k = i,n ( q i -q k ) ∏ k = l ( q n -q k ) , l = i n ∑ k =1 ∂Q ik ∂q i Q -1 ki = ∑ k = i 1 ( q i -q k )  \n/negationslash  \nwhere we are suppressing the summation convention.  \n/square  \nTogether with Lemma 8.5 and Lemma 8.6 this completes the proof of Proposition 8.1.  \nProposition 8.8. L W g ω = g ω where W is given by (7.13).  \nProof. This is straightforward to verify using (7.14, 7.16) and  \nW ( A ij ) = 2 i -2 n -3 2 n +3 A ij W ( ˜ C ij ) = 2 n +1 2 n +3 ˜ C ij W ( ˆ C ij ) = 2 n -3 ˆ C ij .  \n2 n +3  \nProposition 8.9. The metric g ω admits a projectable hyperlagranian foliation U .  \nProof. Clearly U := span { U i 0 ' , U i 1 ' } i =1 ,...,n is an integrable distribution. That µ kl = 0 ensures it is hyper-Lagrangian. Its image under dπ is the Lagrangian foliation for ω given by the span of the b i coordinate fields. /square  \nTheorem 1.4 now follows from Proposition 8.1, Proposition 8.8 and Proposition 8.9.  \nRemark 8.10. The foliation U satisfies a stronger condition than being hyper-Lagrangian: There is a 2 n -2-dimensional family of β -surfaces. They are given by the leaves of span { σ i U i 0 ' , σ i U i 1 ' } where σ i are constants for i = 1 , ..., n .  \nRemark 8.11. U has an interpretation in terms of the Hodge structure of the hyperelliptic curves specified by points in M . The image of dπ ( U ) under the isomorphism µ : TM → E is the rankn sub-bundle with fibre H 1 , 0 (Σ ( a,b ) , C ).", '9. Outlook': "We have constructed a family of complex hyper-Kahler metrics associated with the isomonodromy problem for the deformed polynomial oscillator of degree 2 n + 1. These are compatible with the symplectic structure on a space M of meromorphic quadratic differentials on CP 1 with a single pole of order 2 n +5. If n = 1, then this metric corresponds to the A 2 Joyce structure of [13], and arises from isomonodromic flows defining Painlev'e I. One motivation for this explicit construction is as follows: For general n , the base M can be identified as the universal unfolding of the A 2 n singularity which has a standard Frobenius structure in a family studied by Saito [36] and Dubrovin [18]. In the n = 1 case, the Joyce structure is compatible with this Frobenius structure in the sense of [11]; a Joyce structure induces a bilinear form on the base M which in this case agrees  \n/negationslash  \nwith the Frobenius metric up to a constant scale. In this paper we did not go as far as to check this for general n but the expressions obtained here would be the starting point for such a calculation.  \nWe did not go as far to show that the metrics constructed in § 8 give rise to Joyce structures. We expect the construction to not differ drastically from [13]. After choosing a basis of cycles, a map X ( a,b ) → T ( a,b ) M/H 1 (Σ ( a,b ) , Z ) can be constructed by considering integrals of the differential form  \n/pi1 := Q 1 ( x ) dx 2 y + n ∑ i =1 dx 2( x -q i )  \naround the basis of cycles. That the image is an open dense set follows due to Jacobi inversion. The hyper-Kahler metric can be pushed forward by this map in each fibre since it is invariant under exchange of the q i . The Joyce structure should be obtained by pulling the structure back to TM , and this leads to the lattice invariance property. Finally, Pleba'nski coordinates ζ i and θ i for X are given by the integrals of ydx and /pi1 (defined up to integer multiples of 2 πi ) respectively, around the cycles.  \nThe heavenly potential corresponding to the metric of Theorem 1.4 will be quadratic in half of the fibre coordinates in a suitably adapted Pleba'nski coordinate system. This is a general property (Theorem 1.2) of metrics admitting a projectable hyper-Lagrangian foliation. In dimension four this property is sufficient (Theorem 1.3) to linearise the heavenly equation. It would be interesting to know whether this linearisation generalises to the heavenly system (2.9) in higher dimensions.  \nIn [4, 3] Alexandrov and Pioline interpreted the heavenly equation corresponding to Joyce structures in terms of the thermodynamics Bethe ansatz, and obtained integral representations for the heavenly potential Θ. It would be interesting to put our A 2 n solution in this framework, and understand it as a conformal limit. The connection may arise by the iterative construction of the holomorphic coordinates on the twistor space associated to the A 2 n metric. Rather than employing a recursion operator procedure of [22] to construct such twistor functions, one may attempt to iterate the integral equations of the TBA as in [1] .", "Appendix. Cohomogeneity one Painlev'e I metric": "In § 6.1 we constructed a complex hyper-Kahler metric (6.8) from a Lax pair { l 1 , l 2 } such that the integral curves of l 2 were defined by the Painlev'e I equation. In dimension four all complex hyper-Kahler metrics are conformally anti-self-dual, and null-Kahler in the sense of Definition 2.2.  \nIn [19] another anti-self-dual null-Kahler metric was constructed using the formulation of Painlev'e I as a reduction of the isomonodromy equations and the twistor construction of Hitchin [25]. This led to a question of whether these two metrics were the same, or at least conformally related. In this section we shall settle this and deduce that, despite being anti-self-dual and null-Kahler, these two metrics are different.  \nConsider the isomonodromic Lax pair with irregular singularity of order four  \nM 1 = ∂ ∂λ -( -z y 2 + t 2 -4 y z ) -λ ( 0 y 4 0 ) -λ 2 ( 0 1 0 0 ) (A.1) M 2 = ∂ ∂t -( 0 y 2 0 ) -1 2 λ ( 0 1 0 0 ) ,  \nwhere y = y ( t ) , z = z ( t ). The compatibility condition [ M , M ] = 0 is equivalent to  \n1 2 (A.2) ˙ y = z, ˙ z = 6 y 2 + t so that y = 6 y 2 + t  \nwhich is the Painlev'e I equation. Now replace matrix generators of sl (2 , C ) by leftinvariant vector fields ( l 1 , l 2 , l 3 ) on SL (2 , C ) according to  \n( 1 / 2 0 0 -1 / 2 ) → l 1 , ( 0 1 0 0 ) → l 2 , ( 0 0 1 0 ) → l 3 ,  \nand remove the quadratic term from M 1 by taking the linear combination  \nL 1 = 2 λM 2 -M 1 , L 2 = 2 M 2 .  \nThis results in a Frobenius integrable Lax distribution { L 1 , L 2 } of the form (2.20) on X × CP 1 , where the four-manifold X is a product SL (2 , C ) × C , the four vector fields E ij ' are given by  \nt  \nE 10 ' = -2 zl 1 + y 2 + l 2 -4 yl 3 , E 21 ' = l 2 , (A.3)  \nE 11 ' = -2 + yl 2 , E 20 ' = 2 -(2 yl 2 +4 l 3 )  \n( 2 ) ∂ ∂ ,  \n∂t ∂t  \nand f 1 = -1 , f 2 = 0.  \nTo write the corresponding anti-self-dual metric explicitly in terms of the Painlev'e I transcendent introduce the left-invariant one-forms on SL (2 , C ) such that σ i ( l j ) = δ i j , and  \ndσ 1 = 2 σ 3 ∧ σ 2 , dσ 2 = σ 2 ∧ σ 1 , dσ 3 = σ 1 ∧ σ 3 .  \nThen g , when rescaled by 16 z ( t ) takes the cohomogeneity-one form  \n(A.4) g = σ 1 /circledot ( 12 y 2 +2 t z σ 1 +8 σ 2 -6 σ 3 ) + σ 3 /circledot ( zσ 3 +2 zdt ) .  \nThe 2-form 2 σ 3 ∧ σ 1 is parallel as a consequence of Painlev'e I. Consequently the metric is null-Kahler.  \nIt gives rise to a two-parameter family of α -surfaces in X spanned by vectors in its kernel. This family gives a divisor line bundle in the complex three-fold Z (this is the twistor space whose points are all α -surfaces in X ). This is a canonical divisor - a section s of the 1 / 4 power of the holomorphic canonical bundle of Z - in fact the existence of such section singles out a null-Kahler metric in an anti-self-dual conformal class [19]. The twistor space Z arises as a quotient  \nX r ←- F q -→ Z ,  \nof the correspondence space F = C × SL (2 , C ) × CP 1 by the distribution (2.20). There is an SL (2 , C ) action on the twistor space, and the holomorphic vector fields generating this action become linearly dependent on the divisor s = 0. The divisor is preserved by the action, so there is an element of the Lie algebra that maps to zero under the associated vector bundle homomorphism  \n(A.5) φ : sl (2 , C ) ×Z → T Z .  \nThis element is nilpotent, which distinguishes this from the twistor space of Painlev'e II (where the divisor s = 0 also meets each twistor line to order 4). The inverse of φ is the SL (2 , C ) connection with a pole of order 4 on the divisor { s = 0 } underlying the isomonodromy problem for Painlev'e I.  \nAt this point we can compare the metric (A.4) to the ASD Ricci-flat metric (6.8). Both metrics are anti-self-dual, null-Kahler, and arise from the isomonodomy problem  \nof Painlev'e I but that is where the similarities end. The metric (A.4) has cohomogeneity one, so any conformally related metric will admit SL (2 , C ) acting by conformal isometries. On the other hand the metric (6.8) has only one conformal Killing vector. Therefore the two are neither locally isometric, nor conformally related.", 'References': "- [1] Alday, L. F, Gaiotto, D, and Maldacena, J. (2011) Thermodynamic Bubble Ansatz, JHEP 09 .\n- [2] T. Adamo, L.J. Mason, A. Sharma, Twistor sigma models for quaternionic geometry and graviton scattering , arXiv:2103.16984v1 (2021).\n- [3] S. Alexandrov, B. Pioline, Conformal TBA for resolved conifolds , Ann. H. Poincar'e. 23 , 1909-1949, (2022)\n- [4] S. Alexandrov, B. Pioline, Heavenly metrics, BPS indices and twistors , Lett. Math. Phys. 111 (2021)\n- [5] M. Atiyah, M. Dunajski, L. J. Mason, Twistor theory at fifty: from contour integrals to twistor strings , Proceedings of the Royal Society, 473 (2017).\n- [6] A. Bobenko, Introduction to Compact Riemann Surfaces , Computational Approach to Riemann Surfaces, Lecture Notes in Mathematics, Springer (2013).\n- [7] T N. Bailey, M.G. Eastwood, Complex Paraconformal Manifolds - their Differential Geometry and Twistor Theory , Forum. Math. 3 1 61-104 (1991).\n- [8] T. Bridgeland, Joyce structures on spaces of quadratic differentials , arXiv:2203.17148v2 (2024).\n- [9] T. Bridgeland, Joyce structures on spaces of quadratic differentials I , arXiv:2203.17148v1 (2022).\n- [10] T. Bridgeland, Tau functions from Joyce structures , arXiv:2303.07061v1 (2023).\n- [11] T. Bridgeland, Geometry from Donaldson-Thomas invariants , Integrability, Quantization, and Geometry II. Quantum Theories and Algebraic Geometry, 1-66, Proc. Sympos. Pure Math. Amer. Math. Soc. (2021).\n- [12] T. Bridgeland, F. Del Monte, in preparation.\n- [13] T. Bridgeland, D. Masoero, On the monodromy of the deformed cubic oscillator , Math. Ann. 385 , 193-258 (2023).\n- [14] T. Bridgeland, I.A.B. Strachan, Complex hyperkahler structures defined by Donaldson-Thomas invariants , Lett. Math. Phys. 111 , 54 (2021).\n- [15] T. Bridgeland, I. Smith, Quadratic Differentials as Stability Conditions , Publ.math.IHES 121 , 155-278 (2015).\n- [16] D. Calderbank, Selfdual 4-Manifolds, Projective Surfaces, and the Dunajski-West Construction SIGMA 10 035 (2014).\n- [17] J. Carlson, S. Muller-Stach, C. Peters, Period Mappings and Period Domains , 2nd ed. Cambridge Studies in Advanced Mathematics (2017).\n- [18] B. Dubrovin, Geometry of 2D topological field theories , in Integrable Systems and Quantum Groups , Lecture Notes in Mathematics 1620 , Springer (1996).\n- [19] M. Dunajski, Null Kahler Geometry and Isomonodromic Deformations , Commun. Math. Phys. 391 , 77-105 (2022).\n- [20] M. Dunajski, Solitons, Instantons and Twistors , 2nd edition. Oxford Graduate Texts in Mathematics 31 (2024).\n- [21] M. Dunajski, The Nonlinear Graviton as an Integrable System. DPhil Thesis. Oxford. (1999).\n- [22] M. Dunajski, L. J. Mason, Twistor theory of hyper-Kahler metrics with hidden symmetries. J. Math. Phys. 44 , 3430-3454 (2003).\n- [23] M. Dunajski, K.P. Tod, Einstein-Weyl structures from hyper-Kahler metrics with conformal Killing vectors , Differ. Geom. App. 14 39-55 (2001).\n- [24] M. Dunajski, S. West, Anti-self-dual Conformal Structures with Null Killing Vectors from Projective Structures , Commun. Math. Phys. 272 , 85-118 (2007).\n- [25] N. J. Hitchin, Twistor spaces, Einstein metrics and isomonodromic deformations, J. Differential Geom. 42 , 30-112 (1995).\n- [26] P. Jones, K.P Tod, Minitwistor spaces and Einstein-Weyl spaces , Class. Quant. Grav. 2 ,4, 565-577 (1985).\n- [27] M. Jimbo, T. Miwa, K. Ueno, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients , Physica D 2 ,2, 306-352 (1981)  \n- [28] G. Mahoux, Introduction to the Theory of Isomonodromic Deformations of Linear Ordinary Differential Equations with Rational Coefficients , The Painlev'e Property, One Century Later, CRM Series in Mathematical Physics, Springer,(1999).\n- [29] L.J. Mason, E.T. Newman, A connection between the Einstein and Yang-Mills equations, Comm. Math. Phys., 121 , 659-668 (1989) .\n- [30] L.J. Mason, N.M.J Woodhouse, Integrability, Self-duality and Twistor theory , LMS Monograph, OUP, (1996).\n- [31] K. Okamoto, Polynomial Hamiltonians associated with Painlev'e equations, I. Proc. Japan Acad. Ser. A Math. Sci. 56 : 264-268 (1980).\n- [32] R. Penrose, Nonlinear gravitons and curved twistor theory, Gen. Rel. Grav. 7 , 31-52 (1976).\n- [33] R. Penrose, W. Rindler, Spinors and Space-time, vol. 1: Two-spinor calculus and relativistic fields , Cambridge University Press (1984).\n- [34] R. Penrose, W. Rindler, Spinors and Space-time, vol. 2: Spinor and twistor methods in space-time geometry , Cambridge University Press (1984).\n- [35] J. Pleba'nski, Some solutions of complex Einstein equations J. Math. Phys. 16 2395-2402. (1975).\n- [36] K. Saito, Period mapping associated to a primitive form , Publ. Res. Inst. Math. Sci. 19 (3), (1983).\n- [37] G. A. J. Sparling, K. P. Tod, An example of an H-space J. Math. Phys. 22 331-332 (1981).\n- [38] K. Takasaki, An infinite number of hidden variables in hyper-Kahler metrics. J. Math. Phys. 1515 (1989).\n- 30 ,\n- [39] K. Ueno, Monodromy Preserving Deformation of Linear Differential Equations with Irregular Singular Points Proc. Japan. Acad., 56 Ser. A (1980).  \nDepartment of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK.  \nEmail address : m.dunajski@damtp.cam.ac.uk  \nDepartment of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK.  \nEmail address : tjahm2@cam.ac.uk"}