Toric Ricciflat metrics in dimension four correspond to certain holomorphic vector bundles over a twistor space. We construct these bundles explicitly by exhibiting and characterising their patching matrices for the fiveparameter family of Riemannian ALF metrics constructed by Chen and Teo. The ChenTeo family contains a twoparameter family of asymptotically flat gravitational instantons. The patching matrices for these instantons take a simple rational form. Dedicated to Nick Woodhouse on the occasion of his 75th birthday.
['gravitational instantons', 'twistor theory', 'asymptotically flat toric metrics', 'General Relativity and Quantum Cosmology', 'High Energy Physics - Theory', 'Mathematics - Differential Geometry']
Twistor theory of the ChenTeo gravitational instanton
{'Twistor theory of the Chen-Teo gravitational instanton': 'Maciej Dunajski Department of Applied Mathematics and Theoretical Physics University of Cambridge Wilberforce Road, Cambridge CB3 0WA, UK m.dunajski@damtp.cam.ac.uk  \nand  \nPaul Tod The Mathematical Institute Oxford University Woodstock Road, Oxford OX2 6GG, UK. tod@maths.ox.ac.uk  \n29 May 2024  \nDedicated to Nick Woodhouse on the occasion of his 75th birthday', 'Abstract': 'Toric Ricci-flat metrics in dimension four correspond to certain holomorphic vector bundles over a twistor space. We construct these bundles explicitly, by exhibiting and characterising their patching matrices, for the five-parameter family of Riemannian ALF metrics constructed by Chen and Teo. The Chen-Teo family contains a two-parameter family of asymptotically flat gravitational instantons. The patching matrices for these instantons take a simple rational form.', '1 Introduction': "In [5] Chen and Teo constructed a five-parameter family of Riemannian Ricci-flat four-manifolds with torus symmetry. In the limiting cases this family contains the Riemannian Pleba'nski-Demia'nski solutions [13] and some multi-centred Gibbons-Hawking metrics [10], as well as a new two-parameter family of asymptotically flat gravitational instantons [4] on the four-manifold CP 2 \\ S 1 .  \nIt was shown by Louis Witten [16], and in a different form by Richard Ward [15], that any Lorentzian Ricci flat toric metric in four-dimensions corresponds to a static and axi-symmetric solution of the anti-self-dual Yang Mills equations (ASDYM) with gauge group SL (2 , C ) on flat space-time. These field equations are integrable by the twistor correspondence of Ward [14], and their solutions are encoded in rank-two holomorphic vector bundles over the twistor space PT ≡ CP 3 \\ CP 1 which are trivial on twistor lines. Any such bundle can be defined in terms a patching matrix, and the Ward correspondence gives an algorithm for recovering the ASDYM connection, and eventually the toric vacuum solution from such a matrix. This construction has primarily focused on Lorentzian toric vacuum solutions.  \nThe aim of this paper is to adapt this construction to Riemannian signature, and use it to put the Chen-Teo metrics in the twistor framework. In the next section we shall recall the Chen-Teo metrics. In § 3 we shall use the rod structure of these metrics to construct the patching matrix for the Ward bundle. It is a rational matrix with three simple poles, and asymptotics encoding conserved charges. In § 4 we discuss the reconstruction of the Chen-Teo patching matrix from a general patching matrix with three simple poles. The Ward correspondence for the reduced ASDYM equations is summarised in the Appendix.", 'Acknowledgements': 'We are grateful to James Lucietti for explaing to us how the mass of the Chen-Teo instanton which we compute in § 3.2 relates to the mass calculated in [11].', '2 The Chen-Teo Ricci-flat metrics': "Consider a quartic  \nf = f ( ξ ) = a 4 ξ 4 + a 3 ξ 3 + a 2 ξ 2 + a 1 ξ + a 0 (2.1)  \nwith four real roots and set  \nF = f ( x ) y 2 -f ( y ) x 2 H = ( νx + y )[( νx -y )( a 1 -a 3 xy ) -2(1 -ν )( a 0 -a 4 x 2 y 2 )] G = f ( x )[(2 ν -1) a 4 y 4 +2 νa 3 y 3 + a 0 ν 2 ] -f ( y )[ ν 2 a 4 x 4 +2 νa 1 x +(2 ν -1) a 0 ] .  \n(2.2)  \nLet ( x, y, τ, ϕ ) be local coordinates on an open set of a four-manifold M . The family of metrics  \ng = kH ( x -y ) 3 ( dx 2 f ( x ) -dy 2 f ( y ) -f ( x ) f ( y ) kF dϕ 2 ) + 1 FH ( x -y ) ( Fdτ + Gdϕ ) 2 (2.3)  \nis Ricci-flat [5] for any values of the constant real parameters ( k, ν, a 0 , · · · , a 4 ). Moreover two out of the five quartic coefficients can be fixed by constant scalings of the parameters and coordinates. Thus (2.3) is a five-dimensional family of toric metrics, where the torus symmetry is generated by constant translations of the ( τ, ϕ ) coordinates. In [2] it was subsequently shown, that all metrics in the Chen-Teo family are Hermitian, or equivalently one-sided type D : the self-dual part of the Weyl tensor is of Petrov-Penrose type D . This Hermitian structure is non-Kahler, unless the parameters are chosen so that the metrics are self-dual or anti-self-dual. It is however the case that all Chen-Teo metrics are conformal to Kahler [7]. Further global and local properties of (2.3) have been uncovered in [11, 3, 1].  \nLet r 1 < r 2 < r 3 < r 4 be four real roots of the quartic (2.1) assumed to exist. To avoid curvature singularities of the metric resulting from vanishing of H , Chen and Teo [4, 5] restrict the range of the coordinates ( x, y ) to a rectangle -∞ < r 1 < y < r 2 < x < r 3 < r 4 < 0 (Figure 1)  \nFigure 1. Domain of ( x, y ) dependence in the Chen-Teo metrics.  \n<!-- image -->  \nwith the diagonal x = y corresponding to the boundary at infinity of the asymptotic region. Further constraints must then be imposed on these roots, as well as the parameter ν , to avoid conical singularities and to ensure asymptotic flatness (AF). The limiting cases ν = 1 and ν = -1 correspond to the Pleba'nski-Demia'nski family [13], and the ALE Gibbons-Hawking metrics [10] with three collinear centres respectively.  \nThe AF condition is equivalent to setting the NUT parameter to zero, and this is in turn equivalent to the vanishing of n in (3.12) below. In the rest of this paper, with the exception of § 3.2 we shall consider the Chen-Teo rectangular domain but will not make any restrictions on the parameters.", '3 The Yang equation and its patching matrix': "The Chen-Teo metrics (2.3) can be put in the 'two commuting Killing vectors form'  \ng = Ω 2 ( dr 2 + dz 2 ) + J ij dϕ i dϕ j , i, j = 1 , 2 (3.1)  \nwhere ϕ i = ( ϕ, τ ), the Killing vectors are K i = ∂/∂ϕ i , and the two by two matrix J depends on the coordinates ( r, z ) on the space of orbits of the T 2 action, with  \nr 2 = det( J ) . (3.2)  \nBoth r and z are harmonic on the orbit space and once r is constructed from J , then z is found, up to additive constant, as its harmonic conjugate. The equation (3.2) implies that the matrix J drops its rank on the axis r = 0, where rank( J ) is generically equal to one. The rank drops to zero at points variously called turning points [4], nuts [12] or nodes : these are the isolated points corresponding to the z values where both Killing vectors vanish. The intervals connecting these points on the r = 0 axes are the rods , and there are two semi-infinite rods at the ends of the axis corresponding to the  \nedges of the rectangle from Figure 1 which meet on the diagonal at ( r 2 , r 2 ). The Ricci-flat condition on ( M,g ) gives the Yang equation  \nr -1 ∂ r ( rJ -1 ∂ r J ) + ∂ z ( J -1 ∂ z J ) = 0 (3.3)  \ntogether with a linear equation relating Ω and J .", '3.1 The patching matrix': "The Yang equation (3.3) also arises as a symmetry reduction of ASDYM by a translation and a rotation. It is therefore integrable by the Ward correspondence [14], and the twistor data consists of a patching matrix for a bundle on either the whole twistor space PT = CP 3 \\ CP 1 or its reduction to the non-Hausdorff mini-twistor space [15, 17, 9] (see Appendix A). The algorithm to construct this patching matrix consists of two steps [17]:  \n- 1. For any choice of U (1) ⊂ T 2 generated by the Killing vector K the twist of K is a gradient, i. e.  \ndψ = ∗ ( K ∧ dK ) . (3.4)  \nThen [9]  \nJ ' = 1 V ( 1 -ψ -ψ ψ 2 -V 2 ) , where V ≡ g ( K,K ) (3.5)  \nis another solution to (3.3) with 1 det( J ' ) = -1.  \n- 2. Pick a rod on which K is not identically zero. Then  \nP = J ' (0 , z ) (3.6)  \nis regular on this rod, except for the poles at its end-points, and is the patching matrix for the holomorphic vector bundle over the reduced twistor space.  \nWe shall implement this for the Chen-Teo class (2.3) which contains the four rods corresponding to the edges of the rectangle on Figure 1, with two-semi infinite outer rods R 1 , R 4 . To transform the metric to the ( r, z ) coordinates use  \nz = 2 a 4 x 2 y 2 +2 a 2 xy +2 a 0 + a 3 xy ( x + y ) + a 1 ( x + y ) 2( x -y ) 2  \nr = √ -f ( x ) f ( y ) ( x -y ) 2  \n,  \nand find the z -values at the turning points ( x, y ) = ( r 2 , r 1 ) , ( r 3 , r 1 ) , ( r 3 , r 2 ) to be  \nz 3 = -a 4 ( r 1 r 2 + r 3 r 4 ) / 2 , z 2 = -a 4 ( r 1 r 3 + r 2 r 4 ) / 2 , z 1 = -a 4 ( r 1 r 4 + r 2 r 3 ) / 2 .  \nThe rod structure is now encoded on the z -axis which contains four edges of the rectangle from Figure 1, with the edges meeting at ( x, y ) = ( r 2 , r 2 ) going out to ±∞ (Figure 2).  \nFigure 2. The rod structure of the Chen-Teo family.  \n<!-- image -->  \nWe shall focus on the semi-infinite rod, where x = r 2 . The Killing vector K = √ 1 -ν 2 ∂/∂τ does not vanish on this rod, and  \nV ≡ g ( K,K ) = (1 -ν 2 ) F ( x -y ) H . (3.7)  \nThe corresponding twist potential ψ in (3.4) is defined up to addition of a constant, and we shall choose this constant so that V -1 ψ tends to zero at the infinite end point x = y = r 2 ( or r = 0 , z = ∞ ) of the rod. This yields  \nψ = ( y -x )( a 4 x 2 y 2 (1 -ν ) 2 -a 3 νxy ( x + y ) + a 1 ν ( x + y ) + a 0 (1 -ν ) 2 ) H (3.8)  \nwhere H is given by (2.2). To transform ψ and V to the ( r, z ) coordinates we shall set x = r 2 (this defines rod R 1 ), and solve for y in terms of z as  \ny = 2 r 2 z + a 4 ( r 1 r 2 r 3 + r 1 r 2 r 4 -r 1 r 3 r 4 + r 3 r 2 r 4 ) 2 z + a 4 ( r 1 r 2 + r 3 r 2 + r 4 r 2 -r 2 2 ) .  \nSubstitute this to (3.8) and (3.7) to find formulae for ψ and V which are rational functions of z . Substituting these formulae into (3.5) and (3.6) yields the twistor patching matrix  \nP ( z ) = ( C 1 /C Q/C Q/C C 2 /C ) , (3.9)  \nwhere Q is a quadratic and C, C 1 , C 2 are monic cubics given by  \nC = ( z -z 1 )( z -z 2 )( z -z 3 ) = z 3 + c 2 z 2 + c 1 z + c 0 C 1 = z 3 + c 12 z 2 + c 11 z + c 10 C 2 = -z 3 + c 22 z 2 + c 21 z + c 20 Q = q 2 z 2 + q 1 z + q 0 . (3.10)  \nThe coefficients of c i , c 1 i , c 2 i , q i , i = 0 , 1 , 2 of C, C 1 , C 2 , Q are listed in Appendix B.  \nThis patching matrix has three poles, but two of them do not belong to the rod we are considering. There is a known procedure [8, 9] to pass through a turning point z = z 3 and construct the patching matrix corresponding to the adjacent rod connecting z 3 to z 2 . This can be repeated to construct patching matrices for all four rods starting from P in (3.9). The asymptotics of P near the semiinfinite end of the rod is  \nP ∼ = ( 1 0 0 -1 ) + 1 z ( 2 m 2 n 2 n 2 m ) + O (1 /z 2 ) , (3.11)  \nwhere  \nm = a 4 ( 2 νr 3 2 + ( ν 2 +1 ) ( r 1 + r 3 + r 4 ) r 2 2 -( r 3 r 1 + r 4 r 1 + r 3 r 4 ) ( ν 2 +1 ) r 2 -2 νr 1 r 3 r 4 ) 2 (2 ν 2 -2) r 2 (3.12) n = q 2 2 = ( -2 ν ( r 1 + r 3 + r 4 ) r 2 2 +2 ν ( r 3 r 1 + r 4 r 1 + r 3 r 4 ) r 2 + ( ν 2 +1 ) ( r 4 r 3 r 1 -r 2 3 ) ) a 4 (4 ν 2 -4) r 2 .  \nJames Lucietti has pointed out to us how ( m,n ) are related to the mass and NUT parameter which we shall now explain. To exhibit the ALF property of their metrics Chen and Teo define (˜ τ, ˜ ϕ ) by  \nτ = √ 1 -ν 2 ˜ τ + b ˜ ϕ, ϕ = c ˜ ϕ,  \nwhere  \nc = -√ k 2 r 2 a 4 ( r 2 -r 1 )( r 2 -r 3 )( r 2 -r 4 ) (3.13) b = √ k (1 -ν 2 )( r 2 3 + r 1 r 3 r 4 ) + 2 νr 2 ( r 1 r 2 + r 1 r 3 + r 1 r 4 + r 2 r 3 + r 2 r 4 + r 3 r 4 ) ( r 1 -r 2 )( r 2 -r 3 )( r 2 -r 4 )  \nso that ∂/∂ ˜ τ and ∂/∂ ˜ ϕ generate the asymptotic Euclidean time translation and rotation respectively and moreover ∂/∂ ˜ τ agrees with the Killing vector K we used to compute the patching matrix (3.9). The canonical coordinates (˜ z, ˜ r ) on the space of orbits M/T 2 are related to our ( z, r ) in (3.1) by  \nz = γ -1 ˜ z, r = γ -1 ˜ r, where γ = c √ 1 -ν 2 .  \nMaking the substitution for z in the patching matrix (3.9) removes the dependence of a 4 but introduces the dependence of the scaling parameter k so that the new parameters in the patching matrix are ( k, r 1 , r 2 , r 3 , r 4 , ν ). To maintain the monic form in ˜ z of the cubics C, C 1 , C 2 in (3.9), the quadratic, linear and 0th order terms of C, C 1 , C 2 , Q scale with γ, γ 2 , γ 3 respectively. This results in scalings of ( m,n ) in (3.12)  \n˜ m = γm, ˜ n = γn. (3.14)  \nIn particular ˜ n is the NUT parameter found by Chen-Teo in [5] (compare their formula (3.10)).", '3.2 The Chen-Teo instanton': 'The expression for the patching matrix simplifies considerably in the case when (2.3) is asymptotically flat and regular on the four-manifold M = CP 2 \\ S 1 . It was shown by Chen and Teo [4, 5], that the subclass of metrics (2.3) with these properties is parametrised by ( k, s ) with  \nr 1 = 4 s 2 (1 -s ) 1 -2 s +2 s 2 , r 2 = -1 , r 3 = 1 -2 s s (1 -2 s +2 s 2 ) , r 4 = ∞ where ν = -2 s 2  \nand s ∈ (1 / 2 , √ 2 / 2). We find that the patching matrix is given by (3.9) with  \n3  \nQ = (1 -s ) ( -1 + 2 s ) ( 2 s 2 -1 ) ( 2 s 2 -s +1 ) 2 (2 s 2 +1)(2 s 2 -2 s +1) 4 ( z + s 2 s 2 -s +1 ) C = ( z + 1 2 )( z -2 s 2 ( -1 + s ) 2 s 2 -2 s +1 )( z + -1 + 2 s 2 s (2 s 2 -2 s +1) ) C 1 = z 3 + c 12 z 2 + c 11 z + c 10 C 2 = -z 3 + c 22 z 2 + c 21 z + c 20  \nwith  \nc 10 = -( -1 + 2 s ) ( -1 + s ) ( 4 s 5 -4 s 4 +2 s 3 -4 s 2 +4 s -1 ) (2 s 2 +1)(2 s 2 -2 s +1) 4  \nc 11 = 32 s 12 -128 s 11 +240 s 10 -384 s 9 +616 s 8 -768 s 7 +684 s 6 -448 s 5 +224 s 4 -88 s 3 +30 s 2 -8 s +1 4 s 2 (2 s 2 +1)(2 s 2 -2 s +1) 4  \nc 12 = 4 s 5 -8 s 4 +8 s 3 -6 s 2 +4 s -1 s (2 s 2 -2 s +1) 2  \nc 20 = -2 ( 8 s 5 -16 s 4 +8 s 3 -2 s 2 +2 s -1 ) s 3 ( -1 + 2 s ) ( -1 + s ) (2 s 2 +1)(2 s 2 -2 s +1) 4  \nc 21 = -128 s 12 -512 s 11 +960 s 10 -1408 s 9 +1792 s 8 -1792 s 7 +1368 s 6 -768 s 5 +308 s 4 -96 s 3 +30 s 2 -8 s +1 4 (2 s 2 +1)(2 s 2 -2 s +1) 4  \nc 22 = 8 s 5 -16 s 4 +12 s 3 -8 s 2 +4 s -1 (2 s 2 -2 s +1) 2 .  \nThe asymptotic charges (3.12) are  \nm = (2 s 2 -1)(2 s 2 +1)( s -1)(2 s -1) 4(2 s 2 -2 s +1) , n = 0 . (3.15)  \nThe vanishing of the NUT parameter n ensures asymptotic flatness, while the mass parameter m is positive in the allowed range of s . The constant c in (3.13) is now given by 2 √ kr 2 ( r 2 -r 1 ) -1 ( r 2 -r 3 ) -1 so that the scaled mass parameter (3 . 14)  \n˜ m = √ k (1 + 2 s 2 ) 2 2 √ 1 -4 s 4  \nagrees with the Kunduri-Lucietti mass found in [11] (see their formula (169)).', '4 An ansatz for the patching matrix': 'Conversely, let us consider a general symmetric patching matrix P = P ( z ) with rational coefficients and three simple poles, and such that lim z = ∞ P = diag(1 , -1). The most general form of such matrix is (3.9), with (3.10) but with the cubics C, C 1 , C 2 and quadratic Q having general coefficients. Thus P depends on 12 parameters. The condition  \ndet P = -1 (4.1)  \ngives six conditions on the parameters  \n- 0 = 2 c -c + c\n- 2 12 22 0 = c 12 c 22 + c 2 2 -q 2 2 +2 c 1 -c 11 + c 21 0 = 2 c 1 c 2 + c 11 c 22 + c 12 c 21 -2 q 1 q 2 +2 c 0 -c 10 + c 20 0 = 2 c 0 c 2 + c 1 2 + c 10 c 22 + c 11 c 21 + c 12 c 20 -2 q 0 q 2 -q 1 2 0 = 2 c 0 c 1 + c 10 c 21 + c 11 c 20 -2 q 0 q 1 0 = c 0 2 + c 10 c 20 -q 0 2 .  \nThe first of these equations is solved by setting  \nc 12 = 2 m + c 2 , c 22 = 2 m -c 2 ,  \nwhere the constant m can be identified with the mass in the asymptotic form (3.11). Four of the remaining five equations can be solved for ( c 0 , c 1 , c 10 , c 20 ), and the solutions are unique (as the equations are linear). The final quadratic for q 0 can also be solved, but this introduces branching. Now ( c 10 , c 12 , c 20 , c 22 , c 0 , c 1 , q 0 ) have been determined in terms of six constants  \nc 11 , c 21 , c 2 , q 1 , q 2 , m. (4.2)  \nWe go back to (B.1) and (3.12) and consider the six resulting equations for the Chen-Teo parameters ( a 4 , r 1 , r 2 , r 3 , r 4 , ν ) (the same argument applies if we use the parameters ( k, r 1 , r 2 , r 3 , r 4 , ν ) resulting from the scalings (3.14)). If we could invert the equations, and thus show that these parameters can be determined in terms of the constants (4.2), then we could deduce that the patching matrices with three simple poles and these asymptotics result precisely from the Chen-Teo metrics. A solution to the six equations is the intersection of six hyper-surfaces. Therefore it is enough to show that the six normal vector fields are linearly independent. Computing the gradients of (4.2) and using (B.1) and (3.12), we find that they are in fact linearly dependent everywhere, and the resulting 6 by 6 matrix M of coefficients has rank 5. There is a residual freedom in P which preserves its symmetry and the asymptotic form diag(1 , -1). It is given by  \nP → LPL T , where L = ( cosh t sinh t sinh t cosh t ) ∈ SO (1 , 1)  \nand t is a constant. This results in  \nC 1 ( t ) = C 1 cosh t 2 + C 2 sinh( t ) 2 + Q sinh (2 t ) C 2 ( t ) = C 1 sinh t 2 + C 2 cosh( t ) 2 + Q sinh (2 t ) ,  \nQ ( t ) = Q (cosh ( t ) 2 +sinh( t ) 2 ) + 1 2 ( C 1 + C 2 ) sinh (2 t )  \nand leaves the cubics C and C 1 -C 2 invariant. It is however the case that the rank of the transformed matrix M ( t ) is still 5. Therefore there is one invariant syzygy between the parameters (4.2) coming from the Chen-Teo metrics. Using the combined results of [2] and [3] suggests that the ansatz (3.9) contains some toric-Einstein metrics which are not Hermitian or not regular in that their singularities are worse than conical.', 'A Twistor correspondence for the Yang equation': 'In this Appendix we shall summarise the Ward twistor correspondence [14] for anti-self-dual YangMills (ASDYM) equations in a gauge where they are equivalent to a single non-linear matrix PDE. We shall follow the notation and conventions of [6].', 'A.1 The Yang equation': 'Let ( W,Z, ˜ W, ˜ Z ) be double null coordinates on the complexified Minkowski space M C = C 4 with the holomorphic metric ds 2 and the volume form vol given by  \nds 2 = 2( dZd ˜ Z -dWd ˜ W ) , vol = dW ∧ d ˜ W ∧ dZ ∧ d ˜ Z. (A.1)  \nLet A be an sl (2)-valued connection one-form with the curvature F = dA + A ∧ A and a covariant derivative D = d + A . The ASDYM condition F = -⋆ F takes the form  \nF WZ = 0 , F ˜ W ˜ Z = 0 , F W ˜ W -F Z ˜ Z = 0 . (A.2)  \nThese equations arise as the compatibility conditions [ L, M ] = 0 for an over-determined linear system of PDEs  \nL Ψ = 0 , M Ψ = 0 , where L = D ˜ Z -λD W , M = D ˜ W -λD Z . (A.3)  \nHere λ ∈ CP 1 is the spectral parameter, and Ψ = Ψ( W,Z, ˜ W, ˜ Z,λ ) takes its values in the group SL (2 , C ) (other gauge groups are also possible).  \nThe first equation in (A.2) implies the existence of a gauge such that A W = A Z = 0. The second equation then implies the existence of an SL (2 , C ) valued function J = J ( W,Z, ˜ W, ˜ Z ) such that  \nA = J -1 ∂ ˜ W Jd ˜ W + J -1 ∂ ˜ Z Jd ˜ Z. (A.4)  \nFinally the last equation in (A.2) becomes the so-called Yang equation  \n∂ Z ( J -1 ∂ ˜ Z J ) -∂ W ( J -1 ∂ ˜ W J ) = 0 . (A.5)  \nTherefore the Yang equation is equivalent to ASDYM together with a particular choice of gauge.', 'A.2 Twistor correspondence': 'The ASDYM equations (A.2) are equivalent to the flatness of the connection A on self-dual totally null two-planes ( α -planes in twistor terminology). All such planes in M C are of the form  \nα = W + λ ˜ Z, β = Z + λ ˜ W (A.6)  \n̸  \nand the space of these planes is the projective twistor space PT ≡ CP 3 \\ CP 1 with affine coordinates ( α, β, λ ) covering the open set where λ = ∞ . The vector space of parallel sections of the Yang-Mills bundle E → M C on each α -plane defines a fibre of a holomorphic rank-2 vector bundle E → PT . For any fixed point p ∈ M C the equations (A.6) define a rational curve L p ∼ = CP 1 in the twistor space. The sections of E constant on all α -planes through p can be compared at p , and thus the twistor bundle E is trivial on all lines (A.6).  \n̸  \n̸  \nConversely, let E -→ PT be a rank-2 vector bundle trivial on lines. Cover PT with two open sets: U where λ = ∞ , and ˜ U where λ = 0. Let P U ˜ U = P ( α, β, λ ) be the patching matrix of the bundle which is holomorphic on U ∩ ˜ U . Restrict P to a twistor line (A.6) where the bundle is trivial so that a Riemann-Hilbert splitting  \nP U ˜ U = P U P ˜ U -1 (A.7)  \ncan be performed with P U and P ˜ U holomorphic and non-singular in U and ˜ U respectively. The patching matrix P U ˜ U is constant over the α -planes spanned by l = ∂ ˜ Z -λ∂ W , and m = ∂ ˜ W -λ∂ Z . Therefore the quantities P U -1 l ( P U ) = P ˜ U -1 l ( P ˜ U ) , P U -1 m ( P U ) = P ˜ U -1 m ( P ˜ U ) are global on CP 1 , and by the application of the Liouville theorem there exist sl (2 , C ) valued one-form A on the complexified Minkowski space such that  \nP -1 U l ( P U ) = P ˜ U -1 l ( P ˜ U ) = A ˜ Z -λA W , P -1 U m ( P U ) = P ˜ U -1 m ( P ˜ U ) = A ˜ W -λA Z . (A.8)  \nSetting H = P U ( λ = 0) , ˜ H = P ˜ U ( λ = ∞ ) then gives  \nA = H -1 ∂ Z H dZ + H -1 ∂ W H dW + ˜ H -1 ∂ ˜ Z ˜ H d ˜ Z + ˜ H -1 ∂ ˜ W ˜ H d ˜ W  \nwhich is gauge-equivalent to (A.4) with J = ˜ HH -1 . Applying l and m to the first and the second relation in (A.8) and commuting derivatives then implies that A is a solution of ASDYM, or equivalently that J satisfies the Yang equation.', 'A.3 Symmetry reduction': 'We shall use the cylindrical polar coordinates  \nZ = t + z, ˜ Z = t -z, W = re iθ , ˜ W = re -iθ  \nso that  \nds 2 = 2( dt 2 -dz 2 -dr 2 -r 2 dθ 2 ) ,  \nand assume that the gauge potential is invariant under a 2-dimensional Abelian group generated by a translation ∂ t and a rotation ∂ θ . We chose J = J ( r, z ) in which case the Yang equation (A.5) reduces to equation (3.3) locally characterising the toric Ricci-flat four-manifolds. The Killing vectors K 1 = ∂ t and K 2 = ∂ θ on M C give rise to holomorphic vector fields ( ˆ K 1 , ˆ K 2 ) on PT . These vector fields have zeros, so the reduced twistor space defined to be the corresponding quotient of PT is a one-dimensional non-Hausdorff complex manifold [8, 9, 17]. Rather than taking this quotient we can consider the holomorphic vector bundle E → PT as defined in § A.2 such that the patching matrix (A.7) depends only on one complex variable γ constant along ( ˆ K 1 , ˆ K 2 ) . The incidence relation (A.6) then reduces to  \nγ = z + 1 2 r ( λ -1 λ ) .  \nTherefore knowing the patching matrix at a rod corresponding to a fixed value of r , as in formula (3.9) suffices (note that iP in this formula belongs to SL (2 , C )), by analytic continuation, to determine it completely on an intersection of open sets covering PT . If two solutions of the Yang equation (3.3) are related by the Backlund transformation (3.5), then the corresponding patching matrices are related by a conjugation  \nP → ( 1 0 γ -1 1 ) -1 P ( 1 0 γ -1 1 ) .', 'B Coefficients of the patching matrix': 'Coefficients of the cubics and the quadratic (3.10) 2  \nc 0 = a 4 3 8 ( r 1 r 3 + r 2 r 4 ) ( r 1 r 2 + r 3 r 4 ) ( r 1 r 4 + r 2 r 3 ) c 1 = a 4 2 4 ( ( r 1 r 3 + r 2 r 4 ) ( r 1 r 2 + r 3 r 4 ) + ( r 1 r 2 + r 1 r 3 + r 2 r 4 + r 3 r 4 ) ( r 1 r 4 + r 2 r 3 ) ) (B.1) c 2 = a 4 2 ( r 1 r 2 + r 1 r 3 + r 1 r 4 + r 2 r 3 + r 2 r 4 + r 3 r 4 )  \nc 10 = 1 (4 ν 2 -4) r 2 ( a 3 4 ( ν ( r 3 r 1 + r 4 r 1 + r 3 r 4 ) r 2 2 -r 1 r 3 r 4 ( ν -1) r 2 -r 1 r 3 r 4 ( r 1 + r 3 + r 4 ) ) ) ( ( -νr 3 2 + ν ( r 1 + r 3 + r 4 ) r 2 2 +( r 3 r 1 + r 4 r 1 + r 3 r 4 ) r 2 -r 1 r 3 r 4 ) )  \nc 11 = 1 (4 -4 ν 2 ) r 2 ( a 2 4 ( ν 2 r 4 2 r 2 1 +4 ν 2 r 1 r 4 2 r 3 +4 ν 2 r 1 r 4 2 r 4 -2 r 1 r 3 r 4 r 3 2 ν 2 -ν 2 r 6 2 + ν 2 r 4 2 r 2 3 +4 ν 2 r 4 2 r 3 r 4 + ν 2 r 4 2 r 2 4 -4 r 2 1 r 3 r 4 r 2 2 ν +4 νr 1 r 4 2 r 3 +4 νr 1 r 4 2 r 4 -4 r 1 r 2 3 r 4 r 2 2 ν -4 r 1 r 3 r 2 4 r 2 2 ν +4 νr 4 2 r 3 r 4 -r 2 1 r 2 2 r 2 3 -4 r 2 1 r 3 r 4 r 2 2 -r 2 1 r 2 2 r 2 4 + r 2 1 r 2 3 r 2 4 +2 r 1 r 3 r 4 r 3 2 -4 r 1 r 2 3 r 4 r 2 2 -4 r 1 r 3 r 2 4 r 2 2 -r 2 2 r 2 3 r 2 4 ) ) c 12 = a 4 ( νr 3 2 + ν 2 ( r 1 + r 3 + r 4 ) r 2 2 -( r 3 r 1 + r 4 r 1 + r 3 r 4 ) r 2 -νr 1 r 3 r 4 ) r 2 ( ν 2 -1)  \nc 20 = 1 (4 -4 ν 2 ) r 2 ( a 4 3 ( ( -r 3 r 1 -r 4 r 1 -r 3 r 4 ) r 2 2 -r 1 r 3 r 4 ( ν -1) r 2 + νr 1 r 3 r 4 ( r 1 + r 3 + r 4 ) ) ) ( -r 3 2 +( r 1 + r 3 + r 4 ) r 2 2 + ν ( r 3 r 1 + r 4 r 1 + r 3 r 4 ) r 2 -νr 1 r 3 r 4 ) c 21 = -1 (4 -4 ν 2 ) r 2 ( a 2 4 ( r 2 1 r 2 2 r 2 3 ν 2 +4 r 2 1 r 3 r 4 r 2 2 ν 2 + r 2 1 r 2 2 r 2 4 ν 2 -r 2 1 r 2 3 r 2 4 ν 2 -2 r 1 r 3 r 4 r 3 2 ν 2 +4 r 1 r 2 3 r 4 r 2 2 ν 2 +4 r 1 r 3 r 2 4 r 2 2 ν 2 + r 2 2 r 2 3 r 2 4 ν 2 +4 r 2 1 r 3 r 4 r 2 2 ν -4 νr 1 r 4 2 r 3 -4 νr 1 r 4 2 r 4 +4 r 1 r 2 3 r 4 r 2 2 ν +4 r 1 r 3 r 2 4 r 2 2 ν -4 νr 4 2 r 3 r 4 -r 4 2 r 2 1 -4 r 1 r 4 2 r 3 -4 r 1 r 4 2 r 4 +2 r 1 r 3 r 4 r 3 2 + r 6 2 -r 4 2 r 2 3 -4 r 4 2 r 3 r 4 -r 4 2 r 2 4 ) ) c 22 = a 4 ( νr 3 2 +( r 1 + r 3 + r 4 ) r 2 2 -( r 3 r 1 + r 4 r 1 + r 3 r 4 ) ν 2 r 2 -νr 1 r 3 r 4 ) r 2 ( ν 2 -1) q 2 = ( -2 ν ( r 1 + r 3 + r 4 ) r 2 2 +2 ν ( r 3 r 1 + r 4 r 1 + r 3 r 4 ) r 2 + ( ν 2 +1 ) ( r 4 r 3 r 1 -r 2 3 ) ) a 4 (2 ν 2 -2) r 2 . 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