Elementary Deformations HK/QK. Oscar MACIA. Introduction [0] Physics [8] Twist [20] Example [42]

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2 deformations and the HyperKaehler/Quaternionic Kaehler correspondence. (Joint work with Prof. A.F. SWANN) Dept. Geometry Topology University of Valencia (Spain) June 26, 2014

3 What do we have

4 Berger s list Theorem (Berger, 1955) Let M be a oriented symply-connected n-dimensional Riemannian manifold which is neither locally a product nor symmetric. Then its holonomy group belongs to the following list.

5 Kaehler manifolds Kaehler manifolds are Riemannian 2m-dimensional manifold with holonomy group Hol U(m). They admit mutually compatible Riemannian, Complex and Symplectic structures. 1 acs: I End(T M) : I 2 = Id 2 Hermitian metric: g(ix, IY ) = g(x, Y ) 3 Kaehler 2-form: ω(x, Y ) = g(ix, Y ) non-degenerate. Kaehler condition: LC I = 0 or dω = 0.

6 Hyperkaehler manifolds HyperKaehler manifolds are 4k-dimensional Riemannian manifolds with holonomy group Sp(k). H = K Q H {r = 0, s = 0}. H {I, J, K} : I 2 = J 2 = K 2 = IJK = Id LC I = LC J = LC K = 0, dω I = dω J = dω K = 0. eg., 4D Kaehler & Ricci flat, K3-surfaces, Beauville s S [r] manifolds.

7 Quaternionic Kaehler manifolds Quaternionic Kaehler manifolds are 4k-dimensional Riemannian manifolds with holonomy group Hol Sp(k)Sp(1). Q K s = constant. Q {I, J, K} : I 2 = J 2 = K 2 = IJK = Id Ω = ω 2 I + ω 2 J + ω 2 K LC Ω = 0 eg., Wolf spaces (compact,positive,symmetric), Alekseevsky spaces (non-compact, homogeneous not necessarily symmetric).

8 (First) motivating problem 1 FIND EXPLICIT METRICS WITH SPECIAL HOLONOMY

9 (First) motivating problem 1 FIND EXPLICIT METRICS WITH SPECIAL HOLONOMY 2 HK and QK, in particular

10 Hint The mathematical problems that have been solved or techniques that have arisen out of physics in the past have been the lifeblood of mathematics. Sir Michael F. Atiyah Collected Works Vol. 1 (1988), 19, p.13

11 c-map as Superstring theory T -duality

12 c-map as Superstring theory T -duality (Cecotti, Ferrara, Girardello, 1989; Ferrara, Sabharwal, 1990) c Q 4(n+1) S 2n

13 c-map as Superstring theory T -duality (Cecotti, Ferrara, Girardello, 1989; Ferrara, Sabharwal, 1990) c Q 4(n+1) S 2n In the simplest SU SY case C 2n c H 4n

14 Historic relevance of the c-map for homogeneous QK (Alekseevsky, 1975) 1 Completely solvable Lie groups admitting QK metrics.

15 Historic relevance of the c-map for homogeneous QK (Alekseevsky, 1975) 1 Completely solvable Lie groups admitting QK metrics. 2 All known homogeneous non-symmetric QK manifolds.

16 Historic relevance of the c-map for homogeneous QK (Alekseevsky, 1975) 1 Completely solvable Lie groups admitting QK metrics. 2 All known homogeneous non-symmetric QK manifolds. (Cecotti, Ferrara, Girardello, 1989; Ferrara, Sabharwal, 1990) c Q 4(n+1) S 2n

17 Historic relevance of the c-map for homogeneous QK (Alekseevsky, 1975) 1 Completely solvable Lie groups admitting QK metrics. 2 All known homogeneous non-symmetric QK manifolds. (Cecotti, Ferrara, Girardello, 1989; Ferrara, Sabharwal, 1990) c Q 4(n+1) S 2n (de Witt, Van Proeyen, 1992) Use c-map to complete Alekseevsky s classification.

18 Historic relevance of the c-map for homogeneous QK (Alekseevsky, 1975) 1 Completely solvable Lie groups admitting QK metrics. 2 All known homogeneous non-symmetric QK manifolds. (Cecotti, Ferrara, Girardello, 1989; Ferrara, Sabharwal, 1990) c Q 4(n+1) S 2n (de Witt, Van Proeyen, 1992) Use c-map to complete Alekseevsky s classification. (Cortes, 1996) Complete classification (without c-map).

19 Special Kaehler manifolds I A Special Kaehler manifold is a K manifold (M, g, I, ω) together with a secondary connection (the special connection): R( ) = T( ) = 0, ω I = 0, X IY = Y IX.

20 Special Kaehler manifolds I A Special Kaehler manifold is a K manifold (M, g, I, ω) together with a secondary connection (the special connection): R( ) = T( ) = 0, ω I = 0, X IY = Y IX. A conic special Kaehler manifold C is a SK manifold (M, g, I, ω, ) together with a distinguished vector field X (the conic isometry) satisfying: 1 g(x, X) is nowhere vanishing; 2 X = I = LC X. The conic structure is periodic or quasiregular if it exponentiates to a circle action, regular if the circle action is free.

21 Special Kaehler manifolds II A projective special Kaehler manifold S is a Kaehler quotient S = C// c X = µ 1 (c)/x of a conic SK manifold by a conic isometry X at some level c R, together with the data necessary to reconstrut C up to equivalence. C 0 = µ 1 (c) C C// c X = S

22 N = 2, D = 4 SIMPLE SUPERGRAVITY The field content of N = 2, D = 4 SUGRA: 1 n-vector multiplets (A iˆµ, λi Λ, φ i ) 2 the gravity multiplet (V aˆµ, ψλ, A0ˆµ ). i = 1,..., n and ˆµ = 0, 1, 2, 3. In the σ-model apprach, the scalar fields are interpreted as coordintes of some differentiable manifold. 1 The manifold defined by the vector multiplet has n complex coordinates φ i is projective special Kaehler.

23 Kaluza Klein compactification of the metric field When we reduce from D = 4 to D = 3 by Kaluza Klein compactification the space-time metric (a 4D-tensor g (4) ˆµˆν ) splits as g (4) ˆµˆν = ( e iσ g µν (3) + e 2σ A µ A ν e 2σ A µ e 2σ A ν e 2σ increasing the numer of independent fields: 1 the 3D-metric tensor g (3) µν ; 2 a 3D-vector field A µ ; 3 a new scalar field φ = e 2σ. µ, ν = 0, 1, 2. )

24 Kaluza-Klein compactification on vector fields The n 4D-vector fields Aiˆµ (in vector mulitplets) and the graviphoton A0ˆµ also split into (n + 1) 3D-vectors and (n + 1) scalars, called axions. (A iˆµ, A0ˆµ ) = (Ai µ, A i 3, A 0 µ, A 0 3) 1 Axions ζ Λ = A i 3, A 0 3, Λ = 0,..., n. 2 Nevertheless, 3D-vectors in 3D are T-dual to scalars therefore each A i µ, A 0 µ gives an extra scalar ζ Λ, n + 1 in total. The 3D-gravitational vector A µ also by T -duality gives rise to an extra scalar a.

25 Scalar fields in D = 3. After the Kaluza Klein compactification we have the following list of scalar fields: SUGRA SUSY φ i 2n 2n φ 1 ζ Λ n + 1 n ζ Λ n + 1 n a 1 TOTAL 4(n + 1) 4n Q H

26 The general picture arising from physics C 2(n+1) c H 4(n+1) // cx? S 2n c Q 4(n+1)

27 The Twist Construction (sketch) The Twist Construction associates to a manifolds M with a S 1 -action (generated by X) a new space W of the same dimension with distinguished vector field Y. This construction fits in to a double fibration Y π P π W X M twist W so W is M TWISTED by the S 1 -bndle P.

28 Twists & correspondence (Joyce, 1992; Grantcharov, Poon, 2001) Instanton twists (Hypercomplex, quaternionic, HKT).

29 Twists & correspondence (Joyce, 1992; Grantcharov, Poon, 2001) Instanton twists (Hypercomplex, quaternionic, HKT). (Swann,2007,2010) General twists (T-duality, HKT, KT, SKT,...)

30 Twists & correspondence (Joyce, 1992; Grantcharov, Poon, 2001) Instanton twists (Hypercomplex, quaternionic, HKT). (Swann,2007,2010) General twists (T-duality, HKT, KT, SKT,...) (Haydys, 2008) correspondence { HK symmetry fixing one ω } { QK cirlce action }

31 Twists & correspondence (Joyce, 1992; Grantcharov, Poon, 2001) Instanton twists (Hypercomplex, quaternionic, HKT). (Swann,2007,2010) General twists (T-duality, HKT, KT, SKT,...) (Haydys, 2008) correspondence { HK symmetry fixing one ω } { QK cirlce action } (Hitchin, 2013) Twistor interpretation.

32 Twists & correspondence (Joyce, 1992; Grantcharov, Poon, 2001) Instanton twists (Hypercomplex, quaternionic, HKT). (Swann,2007,2010) General twists (T-duality, HKT, KT, SKT,...) (Haydys, 2008) correspondence { HK symmetry fixing one ω } { QK cirlce action } (Hitchin, 2013) Twistor interpretation. (Alekseevsky, Cortes, Dyckmanns, Mohaupt, 2013) c-map QK 4(n+1) correspondence

33 Objective To give an explanation to the correspondence arising in the c-map... C 2(n+1) c H 4(n+1) // cx? S 2n c Q 4(n+1)

34 Objective To give an explanation to the correspondence arising in the c-map... C 2(n+1) c H 4(n+1) π M // cx twist P π W S 2n c Q 4(n+1)... using the Twist Construction.

35 The Twist Construction in detail (Swann, 2007,2010) π M P π W M W 1 P M a principal S 1 -bundle, with a symmetry Y, connection 1-form θ, curvature π M F = dθ.

36 The Twist Construction in detail (Swann, 2007,2010) π M P π W M W 1 P M a principal S 1 -bundle, with a symmetry Y, connection 1-form θ, curvature π M F = dθ. 2 X X M generating a S 1 -action preserving F, L X F = 0

37 The Twist Construction in detail (Swann, 2007,2010) π M P π W M W 1 P M a principal S 1 -bundle, with a symmetry Y, connection 1-form θ, curvature π M F = dθ. 2 X X M generating a S 1 -action preserving F, L X F = 0 3 X = ˆX + ay X P preserving θ and Y : ˆX H = ker θ, π M ˆX = X and da = X F.

38 The Twist Construction in detail (Swann, 2007,2010) π M P π W M W 1 P M a principal S 1 -bundle, with a symmetry Y, connection 1-form θ, curvature π M F = dθ. 2 X X M generating a S 1 -action preserving F, L X F = 0 3 X = ˆX + ay X P preserving θ and Y : ˆX H = ker θ, π M ˆX = X and da = X F. 4 W = P/X has an action induced by Y.

39 Twist data P π M π W M W M manifold; X X M, circle action; F Ω 2 M closed, X-invariant, with integral periods; a CM with da = X F.

40 H-related tensors. M π M P π W Horizontal distribution: H = ker θ T P α tensor on M is H related to α W on W if Write α H α W. π M α = π W α W W on H = ker θ Lemma (Swann, 2010) Each X-invariant p-form α Ω p M is H-related to a unique p-form α W Ω p W given by π α = W W π α θ π (a 1 X α) M

41 Twist computations Lemma (S 10) If α Ω p M is a X-invariant p-form on M with exterior differential dα, and α H α W then dα W H dα 1 F (X α). a where α W H α, dα W H d W α d W := d 1 a F X is the twisted exterior differential. ( (Λ W, d) H Λ X, d ) M W

42 Twist and complex structures In general, twisting do not preserve integrability. Proposition (S 10) For an invariant complex structure I on M that is H-related to an almost complex structure I W on W we have that I W is integrable iff F Ω 1,1 I (M). F (IA, IB) = F (A, B), A, B T M.

43 Basic example S 2n+1 T 2 π M π W CP n T 2 S 2n+1 S 1 M = CP n T 2 (Kaehler product)

44 Basic example S 2n+1 T 2 π M π W CP n T 2 S 2n+1 S 1 M = CP n T 2 (Kaehler product) X generates one of the circle factors of T 2 = S 1 S 1.

45 Basic example S 2n+1 T 2 π M π W CP n T 2 S 2n+1 S 1 M = CP n T 2 (Kaehler product) X generates one of the circle factors of T 2 = S 1 S 1. Twist data: F = ω F S Fubini Study on CP n. twisting function: a = 1.

46 Basic example S 2n+1 T 2 π M π W CP n T 2 S 2n+1 S 1 M = CP n T 2 (Kaehler product) X generates one of the circle factors of T 2 = S 1 S 1. Twist data: F = ω F S Fubini Study on CP n. twisting function: a = 1. Double bundle fibration: P = S 2n+1 T 2.

47 Basic example π M S 2n+1 T 2 π W CP n T 2 S 2n+1 S 1 M = CP n T 2 (Kaehler product) X generates one of the circle factors of T 2 = S 1 S 1. Twist data: F = ω F S Fubini Study on CP n. twisting function: a = 1. Double bundle fibration: P = S 2n+1 T 2. Twist W = S 2n+1 S 1. (Complex Non-Kaehler)

48 Basic example π M S 2n+1 T 2 π W CP n T 2 S 2n+1 S 1 M = CP n T 2 (Kaehler product) X generates one of the circle factors of T 2 = S 1 S 1. Twist data: F = ω F S Fubini Study on CP n. twisting function: a = 1. Double bundle fibration: P = S 2n+1 T 2. Twist W = S 2n+1 S 1. (Complex Non-Kaehler) 1 F Ω 1,1 I (CP n ), comlex;

49 Basic example π M S 2n+1 T 2 π W CP n T 2 S 2n+1 S 1 M = CP n T 2 (Kaehler product) X generates one of the circle factors of T 2 = S 1 S 1. Twist data: F = ω F S Fubini Study on CP n. twisting function: a = 1. Double bundle fibration: P = S 2n+1 T 2. Twist W = S 2n+1 S 1. (Complex Non-Kaehler) 1 F Ω 1,1 I (CP n ), comlex; 2 b 2 (W ) = 0, non-kaehler.

50 (Second) motivating problem HOW CAN WE GET HYPERKAEHLER, QUATERNIONIC KAEHLER TWISTS?

51 (Second) motivating problem HOW CAN WE GET HYPERKAEHLER, QUATERNIONIC KAEHLER TWISTS? Which TWIST DATA determines W to be HK, QK?

52 Need of deformation dα = 0

53 Need of deformation dα = 0 dα W d W α = dα 1 a F (X α) = 1 F (X α) a

54 Need of deformation dα = 0 dα W d W α = dα 1 a F (X α) = 1 F (X α) a In order to get dα W = 0 we may need to start not from α but from some deformation α N.

55 Need of deformation dα = 0 dα W d W α = dα 1 a F (X α) = 1 F (X α) a In order to get dα W = 0 we may need to start not from α but from some deformation α N. We will exploit the symmetries of M to get the deformation α N (defining the same structure on M.)

56 Symmetries of the HK structure We say that X X M is a symmetry of a HK manifold (M, g, I, J, K) iff 1 X is an isometry of the HK metric: L X (g) = 0. 2 X preserves the linear span I, J, K EndT M, ie., L X (I) = I, J, K etc.

57 Symmetries of the HK structure We say that X X M is a symmetry of a HK manifold (M, g, I, J, K) iff 1 X is an isometry of the HK metric: L X (g) = 0. 2 X preserves the linear span I, J, K EndT M, ie., L X (I) = I, J, K etc. We will say that the symmetry X is rotating if L X (I) = 0, L X (J) = K.

58 deformations Define one-forms (α 0, α A ) : A = I, J, K : Then, α 0 = g(x, ), α A = g(ax, ), A = I, J, K. g α := α A α 2 A is definite semi-positive and proportional to the restriction g HX where HX = X, IX, JX, KX. An elementary deformation g N of a HK mertric g wrt a symmetry X is a new metric of the form g N = fg + hg α where f, h C M.

59 Main theorem Theorem Let (M, g, I, J, K) be a hyperkaehler manifold with non-null rotating symmetry X and Kaehler moment map µ. If dim M 8, then up to homothety, the only twists of elementary deformations g N = fg + hg α of g that are quaternion-kaehler have g N = 1 (µ c) 2 g α 1 µ c g for some constant c. The corresponding twist data is given by F = k(dα 0 + ω I ), a = k(g(x, X) µ + c).

60 Meaning The unicity statement is particularly important, since it shows that previous constructions using different methods by Haydys, Hitchin, Alekseevsky et al, agree. The constant k changes the curvature form: ie., affects the topology of the twist. Constant c affects the local properties of the QK metric.

61 Sketch of proof From g N and (I, J, K) construct Ω N, and impose that an arbitrary twist Ω W of Ω N is QK. Decompose these equations in type components relative to HX and its orthogonal complement. This computation leads eventually to f = f(µ), h = h(µ), h = f where µ is the moment map for X. First consider the relation XF and determine the twist function a. Investigate the condition df = 0 which lads to a ODE to determine f.

62 The hyperbolic plane Step I. Define indefinite PSK structures on open subsets S of the hyperbolic plane RH(2).

63 The hyperbolic plane Step I. Define indefinite PSK structures on open subsets S of the hyperbolic plane RH(2). Real hyperbolic space RH(2) : 2-dimensional solvable Lie group with Kaehler metric of constant curvature. Local basis of one forms {a, b} Ω 1 S : da = 0, db = λa b

64 The hyperbolic plane Step I. Define indefinite PSK structures on open subsets S of the hyperbolic plane RH(2). Real hyperbolic space RH(2) : 2-dimensional solvable Lie group with Kaehler metric of constant curvature. Local basis of one forms {a, b} Ω 1 S : da = 0, db = λa b 1 Metric: g S = a 2 + b 2

65 The hyperbolic plane Step I. Define indefinite PSK structures on open subsets S of the hyperbolic plane RH(2). Real hyperbolic space RH(2) : 2-dimensional solvable Lie group with Kaehler metric of constant curvature. Local basis of one forms {a, b} Ω 1 S : da = 0, db = λa b 1 Metric: g S = a 2 + b 2

66 The hyperbolic plane Step I. Define indefinite PSK structures on open subsets S of the hyperbolic plane RH(2). Real hyperbolic space RH(2) : 2-dimensional solvable Lie group with Kaehler metric of constant curvature. Local basis of one forms {a, b} Ω 1 S : 1 Metric: da = 0, 2 Almost complex structure: db = λa b g S = a 2 + b 2 Ia = b

67 The hyperbolic plane Step I. Define indefinite PSK structures on open subsets S of the hyperbolic plane RH(2). Real hyperbolic space RH(2) : 2-dimensional solvable Lie group with Kaehler metric of constant curvature. Local basis of one forms {a, b} Ω 1 S : 1 Metric: da = 0, 2 Almost complex structure: 3 Kaehler 2-form: db = λa b g S = a 2 + b 2 Ia = b ω S = a b.

68 Local cone structure The PSK manifold S is the Kaehler quotient of a conic SK manifold C (C, g, ω,, X) C 0 S π i C S = C// c X

69 Local cone structure The PSK manifold S is the Kaehler quotient of a conic SK manifold C (C, g, ω,, X) C 0 S Locally C = R >0 C 0. π i C S = C// c X

70 Local cone structure The PSK manifold S is the Kaehler quotient of a conic SK manifold C (C, g, ω,, X) C 0 S Locally C = R >0 C 0. π i C S = C// c X C 0 is the level set µ 1 (c) for the moment map of X. C 0 S is a bundle with connection 1-form ϕ : dϕ = 2π ω S = i ω

71 Metric and Kaehler form on C Lemma Write t for the standard coordinate on R >0 and ˆψ = dt. Write â = tπ a, ˆb = tπ b, ˆϕ = tϕ The (2, 2) pseudo Riemannian metric and the Kaehler form of C = R >0 C 0 are g C = â 2 + ˆb 2 ˆϕ 2 ˆψ 2 = dt 2 + t 2 g C0 The conic isometry satisfies ω C = â ˆb ˆϕ ˆψ. IX = t t

72 LC connection Coframe s θ: Exterior differential d(s θ) : (â, ˆb, ˆϕ, ˆψ) dâ = 1 t dt â d ˆϕ = 1 t (dt ˆϕ + 2â ˆb) d ˆψ = 0 dˆb = 1 t (dt ˆb λâ ˆb) LC connection s ω LC 0 ˆϕ + λˆb ˆb â s ω LC = 1 ˆϕ λˆb 0 â ˆb t ˆb â 0 ˆϕ â ˆb ˆϕ 0

73 Curvature 2-form Curvature 2-form s Ω LC : s Ω LC = 4 λ2 t 2 0 â ˆb 0 0 â ˆb Lemma The pseudo Riemannian metric g C is flat iff λ 2 = 4.

74 Conic special Kaehler condition Symplectic connection in terms of LC connection: Conditions on η. s ω = s ω LC + η. 1 η s θ (from the special condition); 2 i η = η i (anti-complex condition);c 3 t η s = s η (special connection is symplectic) 4 X η = IX η = 0 (η is {3, 0} totally symmetric). Proposition The cone (C, g C, ω C ) over S RH(2) is conic special Kaehler iff λ 2 = 4 3 or λ4 = 4.

75 Hyperkaehler structure on H = T C from the RIGID c-map Step II. HK metric on T C. Hyperkaehler metric: ( g H = â 2 + ˆb 2 ˆϕ 2 ˆψ 2) (Â2 + + ˆB 2 ˆΦ 2 ˆΨ 2) for (Â, ˆB, ˆΦ, ˆΨ) = s α = dx xs ω LC. Hyperkaehler structure: ω I = â ˆb ˆϕ ˆψ Â ˆB + Φ ˆΨ ω J = Â â + ˆB ˆb + ˆΦ ˆϕ + ˆΨ ˆψ ω K = Â ˆb ˆB â + ˆΦ ˆψ ˆΨ ˆϕ.

76 deformation of g H Step III. deformations, twist data, etc. Deformation elements α 0,... α K : α I = I X g H = t ˆψ, α J = I X g H = tˆφ, Moment map: α 0 = Iα I = t ˆϕ α K = Iα J = t ˆΨ µ = 1 2 X 2 = t2 2 Deformed metric on H = T C : g N = 1 µ g H + 1 µ 2 (α2 0 + α 2 I + α 2 J + α 2 K) = 2 t 2 (â2 + ˆb 2 + ˆϕ 2 + ˆψ 2 + Â2 + ˆB 2 + ˆΦ 2 + ˆΨ 2 )

77 Twist data Twisting (curvature) 2-form: Twisting function F = â ˆb +ˆ ˆψ Â ˆB + ˆΦ ˆΨ a = µ = t2 2

78 Flat case λ 2 = 4. Step IV. Computing d W The LC and the special connection coincide.

79 Flat case λ 2 = 4. Step IV. Computing d W The LC and the special connection coincide. X-invariant coframe on C γ = s θ/t = (ã, b, ϕ, ψ) (we can compute the twisted differentials immediatelly) d W ã = 0 d W b = 2 b ã d W ϕ = 2ã b + 2 t 2 F d W ψ = 0

80 Flat case λ 2 = 4. Step IV. Computing d W The LC and the special connection coincide. X-invariant coframe on C γ = s θ/t = (ã, b, ϕ, ψ) (we can compute the twisted differentials immediatelly) d W ã = 0 d W b = 2 b ã d W ϕ = 2ã b + 2 t 2 F d W ψ = 0 In the vertical directions δ = (s α)/t = (Ã, B, Φ, Ψ) is NOT X-invariant.

81 iτ δ = δe ɛ = 1 2 (δ 1 + δ 4, δ 2 δ 3, δ 2 δ 3, δ 1 + δ 4 ) d W ɛ = ɛ ψ ã 0 2 b 0 0 ψ ã 0 2 b 0 0 ψ ã ψ + ã d W ϕ = 2(ϕ ψ + ɛ 13 + ɛ 4 ɛ 2 ) The resulting Lie algebra is isomorphic to the solvable algebra corresponding to the non-compact symmetric space Gr 2 + U(2, 2) (C2,2 ) = U(2) U(2)

82 non-flat case λ 2 = 4 3 Write (Â, ˆB, ˆΦ, ˆΨ) = dx xs ω LC xη. Then, g H, g N and the twist data F, a are the same. Twisted differentials for γ = (ã, b, ϕ, ψ) : d W ã = 0 d W b = 2 3ã b d W ψ = 0 d W ϕ = 2(ϕ ψ Ã B + Φ Ψ)

83

84 Adjusting the vertical directionss we arrive to d W ɛ = ɛ ψid ɛ ã ɛ b We see the structure of solvable algebra associated to G 2 SO(4)

85 Thank you.

86 Bibliography This talk is based on: 1, Swann, A.F., deformations and the hyperkaehler-quaterninic Kaehler correspondence, (2014). 2, Swann, A.F.,Twist geometry of the c-map, (2014). The general theory can be consulted in: 1 Swann, A.F., T is for twist. 2 Swann, A.F., Twisting Hermitian and hypercomplex geometries, Duke Math. J. 155(2), (2010)