Dunajski M. Solitons, Instantons, and Twistors

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236 10 : Anti-self-dual conformal structures

Moreover (10.2.10) and (10.2.11) arise as the integrability condition for the

linear system L

f = L

f = 0, where f = f (w

, ˜w

,λ) and

∂

∂w

− λ

∂



∂w

∂ ˜w

∂

∂ ˜w

+ λ

∂k

∂w

∂

∂λ

. (10.2.12)

As a spin-off we have

Proposition 10.2.2 A four-dimensional Kähler metric is ASD iff its scalar

curvature vanishes.

Proof Any scalar-ﬂat Kähler metric is locally of the form (10.2.8) (as it is

Kähler). Calculating the Ricci scalar shows that it vanishes iff (10.2.10) and

(10.2.11) hold. Thus the metric is ASD. Conversely, if g is ASD and Kähler then

(10.2.10) and (10.2.11) hold by the integrability of the Lax pair. Therefore g

is scalar-ﬂat. 

In view of this result ASD Kähler metrics are often called scalar-ﬂat Kähler.

10.2.3 Null-Kähler structures

The structure we are about to describe does not exist in Riemannian signature

and neutral signature is the only allowed reality condition which admits real

solutions. We shall impose this condition from the start.

A null-Kähler structure [47] on a real four-manifold M consists of an

inner product g of signature (+ + −−) and a real rank-two endomorphism

N : TM → TM parallel with respect to this inner product such that

= 0 and g(NX, Y)+g(X, NY)=0

for all X, Y ∈ TM.

The parallel two-form  = g(N,...) is simple (i.e.  ∧  = 0) therefore it is

SD or ASD by the argument given in Section 7.1.1. We chose the orientation

such that  is SD. The isomorphism 

(M)

∼

Sym



) between the bundle of

SD two-forms and the symmetric tensor product of two spin bundles implies

that the existence of a null-Kähler structure is in four dimensions equivalent

to the existence of a parallel real spinor. If the spinor is ι



then 



. The Ricci identity (9.2.26) implies the vanishing of the curvature

scalar.

In [24] and [47] it was shown that null-Kähler structures are locally given

by one arbitrary function : M → R of four variables, and admit a canonical

10.2 Curvature restrictions and their Lax pairs 237

form

g = dwdx + dzdy −

−

dwdz, (10.2.13)

with N = dw ⊗ ∂/∂y − dz ⊗ ∂/∂x.

Further conditions can be imposed on the curvature of g to obtain non-linear

PDEs for the potential function . Deﬁne

k :=

−

. (10.2.14)

The Ricci-ﬂat condition implies that

k = xP(w, z)+yQ(w, z)+R(w, z),

where P, Q, and R are arbitrary functions of (w, z). In fact the number of the

arbitrary functions can be reduced down to one by redeﬁnition of and the

coordinates. This is the hyper-heavenly equation of Pleba

nski and Robinson

[136] for non-expanding metrics of type [N]×[Any]. (A manifold (M, g)is

called hyper-heavenly if the SD Weyl spinor is algebraically special and the

Einstein equations hold.)

The conformal ASD condition implies a fourth-order PDE for

k =0, (10.2.15)

where

 = ∂

∂

+ ∂

∂

− 2

∂

This equation is integrable: It admits a Lax formulation [L

, L

] = 0 with

=(∂

−

∂

) − λ∂

+ k

∂

and

=(∂

∂

−

∂

)+λ∂

− k

∂

and its solutions can in principle be constructed by twistor methods [47] or

the dressing method [16].

10.2.4 ASD Einstein structures

If there exists a metric g in the ASD conformal class [g] which is Einstein with

non-zero cosmological constant, that is R

=6g

then [139] the coordinates

can be chosen so that

g =



w ˜w

dwd ˜w + K

dwd

z + K

z ˜w

dzd ˜w +(K

+2e

)dzd

where K = K(w, z, ˜w,

z) satisﬁes the Przanowski equation

w ˜w

− K

z ˜w

+(2K

w ˜w

− K

˜w

=0.

238 10 : Anti-self-dual conformal structures

10.2.5 Hyper-Kähler structures and heavenly equations

In Theorem 9.3.3 it was shown that the hyper-Kähler condition on a four-

metric is equivalent to the ASD Ricci-ﬂat condition. The resulting Lax pair

arises as a special case of the hyper-Hermitian Lax pair.

Theorem 10.2.3 [6, 116] Let e



be four independent holomorphic vector

ﬁelds on a four-dimensional complex manifold M, and let ν be a holomorphic

four-form. Put

= e



− λe



and L

= e



− λe



, L

] = 0 (10.2.16)

for every λ ∈ CP

, and

Lie

ν =0, (10.2.17)

then c

−1



is a null tetrad for a hyper-Kähler metric on M, where

= ν(e



, e



, e



, e



Given any four-dimensional hyper-Kähler metric such a null tetrad and four-

form exists.

Proof Given a Lax pair of vector ﬁelds as in Theorem 10.2.3 deﬁne SD two-

forms 



(λ)=ν(L

, L

,.,.)=



We shall show that (λ) is closed for any ﬁxed λ. This will imply that there

exist a closed basis 



of SD two-forms and the Corollary 9.3.4 can be used

to deduce that g is hyper-Kähler. Let d

be a total derivative on M × CP

which

holds λ = const:

(λ)=d

[

ν(L

, L

,. ,. )

]

= d

[

ν(L

,. ,. ,. )

]

=Lie

[

ν(L

,. ,. ,. )

]

− L

[

ν(L

,. ,. ,. )

]

=[L

, L

] ν + L

Lie

(ν) − L

dν)=0.

Conversely, given a hyper-Kähler metric we can choose coordinates

, ˜w

)=(w, z, ˜w,

z) such that the metric is given in terms of a Kähler

potential by (10.2.8). The SD two-forms are given by the complexiﬁcation

of (9.3.32) and the hyper-Kähler condition (9.3.33) gives the ﬁrst heavenly

equation of Pleba

nski [135]:



z ˜w

− 

w ˜w



=1 or

∂



∂w

∂ ˜w

∂



∂w

∂ ˜w

=1. (10.2.18)

10.2 Curvature restrictions and their Lax pairs 239

We can take the tetrad of vector ﬁelds to be







w ˜w

∂

− 

∂

˜w

∂



z ˜w

∂

− 

∂

˜w

∂





∂



∂w

∂ ˜w

∂

∂ ˜w

∂

∂w



. (10.2.19)

These vector ﬁelds preserve the volume form

ν = dw ∧ dz ∧ d ˜w ∧ d

The dual tetrad is



= dw

and e



∂



∂w

∂ ˜w

d ˜w

with the ﬂat solution  = w

˜w

. The Lax pair for the ﬁrst heavenly equation

:=

w ˜w

∂

− 

∂

˜w

− λ∂

and

:=

z ˜w

∂

− 

∂

˜w

− λ∂

(10.2.20)

satisﬁes (10.2.16) and (10.2.17) where the volume form is ν = dw ∧ dz ∧ d ˜w ∧

z. This ends the proof. 

As a spin-off from the proof we have deduced Pleba

nski’s ﬁrst heavenly

equation and its Lax pair. Pleba

nski also gave the alternative local form of

the ASD Ricci-ﬂat condition [135] called the second heavenly equation. This

equation is the special case of the ASD null-Kähler equations (10.2.14 and

10.2.15) corresponding to k =0.

Below we shall derive the second heavenly equation by ﬁxing the residual

gauge freedom in the Lax pair (10.2.3). Consider the hyper-Kähler Lax equa-

tions (10.2.16 and 10.2.17). The Frobenius theorem (Theorem C.2.4) applied

to the equations



, e



]=0 and [e



, e



]+[e



, e



]=0

implies the existence of a complex-valued function and coordinate system

, x

):=(w, z, x, y), such that





∂

−

∂

−∂

∂

−

∂





∂

∂x

∂

∂w

∂

∂x

∂

∂x



(10.2.21)

Finally equation [e



, e



] = 0 implies that satisﬁes second heavenly

equation

−

=0 or

∂

∂w

∂x

∂

∂x

∂

∂x

=0.

(10.2.22)

240 10 : Anti-self-dual conformal structures

The dual frame is given by



= dx

∂

∂x

and e



= dw

and g is of the form (10.2.13) with = 0 deﬁning the ﬂat metric. The Lax pair

corresponding to (10.2.22) is

= ∂

− λ(∂

−

∂

) and

= ∂

+ λ(∂

∂

−

∂

). (10.2.23)

This is a special case of the Lax pair for equation (10.2.15) with k = 0 (to

reach an agreement one needs to make a transformation λ → 1/λ in the ASD

null-Kähler Lax pair).

Example. One solution to the second heavenly equation is

wx + zy

, t = const.

The corresponding metric is given by

g = dwdx + dzdy −

(wx + zy)

(wdz − zdw)

This metric was ﬁrst constructed by Sparling and Tod [149] using the

H-space formalism. The metric appears to be singular on the light cone of the

origin, but this singularity may be removed by a coordinate transformation

[149].

10.2.5.1 Recursion operator

The recursion operator R is a map from the space of linearized solutions of

the ASD Ricci-ﬂat equations to itself. Linearized solutions can be regarded

as vector ﬁelds on the solutions space. Thus R is a natural generalization of

the recursion operator we introduced in Section 3.2.1 in the context of bi-

Hamiltonian systems. We shall present a theory of recursion operator in the

following [45].

We shall ﬁrst identify the space of linearized solutions to the ASD Ricci-

ﬂat equations with the space of solutions to the background coupled wave

equation in two ways as follows.

Lemma 10.2.4 Let 



and 

denote wave operators on the ASD Ricci-

ﬂat background determined by  and , respectively. Linearized solutions to

10.2 Curvature restrictions and their Lax pairs 241

(10.2.18) and (10.2.22) satisfy





δ =0 and 

δ =0. (10.2.24)

Proof In both cases 

= ε



since



√

∂

√

g∂

)=g

∂

+(∂

)∂

but ∂

= 0 for both heavenly coordinate systems. The linearized ﬁrst heav-

enly equation takes the form

∂

∂( + δ)

= ν, where ∂ = e



⊗ e



and

∂ = e



⊗ e



so that d = ∂ +

∂. This implies

∂

∂ ∧ ∂

∂

δ = d

∂

∂ ∧ (∂ −

∂)δ

= d ∗ dδ.

Here ∗ is the Hodge star operator corresponding to g. For the second equation

we make use of the tetrad (10.2.21) and perform coordinate calculations. 

From now on we identify tangent spaces to the spaces of solutions to (10.2.18)

and (10.2.22) with the space of solutions to the curved background wave

equation, W

. We will deﬁne the recursion operator on the space W

Lemma (10.2.4) shows that we can consider a linearized perturbation as an

element of W

in two ways. These two will be related by the square of the

recursion operator. The linearized ASD Ricci-ﬂat metrics corresponding to δ

and δ are



= ι



)

∇

(A1



∇

B)0



δ and h



= o



∇



∇



δ ,

where o



=(1, 0) and ι



=(0, 1) are the constant spin frame associated to the

null tetrads (10.2.21) and (10.2.19).

Given φ ∈ W

we use the ﬁrst of these equations to ﬁnd h

. If we put the

perturbation obtained in this way on the LHS of the second equation and add

an appropriate gauge term we obtain φ



– the new element of W

that provides

the δ which gives rise to

= h

+ ∇

. (10.2.25)

To extract the recursion relations we must ﬁnd V such that h



−

∇

(AA



)

= o



. Take V



= o



∇



δ, which gives

∇

(AA



)

= −ι



)

∇

(A0



∇

B)1



δ + o



∇



∇



δ.

This reduces (10.2.25) to

∇



∇



φ = ∇



∇



. (10.2.26)

Both heavenly formulations use the covariantly constant spin frame (see

Theorem 9.3.3) so ∇



= e



242 10 : Anti-self-dual conformal structures

Deﬁnition 10.2.5 Deﬁne the recursion operator R : W

−→ W



φ = e



Rφ, (10.2.27)

so formally R =(e



)

−1

◦ e



(no summation over the index A).

Remarks

From (10.2.27) and from the ﬁeld equations it follows that if φ belongs to

then so does Rφ.

If R

δ = δ then δ and δ correspond to the same variation in the metric

up to gauge.

The operator φ → e



φ is overdetermined, and its consistency follows from

the wave equation on φ.

This deﬁnition is formal in that in order to invert the operator φ → e



φ we

need to specify boundary conditions.

To summarize

Proposition 10.2.6 [45] Let W

be the space of solutions of the wave equation

on the curved ASD background given by g:

1. Elements of W

can be identiﬁed with linearized perturbations of the heav-

enly equations.

2. There exists a (formal) map R : W

−→ W

given by (10.2.27).

10.2.5.2 Heavenly hierarchies

The generators of higher ﬂows are ﬁrst obtained by applying powers of the

recursion operator to the linearized perturbations corresponding to the evolu-

tion along coordinate vector ﬁelds. This embeds the second heavenly equation

into an inﬁnite system of overdetermined, but consistent, PDEs (which we will

truncate at some arbitrary but ﬁnite level). These equations in turn can be

naturally embedded into a system of equations that are the consistency condi-

tions for an associated linear system that extends (10.2.16) and (10.2.17). This

yields a hierarchy of ﬂows of the ASD Ricci-ﬂat equations [17, 45, 49, 155].

We shall discuss the hierarchy for the second Pleba

nski equation [45]; that

for the ﬁrst arises from a different coordinate and gauge choice. The ﬁrst few

iterations of the recursion operator (10.2.27) with the tetrad (10.2.21) can be

explicitly integrated to give

w −→ y −→ −

−→

−→ ... and

z −→ −x −→ −

−→ −

−→···.

Introduce the coordinates x

, where for i =0, 1, x

= x



are the original

coordinates on M, and for 1 < i ≤ n, x

are the parameters for the new ﬂows

10.2 Curvature restrictions and their Lax pairs 243

(with (2n − 2)-dimensional parameter space X). The propagation of along

these parameters is determined by the recursion relations (10.2.27):

∂

(∂

Bi+1

)=(∂

−

∂

)∂

and

−∂

(∂

Bi+1

)=(∂

∂

−

∂

)∂

. (10.2.28)

The hyper-Kähler hierarchy is the system arising as the consistency conditions

for (10.2.28):

∂

Bj−1

− ∂

∂

Ai −1

+ {∂

Ai −1

,∂

Bj−1

}

=0, i, j =1,...,n.

(10.2.29)

Here {...,...}

is the Poisson bracket with respect to the Poisson structure

∂/∂x

∧ ∂/∂x

=2∂

∧ ∂

Lemma 10.2.7 The linear system for equations (10.2.29) is

f =(−λD

Ai +1

+ δ

) f =0, i =0,...,n − 1, (10.2.30)

where

1. f := f (x

,λ) is a function on CP

× N , where N = M × X.

2. D

Ai +1

:= ∂

Ai +1

+[∂

, V], (V = ε

∂

), and δ

:= ∂

are 4n vector

ﬁelds on N .

Proof This follows by direct calculation. The compatibility conditions for

(10.2.30) are

Ai +1

, D

Bj+1

]=0, (10.2.31)

[δ

,δ

]=0, and (10.2.32)

Ai +1

,δ

] − [D

Bj+1

,δ

]=0. (10.2.33)

It is straightforward to see that equations (10.2.32) and (10.2.33) hold identi-

cally with the above deﬁnitions and (10.2.31) is equivalent to (10.2.29). 

The concept of hierarchy is useful in ﬁnding solutions to the heavenly equa-

tions. This is analogous to the ﬁnite-gap integration of KdV described in

Section 3.4. We say that an ASD Ricci-ﬂat metric admits a hidden symmetry if

the associated heavenly potential is stationary with respect to some direction

in the extended parameter space X.

Example. Let us demonstrate how to use the recursion procedure to ﬁnd

metrics with hidden symmetries. Let ∂

 := φ

be a linearization of the

ﬁrst heavenly equation. We have R : z −→ 

= ∂

. Look for solutions

to (10.2.18) with an additional constraint ∂

 = 0. The recursion relations

(10.2.27) imply 

= 

= 0, therefore

(w, z, ˜w,

z)=wq(˜w,

z)+P(z, ˜w,

z).

244 10 : Anti-self-dual conformal structures

The heavenly equation yields dq ∧ dP ∧ dz = d

z ∧ d ˜w ∧ dz. With the deﬁni-

tion ∂

P = p the metric is

g =2dwdq +2dzdp + Qd z

where Q = −2P

. We adopt (w, z, q, p) as a new coordinate system. The

heavenly equations imply that Q = Q(q, z) is an arbitrary function of two

variables. These are the null ASD plane wave solutions [135]. They have no

Riemannian real sections but restricting to real coordinates and real Q yields

a metric in neutral signature.

10.2.5.3 Hamiltonian and Lagrangian formalisms

In this section we shall investigate the Lagrangian and Hamiltonian formu-

lations of the hyper-Kähler equations in their ‘heavenly’ forms. Rather than

considering the equations as a real system of elliptic or ultra-hyperbolic equa-

tions, we complexify and consider the equations locally as evolving initial data

from a three-dimensional hypersurface δ M and it is this space of initial data

that leads to local solutions on a neighbourhood of such a hypersurface that is

denoted by S and is endowed with a (conserved) symplectic form.

For the ﬁrst equation we have the Lagrangian density



= 



ν −

(∂

∂)





 −

{

,

˜w

}



ν (10.2.34)

and for the second equation



{

}

−

(

)



ν. (10.2.35)

If the ﬁeld equations are assumed, the variation of these Lagrangians will

yield only a boundary term. Starting with the ﬁrst equation, this deﬁnes a

potential one-form P (compare (5.1.10)) on the solution space S and hence a

symplectic structure  = dP on S. Starting with the second we ﬁnd a symplec-

tic structure with the same expression on perturbations δ as we had for δ.

However, since their relation to perturbations of the hyper-Kähler structure

are different, they deﬁne different symplectic structures on S. These are related

by the recursion operator since we have R

δ = δ from the construction

of the recursion operator. In order to see that these structures yield the bi-

Hamiltonian framework, these symplectic structures need to be compatible

with the recursion operator in the sense that (Rφ,φ



)=(φ, Rφ



We shall discuss this using the ﬁrst heavenly formulation (10.2.18) which is

easier as one can use identities from Kähler geometry.

10.2 Curvature restrictions and their Lax pairs 245

Proposition 10.2.8 The symplectic form on the space of solutions S derived

from the boundary term in the variational principle for the ﬁrst Lagrangian is

(δ

, δ

)=



δM

 ∗ d(δ

) − δ

 ∗ d(δ

). (10.2.36)

Proof Varying (10.2.34) we obtain

δL = δ



ν −

(∂

∂)



−

∂

∂ ∧ ∂

∂δ =

∂

∂ ∧ (δ∂

∂ − ∂

∂δ).

We use the relation between the complex and Kähler structures and the ﬁeld

equation to obtain

δL = −

∂

∂ ∧

δd(∂ −

∂) − d(∂ −

∂)δ

dA(δ) −

∂

∂

−∗∂

∂(∂ −

∂)δ(∂ −

∂) + ∗∂

∂(∂ −

∂)(∂ −

∂)δ

dA(δ), where A(δ)= ∗ dδ − δ ∗ d.

Deﬁne the one-form on M:

P(δ)=



δM

A(δ).

The symplectic structure  is the (functional) exterior derivative of P:

(δ

, δ

)=δ

[

P(δ

)

]

− δ

[

P(δ

)

]

− P([δ

, δ

])



δM

 ∗ d(δ

) − δ

 ∗ d(δ

).



Thus  coincides with the symplectic form on the solution space to the wave

equation on the ASD vacuum background.

The existence of the recursion operator allows the construction of an inﬁnite

sequence of symplectic structures. The key property which can be estabished

[45] by an application of Stokes theorem is the following: Let φ, φ



∈ W

and

let  be given by (10.2.36). Then

(Rφ, φ



)=(φ, Rφ



) (10.2.37)

where the recursion operator is deﬁned by (10.2.27). This property guarantees

that the bilinear forms



(φ, φ



) ≡ (R

φ, φ



) (10.2.38)

are skew. Furthermore they are symplectic and lead to the bi-Hamiltonian

formulation. In this context formula (10.2.37) and the closure condition for



are an algebraic consequence of the fact that R comes from two Poisson

structures.