Dunajski M. Solitons, Instantons, and Twistors

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256 10 : Anti-self-dual conformal structures

following equation:

= β

+ ββ

− β A

− β

. (10.3.68)

If the projective structure is ﬂat, that is, A

= 0 and β = P = 0 then (10.3.65) is

Ricci-ﬂat [135], and in fact this is the most general ASD Ricci-ﬂat metric with

a null Killing vector which preserves the pseudo-hyper-Kähler structure. The

twisting metrics (10.3.66) generalize those found in [123]. More generally, if

the projective structure comes from a Riemannian metric on U then there will

always exist a (pseudo-)Kähler structure in the conformal class (10.3.66) if

G = z

/2+γ (x, y)z + δ(x, y) for certain γ and δ [26].

It is interesting that integrable systems are not involved in the null case, given

their ubiquity in the non-null case.

10.3.3 Dispersionless integrable systems

Dispersionless integrable systems can arise from solitonic systems in a follow-

ing way: Let



∂

∂ X



∂

∂ X

+ a

)

∂

n−1

∂ X

n−1

+ ···+ a

) and



∂

∂ X



∂

∂ X

+ b

)

∂

m−1

∂ X

m−1

+ ···+ b

)

be differential operators on R with coefﬁcients depending on local coordinates

=(X, Y, T)onR

. The overdetermined linear system



= A



∂

∂ X



 and 

= B



∂

∂ X





admits a solution (X, Y, T) on a neighbourhood of an initial point (X, Y

, T

)

for arbitrary initial data (X, Y

, T

)= f (X) iff the integrability conditions



= 

,or

− B

+[A, B] = 0 (10.3.69)

are satisﬁed. The non-linear system (10.3.69) for a

,...,a

and b

,...,b

can be solved by a (2+1)-dimensional version of the IST discussed in Chapter 2.

The dispersionless limit [192] is obtained by substituting

∂

∂ X

= ε

∂

∂x

and (X

) = exp [ψ (x

/ε)],

and taking the limit ε −→ 0. In the limit the commutators of differential

operators are replaced by the Poisson brackets of their symbols according to

10.3 Symmetries 257

the relation

∂

∂ X

 −→ (ψ

)

, [A, B] −→

∂ A

∂λ

∂ B

∂x

−

∂ A

∂x

∂ B

∂λ

= {A, B}, and λ = ψ

where A and B are polynomials in λ, with coefﬁcients depending on x

(x, y, t). The dispersionless limit of the system (10.3.69) is

− B

+ {A, B} =0. (10.3.70)

Non-linear DEs of the form (10.3.70) are called dispersionless integrable

systems.

There are several methods of integrating (10.3.70) [15, 46, 63, 64, 96, 97,

98, 111, 112, 155]. We shall focus on the one which makes contact with

EW geometry and twistor theory. The system (10.3.70) is equivalent to the

integrability [L, M] = 0 of a two-dimensional distribution of vector ﬁelds

L = ∂

− B

∂

+ B

∂

and M = ∂

− A

∂

+ A

∂

(10.3.71)

on R

× RP

. This is similar to the EW Lax pair (10.3.42) in Lorentzian

signature.

We shall therefore generalize the notion of the dispersionless integrable

systems by allowing distributions of vector ﬁelds more general than (10.3.71).

The derivatives A

, A

, B

, and B

of the symbols (A, B) of operators can be

replaced by independent polynomials A

, A

, B

, and B

in λ with coefﬁcients

depending on (x, y, t):

L = ∂

− B

∂

+ B

∂

and M = ∂

− A

∂

+ A

∂

, (10.3.72)

where A

, B

are linear in λ and A

, B

are at most cubic in λ . We take the

integrability of this generalized distribution (10.3.72) as our deﬁnition of a

dispersionless integrable system. The deﬁnition is intrinsic in the sense that it

does not refer to an underlying soliton equation.

10.3.3.1 Interpolating integrable system

In Section 10.3 we have seen how certain dispersionless systems (dKP, SU(∞)-

Toda, and Diff(S

) equation) arise from ASD conformal equations in four

dimensions and lead to EW structures. In each case it was possible to choose a

solution of the generalized monopole equation (10.3.44) so that the resulting

ASD conformal structure is Ricci-ﬂat.

It should therefore be possible to obtain these dispersionless equations

directly as reductions of the ASD Ricci-ﬂat condition, for example, in its

second heavenly form (10.2.22). This is indeed the case and the details can

be found in [46, 51, 65]. Here we shall present a dispersionless integrable

258 10 : Anti-self-dual conformal structures

system which arises as a symmetry reduction of (10.2.22) and contains the

dKP equation (10.3.52) and the Diff(S

) equation (10.3.53) as special cases.

This interpolating integrable system is given by [59]

+ w

= 0 and u

+ w

− c(uw

− wu

)+buu

=0, (10.3.73)

where b and c are constants and u,ware smooth functions of (x, y, t). It admits

a Lax pair

L =

∂

∂t

+(cw + bu − λcu − λ

)

∂

∂x

+ b(w

− λu

)

∂

∂λ

and

M =

∂

∂y

− (cu + λ)

∂

∂x

− bu

∂

∂λ

(10.3.74)

with a spectral parameter λ ∈ CP

. A linear combination of L and M is of the

form (10.3.72).

Setting b = 0 and c = −1 gives the Diff(S

) equation (10.3.53) and setting

c = 0 and b = 1 gives the dKP equation (10.3.52). In fact one constant can

always be eliminated from (10.3.73) by redeﬁning the coordinates and it is

only the ratio of b/c which remains. We prefer to keep both constants as it

makes the limits more transparent.

Proposition 10.3.5 [59] The most general (+ + −−) ASD Ricci-ﬂat metric with

a conformal Killing vector whose SD derivative is null can be put in the form

g = e

cτ

[Vh − V

−1

(dτ + η)

], (10.3.75)

where (u,w) solve the system (10.3.73), τ parameterizes the orbits of the

conformal Killing vector K = ∂/∂τ, and

h =(dy + cudt)

−4[dx + cudy−(cw + bu)dt]dt,η= −

dy +

− u

dt,

and V =

The proof uses the second heavenly formalism. The coordinates can be chosen

so that the conformal Killing vector is given by K =(cz + b)∂

+(cx − 2bz)∂

and the ASD Ricci-ﬂat metric is given by (10.2.13) where satisﬁes the

second heavenly equation (10.2.22). One then imposes the conformal Killing

equations on , and (after a series of Legendre transforms and coordinate

transformations) arrives at the statement of Proposition 10.3.5. In particular

the proof explains the origin of the two parameters (b, c) in (10.3.73) as the

conformal symmetry is

K = c × (dilatation) + b × (rotation with null SD derivative).

The details of the proof are given in [59].

10.3 Symmetries 259

10.3.3.2 Manakov–Santini system

The system (10.3.73) is a special case of the Manakov–Santini system

[111, 112]:

− U

+(UU

)

+ V

− V

= 0 and (10.3.76)

− V

+ UV

+ V

− V

=0,

where U = U(x, y, t) and V = V(x, y, t). To see this, notice that the ﬁrst

equation in (10.3.73) implies the existence of v(x, y) such that u = v

,w = −v

and v satisﬁes

− v

+ c(v

− v

)+buv

=0. (10.3.77)

Differentiating the second equation in (10.3.73) and eliminating w yields

− u

− c(v

− v

)+b(uu

)

=0. (10.3.78)

Now assume the generic case when the constants c and b are non-zero, and set

U = bu and V = cv. Then the systems (10.3.77) and (10.3.78) are equivalent

to (10.3.76) with an additional constraint

cU − bV

=0. (10.3.79)

The Manakov–Santini system corresponds to an EW structure:

h =(dy − V

dt)

− 4[dx − (U − V

)dt]dt and (10.3.80)

ω = −V

dy +(4U

− 2V

+ V

)dt.

To verify this set x

=(y, x, t). The (11), (12), (22), and (23) components of

the EW equations hold identically. The (13) component vanishes if the second

equation in (10.3.76) holds, and ﬁnally the (33) component vanishes if both

equations in (10.3.76) are satisﬁed.

In [111, 112] Manakov and Santini have solved the initial value problem

for the system (10.3.76) using their version of IST applicable to Lax pairs

containing vector ﬁelds.

10.3.3.3 Quadric ansatz

The usual way of reducing PDEs to ODEs is to determine a symmetry group of

transformations acting on dependent and independent variables, and reduce

the number of independent variables down to one. This was discussed in

Section 4.3.

Tod [158] has proposed a non-symmetric way of reducing PDEs to ODEs

which can be used to construct solutions to some dispersionless integrable

systems. Our presentation of this follows [48, 50]. Let u = u(x

,...,x

) ∈ R

260 10 : Anti-self-dual conformal structures

be a solution to a PDE

F (u, u

, u

,...,u

ij···k

, x

)=0, (10.3.81)

where u

= ∂u/∂ x

and (x

,..., x

) ∈ R

. The ansatz is to seek solutions con-

stant on a polynomial hypersurface  ⊂ R

, or equivalently to seek symmetric

objects

M(u), M

(u), M

(u),...,M

ij···k

(u),

so that a solution of equation (10.3.81) is determined implicitly by

Q(x

, u):=M(u)+M

(u)x

+ M

(u)x

+ ···+ M

ij···k

(u)x

···x

= C,

(10.3.82)

where C is a constant. Here  should be regarded as the zero locus of a poly-

nomial Q(x

, u) − C in R

.Ifu satisﬁes (10.3.81) and the algebraic constraint

(10.3.82), then so does g

(u), where g

is a ﬂow generated by any section of

T. Note however that vectors tangent to  do not generate symmetries of

(10.3.81), as the choice of  depends on u.

We shall concentrate on the quadric ansatz

Q(x

, u):=M

(u)x

= C. (10.3.83)

This ansatz can be made whenever we have a non-linear PDE of the form

∂

∂x



(u)

∂u

∂x



=0, (10.3.84)

where u is a function of coordinates x

, i =1,..., n. We shall seek solutions

constant on central quadrics or equivalently seek a matrix M(u)=[M

(u)] so

that a solution of equation (10.3.84) is determined implicitly as in (10.3.83).

We differentiate (10.3.83) implicitly to ﬁnd

∂u

∂x

= −

, where

Q =

∂ Q

∂u

. (10.3.85)

Now we substitute this into (10.3.84) and integrate once with respect to u.

Introducing g(u)by

g =

Tr (bM) (10.3.86)

we obtain

− M

=0,

so that as a matrix ODE

M = MbM. (10.3.87)

10.3 Symmetries 261

This equation simpliﬁes if written in terms of another matrix N(u) where

N = −M

−1

(10.3.88)

for then

N = b, (10.3.89)

and g can be given in terms of  = det (N)by

 = ζ = const. (10.3.90)

Restricting to three dimensions with (x

, x

)=(x, y, z), the SU(∞)-Toda

equation (10.3.46) is given by (10.3.84) with

b(u)=

⎛

⎝

10 0

01 0

00e

⎞

⎠

and as was shown in [158], in this case (10.3.89) can be reduced to the PIII

ODE (4.4.11).

For the dKP equation (10.3.52) we have (10.3.84) with

b(u)=

⎛

⎝

−u 01/2

0 −10

1/20 0

⎞

⎠

. (10.3.91)

(Note that −b(u) is the inverse of the EW metric.)

Proposition 10.3.6 Solutions to the dKP equation (10.3.52) constant on the

central quadric (10.3.83) are implicitly given by solutions to PI or PII:

If (M

−1

)

=0then

v − y

[

wv − (α − 1/2)

]

(

α − 1/2

)

+4wv

[

wv − (α − 1/2)

]

+2v

(

+ xy

(

α −1/2

)

− ytv

(

α −1/2

)

−2 txv

= C

[

2wv −(α −1/2)

]

, (10.3.92)

where α is a constant, v is given by

v =

˙w(u) − w(u)

− u,

and w is a solution to

=8w

+8wu +4α,

which is the rescaled PII ODE (4.4.11).

262 10 : Anti-self-dual conformal structures

If (M

−1

)

=0and (M

−1

)

=0then

+ w

− w



˙w

− 4w



− 4xtw

+2wxy +



˙w

− 4w



yt = C ˙w

(10.3.93)

where w(u) satisﬁes

=24w

+8u,

which is the rescaled PI ODE (4.4.11).

If (M

−1

)

=(M

−1

)

=0then



sin (u)

cos(u) − u sin(u)

+ γ

cos(u)



−sin(u)

tx − γ cos(u)

ty = C tan(u)

, (10.3.94)

where γ is a constant.

The constant C can always be set to 0 or 1.

Example. For certain values of α, PII admits particular solutions expressible

in terms of ‘known’ functions. For α = n ∈ Z the PII equation possesses ratio-

nal solutions, and for α = n +1/2 there exists a class of solutions expressible

by Airy functions. For example, if α = 1, then PII is satisﬁed by w = −1/(2u).

Now v = −u, and the coefﬁcients of the quadric become cubic in u:

+16xtu

+(8x

− 4yt)u − (t

+4xy)=C, C = const.

The three roots of this cubic give three solutions to dKP. A root which yields

a real solution is

u(x, y, t)=



A +12t

√

12t

6yt +4x



A +12t

√

−

, (10.3.95)

A = 144 xyt +64x

+ 108 t

− 108 tC,

B = −96 y

t −48 y

+ 216 xyt

+96x

t +81t

−96

− 216 Cxy − 162 Ct

+81C

10.4 ASD conformal structures in neutral signature

The relevant Lie group isomorphism in (+ + −−) signature is

SO(2, 2)

∼

SL(2, R) × SL(2, R)/Z

. (10.4.96)

10.4 ASD conformal structures in neutral signature 263

In the neutral signature the spinor conjugation is a map S → S given by

=(α, β) → ˆι

=(α, β) (10.4.97)

(compare (9.2.17)) so that there exists a notion of real spinors. The isomor-

phism (9.2.15) is replaced by

∼

S ⊗ S



where S and S



are real rank-two vector bundles.

10.4.1 Conformal compactiﬁcation

We shall now describe a conformal compactiﬁcation of the ﬂat neutral metric

2,2

. The natural compactiﬁcation R

2,2

is a projective quadric in RP

.To

describe it explicitly consider [x, y] as homogeneous coordinates on RP

, and

set Q = |x|

−|y|

. Here (x, y) are vectors on R

with its natural inner product.

The cone Q = 0 is projectively invariant, and the freedom (x, y) ∼ (cx, cy),

where c = 0 is ﬁxed to set |x| = |y| = 1 which is S

× S

. We need to quotient

this by the antipodal map (x, y) → (−x, −y) to obtain the conformal compact-

iﬁcation

2,2

=(S

× S

)/Z

Parameterizing the double cover of this compactiﬁcation by stereographic

coordinates we ﬁnd that the ﬂat metric |dx|

−|dy|

on R

3,3

yields the metric

dζ d

(1 + ζ

ζ )

− 4

dχ d ¯χ

(1 + χ ¯χ)

(10.4.98)

on S

× S

. To obtain the ﬂat metric on R

2,2

we would instead consider the

intersection of the zero locus of Q in R

3,3,

with a null hypersurface x

− y

=1.

The metric g

is conformally ﬂat and scalar ﬂat, as the scalar curvature is the

difference between curvatures on both factors. It is also Kähler with respect to

the natural complex structures on CP

× CP

with holomorphic coordinates

(ζ,χ).

10.4.2 Curved examples

Compact neutral hyper-Kähler metrics. The only compact four-dimensional

Riemannian hyper-Kähler manifolds are the complex torus with the ﬂat

metric and K3 with a Ricci-ﬂat Calabi–Yau metric.

To write down explicit examples in neutral signature, consider the follow-

ing pseudo-hyper-Kähler metric

g = dφdy −dzdx − Q(x, y)dy

, (10.4.99)

264 10 : Anti-self-dual conformal structures

for an arbitrary function Q. This is the neutral version of the pp-wave metric

of general relativity [135], and is a special case of (10.3.65), where the

underlying projective structure is ﬂat. It is non-conformally ﬂat for generic Q.

Deﬁne complex coordinates z

= φ + iz and z

= x + iy on C

. By quotienting

the z

and z

planes by lattices one obtains a product of elliptic curves, a

special type of complex torus. If we require Q to be periodic with respect to

the z

lattice, then (10.4.99) descends to a metric on this manifold.

Tod’s scalar-ﬂat Kähler metrics on S

× S

. Consider S

× S

with the con-

formally ﬂat metric (10.4.98), that is, the difference of the standard sphere

metrics on each factor. Thinking of each sphere as CP

and letting ζ and

χ be non-homogeneous coordinates for the spheres, this metric is given by

(10.4.98). As we have already said, g

is scalar ﬂat, indeﬁnite Kähler. The

obvious complex structure J with holomorphic coordinates (ζ,χ) gives a

closed two-form and  := g

(J.,.). Moreover g

clearly has a high degree of

symmetry, since the two-sphere metrics have rotational symmetry. In [161],

Tod found deformations of g

preserving the scalar-ﬂat Kähler property, by

using the Lorentizan version of the expression (10.3.45) for neutral scalar-ﬂat

Kähler metrics with symmetry. Take the explicit solution

1 − t

(1 + x

+ y

)

to the Lorentzian Toda equation (10.3.46) (where z = it), which can be

obtained by demanding u = f

(x, y)+ f

(t). There remains a linear monopole

equation for V. Setting W = V(1 − t

) and performing the coordinate trans-

formation t = cos θ, ζ = x + iy gives

g =4W

dζ d

(1 + ζ

ζ )

− Wdθ

−

sin

(dτ + η)

, (10.4.100)

and W must solve a linear equation. This metric reduces to (10.4.98) for

W = 1 and η = 0, with θ , φ standard coordinates for the second sphere.

Differentiating the linear equation for W and setting Q =

∂W

∂t

, one obtains

the neutral wave equation

∇

Q = ∇

Q, (10.4.101)

where ∇

1,2

are the Laplacians on the two-spheres, and Q is independent of φ,

that is, axisymmetric for one of the sphere angles. Equation (10.4.101) can

be solved using Legendre polynomials, and one obtains non-conformally ﬂat

deformations of (10.4.98) in this way.

Ooguri–Vafa metrics. In [125] Ooguri and Vafa constructed a class of non-

compact neutral hyper-Kähler metrics on cotangent bundles of Riemann

surfaces with genus ≥ 1 using the heavenly equation formalism (10.2.18),

with a (+ + −−) real section of the complexiﬁed space-time. Instead of using

10.5 Twistor theory 265

the real coordinates in (10.2.18) we set

w = ζ, ˜w =

ζ, z = ip, and

z = −i

p, where ζ, p ∈ C

with  = i(p

ζ −

pζ) corresponding to the ﬂat metric. Let  be a Riemann

surface with a local holomorphic coordinare ζ , such that the Kähler metric

on  is h

dζ d

ζ . Suppose that p is a local complex coordinate for ﬁbres of

the cotangent bundle T

∗

.If is the Kähler form for a neutral metric g then

= ∂

∂

 for a function  on the cotangent bundle. Then the equation

det g

= −1

is equivalent to the ﬁrst heavenly equation (10.2.18), and gives a Ricci-ﬂat

ASD neutral metric.

Suppose that  depends only on the globally deﬁned function X = h

which is the length of the cotangent vector corresponding to p. There is a

globally deﬁned holomorphic (2, 0)-form dζ ∧ dp, which is the holomorphic

part of the standard symplectic form on the cotangent bundle, so (ζ, p)

are the holomorphic coordinates in the Pleba

nski coordinate system. The

heavenly equation reduces to an ODE for (X), and for solutions of this

ODE to exist the metric h must have constant negative curvature, so  has

genus > 1. In this case one can solve the ODE to ﬁnd

 =2

√

+ BX+ Aln

√

+ BX− A

√

+ BX+ A

where A and B are arbitrary positive constants. The metric g is well behaved

when X → 0(orp → 0), as in this limit  → ln (X) and g restricts to

dζ d

ζ on  and −h

dpd

p on the ﬁbres. In the limit X →∞the metric is

ﬂat. To see this one needs to chose a uniformizing coordinate τ on  so that h

is a metric on the upper half-plane. Then make a coordinate transformation

= τ

√

p and ζ

√

p. The holomorphic two-form is still dζ

∧ dζ

, and the

Kähler potential  = i(ζ

− ζ

)

√

B yields the ﬂat metric.

10.5 Twistor theory

Theorem 10.1.2 proved in Section 10.1 motivates the following deﬁnition

which generalizes the notion of the twistor space of Deﬁnition 7.2.1 to the

curved case:

Deﬁnition 10.5.1 The twistor space PT of a holomorphic conformal structure

(M, [g]) with ASD Weyl curvature is the manifold of α-surfaces in M.