Dunajski M. Solitons, Instantons, and Twistors

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146 7: Integrability of ASDYM and twistor theory

where f ∈ H

(CP

, O(−n − 2)).

All solutions to the spin n/2 equation

∂



···A

are given by

···A

2πi





∂

∂ω

···∂ω

π · dπ,

where f ∈ H

(CP

, O(n − 2)).

The details of the Penrose correspondence are presented in [12, 132, 175, 186].

In our next example we shall return to the ASDYM equations.

Example. Atiyah–Ward ansatz for SL(2, C) ASDYM. One way to construct

holomorphic vector bundles is to produce extensions of line bundles, which

comes down to using upper triangular matrices as patching functions.

Let E be a rank-two holomorphic vector bundle over PT which arises as

an extension of a line bundle L

→ PT by another line bundle L

→ PT :

0 −→ L

−→ E −→ L

−→ 0. (7.2.25)

Let us assume det F = 1 so that the gauge group is SL(2, C). This implies

= L ⊗ O(−k) and L

= L

∗

⊗ O(k),

where L is the line bundle from the ASD Maxwell example, and O(k)isthe

pull-back of a line bundle from CP

to PT . Therefore

F =





0 λ

−k

− f



, (7.2.26)

where  and f are holomorphic, and homogeneous of degree 0, and

 ∈ H

(PT , O(L

⊗ L

−1

)) = H

(PT , L

⊗ O (2k))

classiﬁes all extensions. The resulting solution to (6.4.23) depends on two

arbitrary functions of three variables. The relatively simple form of F corre-

sponds to very complicated (as k increases) formulae for A. Let us work out

the details if k =1, f =0.Let

 =

∞



−∞

= 

−

+ φ

+ 

Since δ

 = 0 we have the recursion relations

∂

i+1

= ∂

and ∂

˜w

i+1

= ∂

, (7.2.27)

7.2 Twistor correspondence 147

from which it follows by cross-differentiating that each φ

) satisﬁes the

scalar wave equation (7.2.22).

One can take the splitting matrices to be

H =

√



−λ

−1



−(φ

+ 

)

1 λ



and

H =

√



+ 

−

λ

−

−1



and use the Lemma 7.2.3. This implies

B =

√



−φ



and

B =

√



−1



and ﬁnally

A =

2φ



(∂ −

∂)φ 2(φ

dw + φ

˜w

dz)

2(φ

z + φ

d ˜w)(

∂ − ∂)φ



, (7.2.28)

where φ = φ

is any solution to the wave equation. This solution is invariant

under the Euclidean reality conditions. If φ is a harmonic function on R

recovered the anzatz (6.4.26) with ρ = φ.

If k > 1 then A is given in terms of a solution to the linear

zero-rest-mass ﬁeld equations with higher helicity. In general one can

show

Theorem 7.2.5 (Atiyah–Ward [8]) Every SU(2) ASDYM instanton (with the

boundary conditions (6.4.21)) over R

arises from some ansatz (7.2.26).

There is a relationship between the Atiyah–Ward ansatz and the dressing

method described in Section 3.3.2. See [154] for details.

Exercises

1. Show that the two-forms

= dw ∧ dz,ω

= dw ∧ d ˜w − dz ∧ d

z, and ω

= d

z ∧ d ˜w

span the space of SD two-forms in C

, where

=2(dzd

z − dwd ˜w) and vol = dw ∧ d ˜w ∧ dz ∧ d

Show that a two-form F is ASD iff F ∧ ω

=0.

2. Show that in four dimensions ∗∗F = ±F where the sign depends on the

signature of the metric on R

Show that in the U(1) theory F →∗F interchanges the electric and

magnetic ﬁelds with factors of ±1or±i.

148 7: Integrability of ASDYM and twistor theory

3. Use the Lax pair formulation of ASDYM to

(a) Deduce the existence of a gauge such that A = A

˜w

d ˜w + A

(b) Deduce the existence of a g-valued function K = K(w, z, ˜w,

z) such that

˜w

= ∂

K and A

= ∂

∂

K − ∂

∂

˜w

K +[∂

K,∂

K]=0.

What is the residual gauge freedom in K?

4. Show that

= 0 iff ξ

,ρ

are proportional.

A vector in C

is null iff V



= λ



for some spinors λ

,ξ



= τ

(AB)

τε

where (. . . ) denotes symmetrization, and τ should be

determined.

The ASDYM equations are of the form

∂

B(B



)

+ A

B(B



)

and are equivalent to [L

, L

] = 0, where L

= π



(∂



+ A



) for some

constant spinor π



5. Show that the factorization of the patching matrix F =



−1

in the

Penrose–Ward correspondence is unique up to multiplication of H and

H on the right by a non-singular matrix g depending on the space-time

coordinates, but not λ. Show that different choices of factorization give

gauge-equivalent connections.

6. Let the patching matrix for a rank-two Ward bundle over the twistor space

PT be given by

F =



1 f



where f = f (ω

,π



) is en element of H

(PT , O). Find the YM potential A

in terms of the ASD electromagnetic ﬁeld generated by f .

Symmetry reductions

and the integrable

chiral model

8.1 Reductions to integrable equations

Most integrable systems arise as symmetry reductions of ASDYM, where the

potential one-form is assumed not to depend on one or more coordinates on

the space-time. The resulting system will admit a (reduced) Lax pair with

a spectral parameter as well as a twistor correspondence, and thus will be

integrable. This leads to a classiﬁcation of those integrable systems that can

be obtained by reduction from the ASDYM equations as well as a uniﬁcation

of the theory of these equations by reduction of the corresponding theory for

the ASDYM equations. The programme of reducing the ASDYM equations to

various integrable equations has been proposed and initiated by Ward [171]

and fully implemented in the monograph [118].

The general scheme and classiﬁcation of reductions involves the following

choices:

1. A subgroup H of the complex conformal group PGL(4, C).

The complex conformal group consists of linear transformations ρ of M

such that

∗

(ds

)=

and ρ

∗

(vol) = 

(vol)

for some  : M

→ C. It is a symmetry group of ASDYM, as conformal

transformations map ASD two-forms to ASD two-forms, and therefore

preserve equations (6.4.23). It has 15 generators which are the conformal

Killing vectors, that is, solutions to

∂

(µ

ν)

µν

∂

These generators are given by

= T

+ L

µν

+ Rx

+ x

− 2S

, (8.1.1)

where the constant coefﬁcients T

, L

µν

= −L

νµ

, R, and S

label transla-

tions, Lorentz rotations, dilatations, and special conformal transformations,

150 8: Symmetry reductions and the integrable chiral model

respectively. It can be shown (see e.g. [118]) that the conformal group of

complexiﬁed Minkowski space is isomorphic to the projective general linear

group PGL(4, C).

The reduction is performed by assuming that the components of A do not

depend on one or more variables which parameterize the orbits of some set

of generators of the conformal group. The chosen generators must form a

Lie sub-algebra of the conformal algebra, as otherwise the reduction would

not be consistent. These generators integrate to a subgroup H ⊂ PGL(4, C).

2. A real section.

To obtain hyperbolic equations in lower dimensions we need to work

with ASDYM in the neutral signature. For elliptic reductions one chooses

the Euclidean reality conditions. Once the choice is made, the reductions

are partially classiﬁed by rank and signature of the metric tensor on M

restricted to the space of orbits of H.

3. The gauge group G.

4. Canonical forms of the Higgs ﬁelds.

Any generator of X ∈ H will correspond to a Higgs ﬁeld:

 = X

The gauge transformation g ∈ G is also invariant in the sense that X(g)=0,

so (6.1.1) reduces to

 −→ 



= gg

−1

This transformation can be used to put  into a canonical form which

depends on the Jordan normal form.

Here we are assuming that the inﬁnitesimal action of H on M

is free,

and the invariant gauges exist. If the action is not free (e.g if H = SO(3, C))

the invariant gauges do not have to exist, which leads to additional compli-

cations. See [118] for the full discussion.

Below we shall give several examples of symmetry reductions. In all cases H

will be an abelian subgroup of the conformal group generated by translations

in M

and we will not need to worry about non-trivial lifts of H to the YM

bundle and non-invariant gauges.

Example. Let us impose the Euclidean reality conditions and consider a

reduction of ASDYM by a one-dimensional group of translations. The sim-

plest way to impose a symmetry is to drop the dependence on x

. In this

case one must also restrict the gauge transformations so that they too do not

depend on x

and this implies that the component

− = ∂

8.1 Reductions to integrable equations 151

transforms homogeneously under gauge transformations  → gg

−1

.We

have

= −∂

 − [ A

,]=−D

,

and the ASDYM equations (6.4.23) reduce down to

ijk

= D



which are the Bogomolny equations (6.3.16). One could now guess that the

solutions to the Bogomolny equations with boundary conditions (6.3.11)

give rise to solutions of ASDYM with ﬁnite action by

A = −(x

)dx

+ A

)dx

This is not the case. The resulting YM potential is constant in the x

direc-

tions so the (6.3.11) does not hold.

In the next three examples we shall choose the neutral reality conditions where

all coordinates (z,

z,w, ˜w) are real. The fourth example uses Euclidean reality

conditions.

Example. Consider the SU(2) ASDYM in neutral signature and choose a

gauge A

=0.LetT

,α =1, 2, 3be2× 2 constant matrices such that

, T

]=−

αβγ

Then ASDYM equations are solved by the ansatz

= 2 cos φ T

+2sinφ T

, A

˜w

=2T

, and A

= ∂

φ T

provided that φ = φ(z,

z) satisﬁes

+4sinφ =0

which is the Sine-Gordon equation.

Example. In Section 3.3.3 we have obtained a zero-curvature representation

of the KdV equation given by (3.3.21). We shall extend this representation

to the Lax pair of ASDYM in neutral signature with two symmetries, exactly

one of which is null. First we need to get rid of the quadratic term in

λ from the KdV Lax pair. This is easily done, as [∂

− U

,∂

− V

]=0 is

equivalent to

[∂

− U

,∂

− V

− 4λ(∂

− U

)]=0.

For convenience we replace λ by λ/4 in this last expression so that it takes

the form

[∂

− A + λB,∂

+ C + λ(−∂

+ D)] = 0

152 8: Symmetry reductions and the integrable chiral model

where (A, B, C, D) are matrices given by

A =



u 0



, B =



1/40



C =



−2u

+ u

−u



, and D =



u/20



, (8.1.2)

where u = u(x, t). This can be extended to the ASDYM Lax pair. Introduce

two auxiliary coordinates (p, q) and set

L = ∂

+ ∂

− A + λ(∂

+ B) and M = ∂

+ C + λ(∂

− ∂

+ D). (8.1.3)

Then the KdV equation (2.1.1) is equivalent to [L, M] = 0. But the Lax pair

(8.1.3) is in the form (7.1.6), and so it gives rise to a solution to the ASDYM

equations, where the underlying metric is

= dp

− dx

− 4dqdt.

The solution admits two translational symmetries: ∂/∂q, which is null, and

∂/∂p. The Higgs ﬁeld B corresponding to the null symmetry is nilpotent,

which is a gauge-invariant property. This fact was used by Mason and

Sparling to establish a converse to this construction

Proposition 8.1.1 (Mason–Sparling [117]) Any solution to the ASDYM

equations with the gauge group SL(2, R) and invariant under two transla-

tions exactly one of which is null, and such that the Higgs ﬁeld corresponding

to the null translation is nilpotent is gauge equivalent to (8.1.2).

The proof given in [117] comes down to exploring the gauge freedom in

reduced ASDYM. First a gauge is chosen so that the Higgs ﬁeld B is given by

a constant nilpotent matrix, and then it is shown that the ASDYM equations

and the residual gauge freedom imply the form (8.1.2) for (A, C, D). Note

that the gauge choices made by Mason and Sparing are different than the

ones we use in (8.1.2).

Example. This example deals with a reduction of SL(3, R) ASDYM. We

shall require that the connection possesses two commuting translational

symmetries X

, X

which in our coordinates are X

= ∂

and X

= ∂

˜w

. Direct

calculation shows [58] that the ASDYM equations are solved by the follow-

ing ansatz for Higgs ﬁelds Q = A

and P = A

˜w

, and gauge ﬁelds A

and A

⎛

⎜

⎝

000

⎞

⎟

⎠

, A

˜w

⎛

⎜

⎝

001

000

⎞

⎟

⎠

8.1 Reductions to integrable equations 153

⎛

⎜

⎝

1 −φ

010

⎞

⎟

⎠

, and A

⎛

⎜

⎝

0 e

−2φ

00e

00 0

⎞

⎟

⎠

, (8.1.4)

iff φ(z,

z) satisﬁes the Tzitzéica equation [164]

= e

− e

−2φ

. (8.1.5)

This equation ﬁrst arose in a study of surfaces in R

for which the ratio of the

negative Gaussian curvature to the fourth power of a distance from a tangent

plane to some ﬁxed point is a constant. Tzitzéica has shown [164] that if z

and

z are coordinates on such a surface in which the second fundamental

form is off-diagonal, then there exists a real function φ(z,

z) such that the

Peterson–Codazzi equations reduce to (8.1.5).

This reduction can also be achieved in a gauge-invariant manner and the

analogue of Proposition 8.1.1 can be established.

Proposition 8.1.2 [58] Let A be a solution of the SL(3, R) ASDYM equations

in the neutral signature invariant under two non-null translations X

, X

such that the metric on the space of orbits of these translations has signature

(+−). Then the coordinates can be chosen so that X

= ∂

and X

= ∂

˜w

, and

(Q = A

, P = A

˜w

, A

) can be transformed into (8.1.4) by a gauge and

coordinate transformation iff the following conditions hold:

1. P and Q have minimal polynomial t

, with Tr(PQ) =0.

2. Tr

=0=Tr

and Tr

> 0.

3. Tr [(PQ)

+(PQ)

P)(D

Q) − PQ(D

P)QP(D

Q)] = 0.

The details are more complicated than in the KdV case. This is because there

are more normal forms in the SL(3) case, and additional gauge-invariant con-

ditions (2) and (3) need to be imposed to select the normal forms leading to

the Tzitzéica equation. Dropping the condition (3) leads to the Z

reduction

of the two-dimensional Toda chain [119]. See [58] for details.

Example. By imposing three translational symmetries one can reduce

ASDYM to an ODE. Choose the Euclidean reality condition, and assume

that the YM potential is independent on x

, x

, and x

Select a gauge A

= 0, and set A

= 

, where the Higgs ﬁelds 

are real

g-valued functions of x

= t. The ASDYM equations reduce to the Nahm

equations



=[

,



=[

,

], and



=[

,

]. (8.1.6)

154 8: Symmetry reductions and the integrable chiral model

These equations admit a Lax representation which comes from taking a

linear combination of L and M in (7.1.6). Let

A(λ)=(

+ i

)+2

λ − (

− i

)λ

Then

A =[

− i

,

]+2[

,

]λ − [

+ i

,

]λ

=[A, −i

+ i(

− i

)λ]

=[A, B], where B = −i

+ i(

− i

)λ. (8.1.7)

This representation reveals the existence of many conserved quantities for

(8.1.6). We have

Tr( A

)=Tr(p[A, B]A

p−1

)=pTr( ABA

p−1

− BA

)=0

by the cyclic property of trace. Therefore all the coefﬁcients of the polynomi-

als Tr[A(λ)

] for all p are conserved (which implies that the whole spectrum

of A(λ) is constant in t).

Taking G = SU(2) and setting 

(t)=iσ

(t), where σ

are the Pauli

matrices reduces the problem to the Euler equations

˙w

= w

, ˙w

= w

, and ˙w

= w

(˙w

)

=(w

− C

)(w

− C

)

(where C

and C

are constants) which is solvable by Weierstrass elliptic

function.

8.2 Integrable chiral model

We have seen that symmetry reductions of the ASDYM equations in neutral

signature can lead to hyperbolic equations in lower dimensions, where one can

make direct contact with time evolution of moving solitons. In this section we

shall analyse one such reduction in detail.

Consider the ASDYM equations in neutral signature invariant under the

action of one-parameter group of non-null translations. The underlying metric

on R

2,2

is of the form (7.1.1) where all the coordinates are real. Let us assume,

without loss of generality, that the metric on the three-dimensional space of

orbits has signature (2, 1). Therefore we can choose the coordinates such that

8.2 Integrable chiral model 155

the Killing vector is K = ∂/∂τ and

z =

x − τ

√

z =

x + τ

√

,w=

t + y

√

, and ˜w =

t − y

√

where (t, x, y,τ) are all real. With these choices the Higgs ﬁeld is  = ∂

A =

− A

√

2 and the ASDYM equations become

 = F

, D

 = F

, and D

 = F

D = ∗F, (8.2.8)

where ∗ is the Hodge operator on R

2,1

taken with respect to the Minkowski

metric

= η

µν

= −dt

+ dx

+ dy

The ﬁrst-order Yang–Mills–Higgs system (8.2.8) was introduced by Ward

[173]. The unknowns (A = A

dt + A

dx + A

dy,) are a one-form and a func-

tion which depend on local coordinates x

=(t, x, y), and take values in the Lie

algebra of gauge group G = U(n). They are subject to gauge transformations

A −→ gAg

−1

− dg g

−1

,−→ gg

−1

, and g = g(x, y, t) ∈ U(n).

(8.2.9)

The system (8.2.8) formally resembles the BPS equations (6.3.16) arising in

the study of non-abelian monopoles. We shall however see that the Lorentzian

reality conditions dramatically change the overall structure and the behaviour

of solutions.

There are not any known examples of Lorentz-invariant integrable equa-

tions admitting time-dependent soliton solutions in 2 + 1 dimensions, but the

system (8.2.8) almost does the job: it is Lorentz invariant and in the next

section moving soliton solutions will be written down explicitly. It cannot

however be regarded as a genuine soliton system, because the energy functional

associated to the Lagrangian density

L =

Tr( F

µν

) − Tr( D

D

) (8.2.10)

is not positive-deﬁnite and its density vanishes on all solutions to (8.2.8).

Note that L cannot be regarded as a Lagrangian for (8.2.8) as the sys-

tem (8.2.8) is ﬁrst order and the Euler–Lagrange equation arising from L

are second order. There is however a connection: the second-order Euler–

Lagrange equations are satisﬁed by solutions to the ﬁrst-order system

(8.2.8).