arX

iv:2

007.

0842

4v5

[m

ath-

ph]

13

Jan

2021

Nonlocal symmetry of CMA generates ASD

Ricci-flat metric with no Killing vectors

M. B. Sheftel1

1 Department of Physics, Bogazici University

34342 Bebek, Istanbul, Turkey ∗

Abstract

The complex Monge-Ampere equation (CMA) in a two-componentform is treated as a bi-Hamiltonian system. I present explicitly the firstnonlocal symmetry flow in each of the two hierarchies of this system.An invariant solution of CMA with respect to these nonlocal symme-tries is constructed which, being a noninvariant solution in the usualsense, does not undergo symmetry reduction in the number of inde-pendent variables. I also construct the corresponding 4-dimensionalanti-self-dual (ASD) Ricci-flat metric with either Euclidean or neutralsignature. It admits no Killing vectors which is one of characteris-tic features of the famous gravitational instanton K3. For the metricwith the Euclidean signature, relevant for gravitational instantons, Iexplicitly calculate the Levi-Civita connection 1-forms and the Rie-mann curvature tensor.

MSC : 35Q75, 83C15, 37K05, 37K10.

1 Introduction

In his pioneer paper [1], Plebanski demonstrated that anti-self-dual (ASD)Ricci-flat metrics on four-dimensional complex manifolds are completely de-termined by a single scalar potential which satisfies his first or second heav-enly equation. Such metrics are solutions to complex vacuum Einstein equa-tions with zero cosmological constant. Real four-dimensional hyper-KahlerASD metrics

ds2 = u11dz1dz1 + u12dz

1dz2 + u21dz2dz1 + u22dz

2dz2 (1.1)

∗E-mail: mikhail.sheftel@boun.edu.tr

1

that solve the vacuum Einstein equations with either Euclidean or ultra-hyperbolic signature are governed by a scalar real-valued Kahler potentialu = u(z1, z2, z1, z2) which satisfies elliptic or hyperbolic complex Monge-Ampere equation (CMA)

u11u22 − u12u21 = ε (1.2)

with ε = ±1 respectively. Here u is a real-valued function of the two complexvariables z1, z2 and their conjugates z1, z2, the subscripts denoting partialderivatives with respect to these variables, e.g. u11 = ∂2u/∂z1∂z1 andsuchlike. A modern proof of this result one can find in the books by Masonand Woodhouse [2] and Dunajski [3].

To illustrate this property, we introduce the coframe of one-forms

ω1 =1

√u11

(u11dz1 + u21dz2), ω1 =1

√u11

(u11dz1 + u12dz2)

ω2 =1

√u11

dz2, ω2 =1

√u11

dz2. (1.3)

The metric (1.1) takes the canonical form

ds2 = ω1 ⊗ ω1 + εω2 ⊗ ω2 (1.4)

where complex Monge-Ampere equation (1.2) has been used. Equation (1.4)makes obvious the claim about the signature of the metric.

We are mostly interested in ASD Ricci-flat metrics that describe grav-itational instantons which asymptotically look like a flat space, so thattheir curvature is concentrated in a finite region of a Riemannian space-time (see [3, 4] and references therein). The most important gravitationalinstanton is K3 which geometrically is Kummer surface [5], for which anexplicit form of the metric is still unknown while many of its properties andexistence had been discovered and analyzed [4–7]. A characteristic featureof the K3 instanton is that it does not admit any Killing vectors, that is, nocontinuous symmetries, which implies that the metric potential should be anoninvariant solution of CMA equation. As opposed to the case of invariantsolutions, for noninvariant solutions of CMA there should be no symmetryreduction [8] in the number of independent variables. In this paper weachieve this goal by utilizing the invariance under nonlocal symmetries.

The main result of the paper has the form

u = − γ

c0c21

e−c1x +1

3c20c21c

23µ

2a3− c2

c1x+

γ

c0c1c3(y sin θ + z cos θ)

+γ

c0c21c23

(

σ0 −γν

3c0

)

+εν

4c21+ ρ1t+ ρ2 + r(y, z), v = ut (1.5)

2

wherea = ∓Γ−1/2, Γ = 2c0µ

(

c23e−c1x − σ(y, z, t)

)

+ ν2 (1.6)

with an arbitrary harmonic function r(y, z) and σ, σ0 defined in (7.5), (7.12),respectively, γ = ν/µ, µ and ν defined by (6.47) and (6.51), θ = −c1t+ θ0,c0, c1, c2, c3, ρ1, ρ2, θ0 are arbitrary real constants.

Theorem 1.1 The expressions (1.5), (1.6) for u and v provide a solution of

the CMA in its real two-component form (2.4). This solution by construc-

tion is invariant under the first nonlocal symmetry flow of the CMA in each

of the two hierarchies of this equation. With the generic set of constants, this

solution is noninvariant under point symmetries of CMA and consequently

it experiences no reduction in the number of independent variables.

Noninvariance of the solution for the Kahler potential of the metric impliesthe absence of point symmetries (Killing vectors) for the metric itself sinceno reduction in the number of independent variables obviously takes placealso for the metric coefficients in Section 8.

The noninvariance of the solution (1.5) under point symmetries looks ob-vious from the absence of symmetry reduction in the solution which clearlydepends on four independent combinations of variables t, x, y, z (two differ-ent combinations of y and z enter expressions for v = ∂u/∂t and there isobviously no reduction in either x or t). The outline of a more rigorousapproach to the proof of noninvariance is formulated in the Appendix.

As far as we know, there are no published examples of noninvariant solu-tions of CMA and the corresponding gravitational metrics with no Killingvectors with the exception of our earlier paper [9], and there was still noprogress in the explicit construction of K3 instanton metric. Instructivereviews of the present state of the theory of gravitational instantons can befound in the book [3] and review [4].

The paper is organized as follows. In Section 2, we convert CMA equa-tion into real variables and two-component form. In Section 3, we exhibittwo alternative bi-Hamiltonian representations of CMA which we discov-ered earlier [10]. In Section 4, we explicitly construct first nonlocal flows ineach hierarchy of CMA system related to its two bi-Hamiltonian structures.

In Section 5, we formulate the invariance conditions with respect to bothnonlocal symmetries appearing in each of the two alternative hierarchies. Inthis way, we keep in the invariance conditions the obvious discrete symme-try relating two bi-Hamiltonian structures. Thus, alternatively we can saythat we impose invariance conditions under one nonlocal symmetry and thediscrete symmetry. In the following, for simplicity we restrict ourselves to

3

the invariance under the special first nonlocal symmetry by setting Φ = 0and χ = 0.

In Section 6, we show in detail how a careful analysis of integrabilityconditions specifies various functional parameters in the invariance equationswith no additional assumptions made. In Section 7, we integrate completelyall the obtained equations and end up with a noninvariant solution of CMAwhich is a general form of the solution invariant under the special firstnonlocal symmetry in each hierarchy. In Section 8, we use this solution forconstructing the corresponding (anti-)self-dual Ricci-flat metric with eitherEuclidean or neutral signature.

In Section 9 we calculate Levi-Civita connection 1-forms for Euclideansignature by solving Cartan structure equations in the case of no torsion. InSection 10 we obtain the corresponding Riemann curvature tensor compo-nents. These results should be helpful for the future analysis of singularitiesand asymptotic behavior of the metric. We note that the results for the con-nection and Riemann curvature are not only related to our solution but arealso valid more generally for the Kahler metric (8.2) generated by any solu-tion of CMA which may be helpful for future researchers dealing with thissubject. In the Appendix, we give the full set of point symmetries of CMAand outline the approach needed for the rigorous proof of noninvariance ofits solutions.

2 Real variables and 2-component form of CMA

In our earlier paper [10] we presented bi-Hamiltonian structure of the two-component version of (1.2), which by Magri’s theorem [11] proves that it isa completely integrable system in four dimensions.

We impose additional reality condition for all the objects in the theory.The transformation from complex to real variables has the form

t = z1 + z1, x = i(z1 − z1), y = z2 + z2, z = i(z2 − z2). (2.1)

We introduce the notation

a = ∆(u) = uyy + uzz, b = uxy − vz, c = vy + uxz, Q =b2 + c2 + ε

a(2.2)

where v = ut is the second component of the unknown and ∆ = D2y +D2

z isthe two-dimensional Laplace operator (letter subscripts denote correspond-ing partial derivatives). The definitions (2.2) imply the relations

ax = by + cz, cy − bz = ∆(v) ≡ at. (2.3)

4

The CMA equation (1.2) in the real variables becomes

(utt + uxx)∆(u)− b2 − c2 − ε = 0

or in the two-component form

(

utvt

)

=

(

vQ− uxx

)

(2.4)

which we will call CMA system.The metric (1.1) in real variables reads

ds2 = (vt + uxx)(dt2 + dx2) + a(dy2 + dz2)

− 2b(dt dz − dxdy) + 2c(dt dy + dxdz). (2.5)

The coframe of one-forms becomes

Ω1 =1√

vt + uxx[(c+ ib)(dy + idz) + (vt + uxx)(dt+ idx)]

Ω2 =dy + idz√vt + uxx

(2.6)

with the metricds2 = Ω1 ⊗ Ω1 + εΩ2 ⊗ Ω2. (2.7)

3 Bi-Hamiltonian representations of CMA system

The CMA system (2.4) can be put in the Hamiltonian form

(

utvt

)

= J0

(

δuH1

δvH1

)

(3.1)

where δu and δv are Euler-Lagrange operators [8] with respect to u and v.Here J0 is the Hamiltonian operator

J0 =

0 1a

−1a

1a(cDy +Dyc− bDz −Dzb)

1a

(3.2)

determining the structure of Poisson bracket and H1 is the correspondingHamiltonian density

H1 =1

2[v2∆(u)− uxx(u

2y + u2z)]− εu. (3.3)

5

The first real recursion operator has the form

R1 =

(

0 0QDz − cDx b

)

+ (3.4)

∆−1

Dy

(

−aDx + bDy + cDz

)

+Dz

(

cDy − bDz

)

−Dza

Dx

[

Dy

(

cDy − bDz

)

+Dz

(

aDx − bDy − cDz

)]

−DxDya

where means operator multiplication, and the second recursion operatorreads

R2 =

(

0 0bDx −QDy c

)

+ (3.5)

∆−1 (

Dy(bDz − cDy) +Dz(−aDx + bDy + cDz) Dya

Dx

[

Dy(−aDx + bDy + cDz) +Dz(cDy − bDz)]

−DxDza

)

.

The two recursion operators R1 and R2 generate two alternative secondHamiltonian operators J1 = R1J0 and J1 = R2J0

J1 = R1J0 = ∆−1 (

Dz −DxDy

DxDy D2xDz

)

+ (3.6)

0 ba

− ba

ca2

(

bDy − aDx

)

+(

Dyb−Dxa)

ca2

+Q−

2a Dz +DzQ−

2a

where Q− = (c2 − b2 + ε)/a, and

J1 = R2J0 = ∆−1 (

Dy DxDz

−DxDz D2xDy

)

+ (3.7)

0 − ca

ca

ba2

(

cDz − aDx

)

+(

Dzc−Dxa)

ba2

+Q−

2a Dy +DyQ−

2a

where Q− = (b2 − c2 + ε)/a.The flow (3.1) can be generated by the Hamiltonian operator J1 from

the Hamiltonian density

H0 = zv∆(u) + uxuy (3.8)

so that CMA in the two-component form (3.1) is a bi-Hamiltonian system

[11](

utvt

)

= J0

(

δuH1

δvH1

)

= J1

(

δuH0

δvH0

)

. (3.9)

6

The same flow (3.1) can also be generated by the Hamiltonian operatorJ1 from the Hamiltonian density

H0 = yv∆(u)− uxuz (3.10)

which yields another bi-Hamiltonian representation of the CMA system(3.1)

(

utvt

)

= J0

(

δuH1

δvH1

)

= J1

(

δuH0

δvH0

)

. (3.11)

The relation between the two bi-Hamiltonian structures is realized by thediscrete symmetry z 7→ y, y 7→ −z.

4 Nonlocal flows

The first nonlocal flows of each hierarchy of CMA system are generated byJ1 and J1 acting on the vector of variational derivatives of H1

(

uτ2vτ2

)

= J1

(

δuH1

δvH1

)

(4.1)

(

uτ2 ′

vτ2 ′

)

= J1

(

δuH1

δvH1

)

(4.2)

where τ2, τ2′ are time variables of the flows (4.1), (4.2), respectively. Usingthe expressions (3.6), (3.7) and (3.3) for J1, J

1 and H1 we obtain explicitexpressions for the flows (4.1) and (4.2)

uτ2 = ∆−1

Dz(auxx − u2xy − u2xz − ε)−DxDy(av)

+ uxyv

uτ2 ′

= ∆−1

Dy(auxx − u2xy − u2xz − ε) +DxDz(av)

− uxzv. (4.3)

Second components of these flows are time derivatives of (4.3), vτ2 = Dt[uτ2 ],vτ

2 ′= Dt[uτ

2 ′], so that the flows (4.1) and (4.2) commute with the flow

(2.4) of CMA system and hence they are nonlocal symmetries of the CMAsystem.

5 Invariance conditions with respect to nonlocal

symmetries

Solutions invariant with respect to nonlocal symmetries are determined bythe conditions uτ2 = 0 and uτ

2 ′= 0 which due to (4.3) take the explicit

7

form (with a replaced by ∆[u] according to (2.2))

Dz

[

∆[u]uxx − u2xy − u2xz]

−DxDy [v∆[u]] + ∆ [vuxy] = 0

Dy

[

∆[u]uxx − u2xy − u2xz]

+DxDz [v∆[u]]−∆ [vuxz] = 0. (5.1)

Here we impose both invariance conditions (5.1) on solutions of the CMAsystem in order to keep the discrete symmetry z 7→ y, y 7→ −z between thetwo bi-Hamiltonian structures. Differentiating the first and second equations(5.1) with respect to y and z, respectively, and taking the difference of theresults yield the integrability condition

∆ Dy[vuxy] +Dz[vuxz]−Dx[v∆[u]] = 0

or, equivalently

Dy[vuxy] +Dz[vuxz]−Dx[v∆[u]] = Φ(x, y, z, t)

⇐⇒ vyuxy + vzuxz − vx∆[u] = Φ, where ∆[Φ] = 0.

(5.2)

On account of the relation (5.2) each of the relations (5.1) becomes

vyuxz − vzuxy = ∆[u]uxx − u2xy − u2xz + χ(x, y, z, t) (5.3)

where Φy = χz and Φz = −χy and hence ∆[Φ] = ∆[χ] = 0. Thus, we end upwith the system of two equations (5.2) and (5.3) linear in derivatives of v.Solving this system algebraically for vy and vz and denoting δ = u2xy + u2xz,we obtain

vy =1

δ∆[u](vxuxy + uxxuxz)− δuxz +Φuxy + χuxz

vz =1

δ∆[u](vxuxz − uxxuxy) + δuxy +Φuxz − χuxy . (5.4)

In the following for simplicity we set Φ = 0, χ = 0 and refer to this case asthe invariance under special first nonlocal symmetries. In the following it isconvenient to introduce the quantity

w =δ

∆[u]− uxx. (5.5)

Equations (5.4) become

vy =uxyvx − uxzw

w + uxx, vz =

uxzvx + uxyw

w + uxx(5.6)

8

with the immediate consequences

vx =1

∆[u](uxyvy + uxzvz), w =

1

∆[u](uxyvz − uxzvy). (5.7)

On account of the equations (5.5) and (5.6), the real two-componentform (2.4) of CMA becomes

vt =v2x − uxxw

w + uxx+

ε

∆[u]. (5.8)

Integrability condition (vy)z − (vz)y = 0 of equations (5.6) yields

(uxzwy − uxywz)vx = uxx(uxywy + uxzwz)−∆[u](w + uxx)wx. (5.9)

6 Further integrability conditions

It is convenient to take equation (5.9) in the form

(uxzvx − uxxuxy)wy = (uxyvx + uxxuxz)wz −∆[u](w + uxx)wx. (6.1)

The integrability conditions (vt)y = (vy)t and (vt)z = (vz)t with the use of(6.1) simplify to

∆[uy] =uxy

w + uxx∆[ux], ∆[uz] =

uxzw + uxx

∆[ux] (6.2)

with the integrability condition (∆[uy])z − (∆[uz])y = 0 resulting in

uxzwy − uxywz = 0. (6.3)

On account of (6.3) equation (5.9) becomes

uxx(uxywy + uxzwz)−∆[u](w + uxx)wx = 0. (6.4)

With the aid of the definition (5.5), equations (6.3) and (6.4) can be put inthe form

wy =uxyuxx

wx, wz =uxzuxx

wx (6.5)

with the integrability condition (wy)z = (wz)y identically satisfied. Differ-entiating the definition (5.5) of w with respect to t we obtain

wt =vxuxx

wx. (6.6)

9

The integrability conditions (wt)y = (wy)t and (wt)z = (wz)t of equations(6.5) and (6.6) are identically satisfied.

Equations (6.5) are integrated by the method of characteristics with theresult w = w(ux, t) whereas (6.6) further implies w = w(ux). The remainingequations (6.2) take the form of two first-order PDEs for ∆[u]

(uxx + w(ux))(∆[u])y − uxy(∆[u])x = 0

(uxx + w(ux))(∆[u])z − uxz(∆[u])x = 0. (6.7)

We start with the combination of these equations

uxz(∆[u])y − uxy(∆[u])z = 0 (6.8)

which is integrated by the method of characteristics to give∆[u] = f(ux, x, t). Then equations (6.7) are satisfied by the solution

∆[u] = f(ζ, t), where ζ = ω(ux) + x and ω(ux) =

∫

duxw(ux)

. (6.9)

Equation (5.5) becomes

δ ≡ u2xy + u2xz = A2, where A =

√

f

(

uxx +1

ω′

)

. (6.10)

We rewrite (6.10) in the form

uxy = A sin θ, uxz = A cos θ (6.11)

where we have introduced the new unknown θ = θ(x, y, z, t). The definitionof A in (6.10) implies

Ay = a sin θ +f

2cos θ · θx, Az = a cos θ − f

2sin θ · θx

Ax =fζA

3

2f2ω′ − A

2

ω′′

ω′2+

f

2A

(

ω′′

ω′3+ uxxx

)

(6.12)

where

a =3

4

(

fζfA2ω′ − f

ω′′

ω′2

)

+f2

4A2

(

ω′′

ω′3+ uxxx

)

. (6.13)

The integrability condition (uxy)z − (uxz)y = 0 of the system (6.11) has theform

sin θ · θy + cos θ · θz =f

2Aθx. (6.14)

10

The first equation ∆[u] = f(ζ, t) in (6.9) implies

∆[ux] = (uxy)y + (uxz)z = (A sin θ)y + (A cos θ)z = fζ(ω′uxx + 1) (6.15)

which finally results in the equation

cos θ · θy − sin θ · θz =fζA

fω′ − a

A. (6.16)

We solve algebraically the system of equations (6.14) and (6.16) to obtain

θy =f

2Asin θ · θx + b cos θ, θz =

f

2Acos θ · θx − b sin θ (6.17)

where

b =fζfAω′ − a

A. (6.18)

Integrability condition for the system (6.17) reads

(θy)z − (θz)y = bz cos θ + by sin θ −f

2Abx + b2 +

f2

4A2θ2x = 0 (6.19)

or in the explicit form

(

f2

2A3uxxx −

fζ2f

Aω′ +f2

2A3

ω′′

ω′3− 3f

2A

ω′′

ω′2

)2

+f2

A2θ2x = 0. (6.20)

Reality condition for (6.20) implies that both quadratic terms vanish sepa-rately

uxxx =fζf3

A4ω′ − ω′′

ω′3+

3A2

f

ω′′

ω′2(6.21)

and θx = 0 ⇐⇒ θ = θ(y, z, t). Using (6.21) in (6.13) for a and then in(6.18) for b we obtain

b = 0, a =fζfA2ω′ (6.22)

and (6.17) with θx = 0 implies θy = θz = 0, so that θ = θ(t). Equations(6.12) become

Ay = a sin θ, Az = a cos θ, Ax =fζf2

A3ω′ +Aω′′

ω′2. (6.23)

From the definition (6.10) of A it follows

uxx =A2

f− 1

ω′(6.24)

11

Since f = f(ζ, t) where ζ = ω(ux) + x, we have

fx =fζfA2ω′, fy = fζω

′A sin θ fz = fζω′A cos θ (6.25)

and hence Ay/A = fy/f , Az/A = fz/f , so that A = α(x, t)f . Then the lastequation in (6.23) implies

Ax

A=

fxf

+ω′′

ω′2=

fxf

+αx(x, t)

α=⇒

(

ω′′

ω′2

)

y

= 0,

(

ω′′

ω′2

)

z

= 0 (6.26)

and hence(

ω′′

ω′2

)

′

= 0 =⇒ 1

ω′= c1ux + c2 ⇐⇒ ω′ =

1

c1ux + c2. (6.27)

From (6.26) and (6.27) it follows that

α(x, t) = c3(t)e−c1(t)x, A = c3(t)e

−c1xf(ζ, t) (6.28)

while (6.24) and (6.11) imply

uxx = c23(t)e−2c1xf − (c1ux + c2)

uxy = c3e−c1xf sin θ, uxz = c3e

−c1xf cos θ (6.29)

while (6.21) takes the form

uxxx = c43(t)e−4c1xffζc1ux + c2

− 3c23(t)c1e−2c1xf + c1(c1ux + c2). (6.30)

It is easy to check that the integrability condition (uxx)x = uxxx of equations(6.29) and (6.30) is identically satisfied.

Expressions (5.6) and (5.8) for derivatives of v take the form

vy =ec1x

c3(t)[vx sin θ − (c1ux + c2) cos θ]

vz =ec1x

c3(t)[vx cos θ + (c1ux + c2) sin θ]

vt =e2c1x

c23(t)f[v2x + (c1ux + c2)

2]− (c1ux + c2) +ε

f. (6.31)

Integrability condition (vt)x = (vx)t where vx is determined by (5.7) in theform

vx = c3(t)e−c1x(vy sin θ + vz cos θ)

12

yields the relation

(c1 + θ′(t))(c1ux + c2)−c′3(t)

c3(t)vx = 0. (6.32)

If c′3(t) 6= 0, (6.32) can be solved for vx and the integrability conditions(vy)x = (vx)y, (vz)x = (vx)z imply c1c3 = 0 and, since c1 6= 0 (otherwise(6.32) leads to a reduction), we have c3 = 0 which contradicts to our as-sumption c′3(t) 6= 0.

Thus, we proceed to the opposite case of constant c3 in (6.32) whichimplies θ(t) = −c1t + θ0 with vx remaining undetermined. Then we checkthat all the integrability conditions (vy)t = (vt)y, (vz)t = (vt)z and (vy)z =(vz)y are identically satisfied.

Differentiating first two equations (6.31) with respect to x we obtain

vxy =ec1x

c3(vxx + c1vx) sin θ − c1c3e

−c1xf cos θ

vxz =ec1x

c3(vxx + c1vx) cos θ + c1c3e

−c1xf sin θ. (6.33)

Another type of integrability conditions arises from the utilization of therelation ut = v. We differentiate with respect to t equations (6.29). Thefirst equation in (6.29) yields

vxx = −c1vx + c23e−2c1x

(

fζvxc1ux + c2

+ ft

)

(6.34)

while the two last equations give the same results as (6.33). The sameresult (6.34) we obtain by differentiating with respect to t the equation∆[u] = f(ζ, t). By using the result (6.34) in (6.33) we obtain

vxy = c3e−c1x

[(

fζvxc1ux + c2

+ ft

)

sin θ − c1f cos θ

]

vxz = c3e−c1x

[(

fζvxc1ux + c2

+ ft

)

cos θ + c1f sin θ

]

. (6.35)

The two-component real form (5.8) of CMA equation becomes

vt =e2c1x

c23f[v2x + (c1ux + c2)

2] +ε

f− (c1ux + c2) (6.36)

together with its derivative with respect to x

vtx =fζf

[

v2xc1ux + c2

− (c1ux + c2)−εc23e

−2c1x

c1ux + c2

]

+ 2ftfvx + 3c1(c1ux + c2)− c1c

23e

−2c1xf. (6.37)

13

Differentiating (6.37) with respect to x we discover the integrability condi-tion (vt)xx − (vxx)t = 0 of (6.36) and (6.34) in the form

fttf − 2f2t − 4c21f

2 + fζζf − 2f2ζ + 4c1fζf

+εc23e

−2c1x

(c1ux + c2)2(fζζf − 2f2

ζ − c1fζf) = 0. (6.38)

Since ux and x in equations for f are allowed only in the combination ζ =ln(c1ux + c2)/c1 + x + ζ0, which is the independent variable in f(ζ, t), theequation (6.38) splits into two equations

fζζf − 2f2ζ − c1fζf = 0 (6.39)

fttf − 2f2t − 4c21f

2 + fζζf − 2f2ζ + 4c1fζf = 0. (6.40)

Using (6.39) in the equation (6.40) we simplify the latter equation to theform

fttf − 2f2t − 4c21f

2 + 5c1fζf = 0. (6.41)

Equation (6.39) by the substitution g = fζ/f is reduced to the first-orderseparable equation gζ−g(g+c1) = 0 with the solution g = c1/(γ(t)e

−c1ζ−1).The corresponding general solution for f is given by the quadrature

ln f = ln f0(t) + c1

∫

dζ

γ(t)e−c1ζ − 1

or in the explicit form

f =f0(t)

γ(t)− ec1ζ. (6.42)

In the original variables this becomes

f =f0(t)

γ(t)− c0ec1x(c1ux + c2). (6.43)

Next we substitute the expression (6.42) for f in the equation (6.41) andafter cancelation of the common factor γ − ec1ζ we obtain the result

(f ′′

0 f0 − 2f ′

02 − 4c21f

20 )(γ − ec1ζ)

+ 2f0f′

0γ′ − f2

0γ′′ + 5c21f

20 e

c1ζ = 0 (6.44)

where primes denote derivatives of functions depending on t only. Termswith the first and zeroth powers of ec1ζ should vanish separately. The termswith ec1ζ yield the equation

f ′′

0 f0 − 2f ′

02 − 9c21f

20 = 0 (6.45)

14

with the general solution

f0 =1

µ(t), where µ′′ = −9c21µ (6.46)

or, explicitly,µ(t) = µ1 cos(3c1t) + µ2 sin(3c1t) (6.47)

with arbitrary real constants µ1 and µ2. The remaining equation consists ofthe terms in (6.44) without ec1ζ which with the use of (6.46) becomes

γ′′ + 2µ′

µγ′ − 5c21γ = 0. (6.48)

For the integration of (6.48) we use its two commuting Lie point symmetries

X1 = γ∂γ , X2 = −∂t +µ′

µγ∂γ . (6.49)

Using the appropriate algorithm for the case G2Ia from H. Stephani’s book[12] we easily obtain the general solution

γ(t) =ν(t)

µ(t), where ν ′′ = −4c21ν (6.50)

or, explicitlyν(t) = ν1 cos(2c1t) + ν2 sin(2c1t) (6.51)

whereas µ is defined in (6.47) and ν1, ν2 are arbitrary real constants.Using our results in (6.42) we obtain the final result for f

f(ζ, t) =1

ν(t)− µ(t)ec1ζ(6.52)

or, alternatively, by using ec1ζ0 = c0 in the definition ζ = ln(c1ux + c2)/c1 +x+ ζ0 coming from (6.9) and (6.27)

f =1

ν(t)− µ(t)ec1xc0(c1ux + c2)(6.53)

where µ(t) is defined by (6.46) and ν(t) defined in (6.50). It turns outthat the integrability conditions (vxt)y = (vxy)t and (vxt)z = (vxz)t of theequations (6.35) and (6.37) with f defined by (6.53) are identically satisfiedwithout further constraints on the unknowns vx, ux, µ and ν.

15

7 Integration of equations

After making sure that the integrability conditions of our equations aresatisfied we may proceed to the integration of these equations. We startwith the integration of the first equation in (6.29) with respect to x withthe result

1

2c0µe

2c1x(c1ux + c2)2 − νec1x(c1ux + c2) = c23e

−c1x − σ(y, z, t)

which can be rewritten as

c1ux + c2 =1

c0µe−c1x

(

ν ± Γ1/2)

(7.1)

whereΓ = 2c0µ

(

c23e−c1x − σ(y, z, t)

)

+ ν2 (7.2)

with σ(y, z, t) playing the role of the constant of integration with respectto x. Henceforth we are free to use either upper or lower sign in all theformulas. Using (7.1) together with the notation (7.2) in the formula (6.53)for f we finally obtain

a = ∆(u) = f = ∓Γ−1/2 (7.3)

so that (7.1) becomes

ux =1

c0c1e−c1x

(

γ − 1

µa

)

− c2c1. (7.4)

Next we utilize the result (7.4) in the second and third equations in(6.29), using Γy = −2c0µσy and Γz = −2c0νσz to obtain σy = c1c3 sin θ andσz = c1c3 cos θ and hence

σ = c1c3(y sin θ + z cos θ) + σ0(t) (7.5)

which implies σt = c21c3(z sin θ − y cos θ) + σ′

0(t). The result (7.5) should beused in the definition (7.2) of Γ.

Total derivative of (7.4) with respect to t yields

vx =e−c1x

c0c1

(

γ − 1

µa

)

t

. (7.6)

Integrating (7.4) with respect to x we obtain

u = − γ

c0c21e−c1x +

1

3c20c21c

23µ

2a3− c2

c1x+ ρ(y, z, t) (7.7)

16

with the “constant of integration” ρ(y, z, t). Differentiation of (7.7) withrespect to t yields the result

v = ut = − γ′

c0c21e−c1x +

1

3c20c21c

23

(

1

µ2a3

)

t

+ ρt. (7.8)

Now we use the expression (7.8) to compute vy and vz in the first twoequations (6.31) which imply the following equations

ρty(y, z, t) =1

c0c1c3(γ sin θ)′ , ρtz(y, z, t) =

1

c0c1c3(γ cos θ)′

with the final result

ρ(y, z, t) =γ

c0c1c3(y sin θ + z cos θ) + ρ0(t) + r(y, z) (7.9)

with arbitrary ρ0(t) and r(y, z). Equation ∆[u] = f implies ∆[r(y, z)] = 0.We differentiate v in (7.8) with respect to t and use vt in the third

equation of (6.31) which is equivalent to our basic complex Monge-Ampereequation in the two-component form (2.4) (since ut ≡ v by construction).This equation after numerous cancelations, an appropriate use of equations(6.47), (6.51) and (6.48) for µ, ν and γ, respectively, and taking two quadra-tures with respect to t yields the result

ρ0(t) =γ

c0c21c23

(

σ0 −γν

3c0

)

+εν

4c21+ ρ1t+ ρ2 (7.10)

with arbitrary real constants ρ1 and ρ2 and σ0(t) satisfying the equation

σ′′

0 (t) + c21σ0(t) = −εc0c21c

23µ. (7.11)

The solution to (7.11) reads

σ0(t) =εc0c

23

8µ+A cos c1t+B sin c1t (7.12)

where µ = µ1 cos 3c1t+ µ2 sin 3c1t, A and B are arbitrary real constants.By using a, σ, σ0, ρ and ρ0 defined in (7.3), (7.5), (7.12), (7.9), and

(7.10), respectively, in (7.7) and (7.8) for u and v we arrive at the solution(1.5), (1.6) announced at the Introduction.

17

8 The metric

All our equations being completely solved, we can explicitly construct thecorresponding metric (2.5). By using the definition of Q from (2.2), we usethe CMA system in the two-component form

ut = v, vt + uxx = Q ≡ (b2 + c2 + ε)/a (8.1)

so that for any solution of (8.1) the metric (2.5) becomes

ds2 = Q(dt2 + dx2) + a(dy2 + dz2)

− 2b(dt dz − dxdy) + 2c(dt dy + dxdz). (8.2)

By introducing the coframe of 1-forms

e0 = Q1/2dt+Q−1/2(cdy − bdz)

e1 = Q1/2dx+Q−1/2(bdy + cdz)

e2 = Q−1/2dy, e3 = Q−1/2dz. (8.3)

we convert the metric (8.2) into the diagonal quadratic form

ds2 = (e0)2 + (e1)2 + ε[

(e2)2 + (e3)2]

, ε = ±1. (8.4)

For our solution, the metric coefficients in (8.2) and (8.3) are defined asfollows

a = ∆(u) = ∓Γ−1/2 (8.5)

− b = vz − uxy =1

c0c1c3

[(

γ − 1

µa

)

cos θ

]

t

− c3e−c1xa sin θ (8.6)

c = vy + uxz =1

c0c1c3

[(

γ − 1

µa

)

sin θ

]

t

+ c3e−c1xa cos θ (8.7)

Q =1

a

1

c20c21c

23

[(

γ − 1

µa

)

t

]2

+1

c20c23

(

γ − 1

µa

)2

− 2a

c0e−c1x

(

γ − 1

µa

)

+ c23a2e−2c1x + ε

. (8.8)

Here Γ is defined in (7.2), γ = ν/µ with µ, ν and σ given in (6.47), (6.51)and (7.5), respectively, θ = −c1t+ θ0.

Since σ and σt involve two different combinations of y and z and thereis obviously no reduction in either x or t, there is no symmetry reduction

18

of the metric (2.5) in the number of independent variables and hence thismetric does not admit any Killing vectors.1

Thus, we have obtained the Ricci-flat (anti-)self-dual metric withoutKilling vectors. It is generated by a non-invariant solution of the complexMonge-Ampere equation determined solely by its invariance under the pairof special first nonlocal symmetries of CMA without any additional as-sumptions. We see that such an invariance does not lead to a reductionin the number of independent variables in the solution, as opposed to theinvariance under Lie point symmetries. This explains our particular interestfor nonlocal symmetry flows in the hierarchies of bi-Hamiltonian systems ofMonge-Ampere type which we constructed recently [13].

9 Levi-Civita connection 1-forms

Here we restrict ourselves to the case of the Euclidean signature ε = +1important for gravitational instantons..

For Levi-Civita connection 1-forms ωab, Maurer-Cartan structure equa-tions in the absence of torsion imply

dea = ca0ie0 ∧ ei + ca23e

2 ∧ e3 + ca31e3 ∧ e1 + ca12e

1 ∧ e2 = −ωab ∧ eb

ωab = −ωba (9.1)

where ∧ denotes wedge (exterior) product and summation over the dummyindex i from 0 to 3 is assumed. By solving equations (9.1), one obtains theexpressions for the connection 1-forms in terms of the coefficients cabj(see [4], pp. 377, 378)

ω01 = −c001e

0 − c101e1 +

1

2(c012 − c102 − c201)e

2 − 1

2(c031 + c301 + c103)e

3

ω02 = −c002e

0 − 1

2(c012 + c102 + c201)e

1 − c202e2 +

1

2(c023 − c203 − c302)e

3

ω03 = −c003e

0 +1

2(c031 − c301 − c103)e

1 − 1

2(c023 + c203 + c302)e

2 − c303e3

ω23 =

1

2(c302 − c023 − c203)e

0 +1

2(c231 + c312 − c123)e

1 − c223e2 − c323e

3

ω31 =

1

2(c103 − c031 − c301)e

0 − c131e1 +

1

2(c312 + c123 − c231)e

2 − c331e3

ω12 =

1

2(c201 − c012 − c102)e

0 − c112e1 − c212e

2 +1

2(c123 + c231 − c312)e

3.

(9.2)

1An outline of the rigorous proof is given in the Appendix.

19

Coefficients cabj are determined by taking exterior derivatives of ea in (8.3).To exhibit the results in a compact form, we introduce the notation

Ψ = Q−1/2 and define

K = (bΨ)t + cΨx + (Ψ−1)z, L = (cΨ)t − bΨx − (Ψ−1)y

M = bΨt − cΨx − (Ψ−1)z, N = cΨt + bΨx + (Ψ−1)y.

(9.3)

Then the only nonzero coefficients have the form

c001 = Ψx, c002 = L, c003 = −K, c031 = (bΨ)x + bΨx

c012 = (cΨ)x + cΨx, c023 = −2Ψx, c101 = −Ψt, c102 = (bΨ)t + bΨt

c103 = (cΨ)t + cΨt, c131 = −K, c112 = −L, c123 = 2Ψt, c202 = Ψt

c212 = Ψx, c223 = −M, c303 = Ψt, c331 = −Ψx, c323 = −N. (9.4)

Using the relations inverse to (8.3)

dt = Ψ(e0 − ce2 + be3), dx = Ψ(e1 − be2 − ce3)

dy = Ψ−1e2, dz = Ψ−1e3 (9.5)

we obtain the final form of the solution (9.2) for connection 1-forms

ω23 = Ψxe

0 −Ψte1 +Me2 +Ne3 = −ω0

1

ω31 = Le0 +Ke1 +Ψte

2 +Ψxe3 = −ω0

2

ω12 = −Ke0 + Le1 −Ψxe

2 +Ψte3 = −ω0

3. (9.6)

Due to the relations between ωab in (9.6), the equations (9.1) imply

de0 = −(ω01 ∧ e1 + ω0

2 ∧ e2 + ω03 ∧ e3)

de1 = ω01 ∧ e0 + ω0

3 ∧ e2 − ω02 ∧ e3

de2 = ω02 ∧ e0 − ω0

3 ∧ e1 + ω01 ∧ e3

de3 = ω03 ∧ e0 + ω0

2 ∧ e1 − ω01 ∧ e2. (9.7)

10 Riemann curvature

Riemann curvature 2-forms are determined by the Cartan equations

Rab = dωa

b + ωac ∧ ωc

b where Rab =

1

2Ra

bcdec ∧ ed (10.1)

20

and Rabcd are the Riemann tensor components. Due to the relations between

connection 1-forms in (9.6), we obtain from (10.1)

R01 = −R2

3 = dω01 + 2ω0

2 ∧ ω03

R02 = R1

3 = dω02 + 2ω0

3 ∧ ω01

R03 = −R1

2 = dω03 + 2ω0

1 ∧ ω02. (10.2)

The symmetry Rab = −Rb

a implies that the formulas (10.2) determine inprinciple all nonzero curvature 2-forms. Utilizing explicit expressions (9.6)for connection forms we obtain the following expressions for the Riemann

21

tensor components

R0101 = ΨttΨ−Ψ2

t +ΨxxΨ−Ψ2x + 2(K2 + L2)

R0102 = Ψ−1Ψxy −Ψ(Ψtxc+Ψxxb+Mt) + 3(ΨtK −ΨxL)

R0103 = Ψ−1Ψxz +Ψ(Ψtxb−Ψxxc−Nt) + 3(ΨtL+ΨxK)

R0112 = −Ψ−1Ψty +Ψ(Ψtxb+Ψttc−Mx)− 3(ΨtL+ΨxK)

R0113 = −

[

Ψ−1Ψtz +Ψ(Ψttb−Ψtxc+Nx)− 3(ΨtK −ΨxL)]

R0123 = Ψ−1(Mz −Ny) + Ψ [b(Mt +Nx) + c(Nt −Mx)]

+M2 +N2 + 4(Ψ2t +Ψ2

x)

R0201 = Ψ(Lx −Kt) + 3(ΨtK −ΨxL)

R0202 = Ψ−1Ly −Ψ(cLt + bLx)− (K2 + L2 + 3KM)

−ΨttΨ−Ψ2t + 2Ψ2

x

R0203 = Ψ−1Lz +Ψ(bLt − cLx)− 3(ΨtΨx +KN)−ΨtxΨ

R0212 = Ψ−1Ky −Ψ(cKt + bKx)− 3(ΨtΨx − LM)−ΨtxΨ

R0213 = Ψ−1Kz +Ψ(bKt − cKx)−K2 − L2 + 3LN

−ΨxxΨ−Ψ2x + 2Ψ2

t

R0223 = Ψ−1(Ψtz −Ψxy) + bΨ(Ψtt +Ψxx) + 2(ΨxL−ΨtK)

−ΨtM −ΨxN. (10.3)

R0301 = −Ψ(Kx + Lt) + 3(ΨxK +ΨtL)

R0302 = −Ψ−1Ky +Ψ(cKt + bKx) + ΨΨtx + 3(ΨtΨx − LM)

R0303 = −Ψ−1Kz −Ψ(bKt − cKx)−K2 − L2 − 3LN

−ΨΨtt −Ψ2t + 2Ψ2

x

R0312 = Ψ−1Ly −Ψ(cLt + bLx) +K2 + L2 − 3KM

+ΨΨxx +Ψ2x − 2Ψ2

t

R0313 = Ψ−1Lz +Ψ(bLt − cLx)−ΨΨtx − 3(ΨtΨx +KN)

R0323 = −Ψ−1(Ψxz +Ψty) + cΨ(Ψtt +Ψxx)

−2(ΨxK +ΨtL) + ΨxM −ΨtN.

Here Ψ = Q−1/2 and the letter subscripts denote partial derivatives.We note that formulas (9.6) for connection forms and (10.3) for the

Riemann curvature are valid for any hyper-Kahler metric of the form (8.2),so that they can be applied for any solution of the complex Monge-Ampereequation in the two-component form (2.4).

For our solution, expressions (8.5), (8.6), (8.7), (8.8) for a, b, c, Q shouldbe used.

22

11 Conclusion

Our search for non-invariant solutions to the elliptic complex Monge-Ampereequation has been motivated by the fundamental problem of obtaining ex-plicitly the metric of the gravitational instanton K3, since it should notadmit any Killing vectors (continuous symmetries). Recently, we producedan example of such a metric, though not an instanton one, by combiningour previous approaches to the problem and choosing at random a veryparticular solution to resulting equations [9].

To the best of our knowledge, there are no more published results onnoninvariant solutions of CMA and corresponding metrics of the form (8.2)with no Killing vectors.

Here we have demonstrated that a general requirement of invariance un-der nonlocal symmetries of CMA yields solutions which are not invariantwith respect to any point symmetries and therefore no symmetry reductionresults in the number of independent variables. We have explicitly con-structed such a solution by a meticulous analysis of all integrability condi-tions of the invariance equations which made possible a complete integrationof these equations with no additional assumptions. Thus, we have obtainedthe most general form of the solution of CMA which is invariant underthe special first nonlocal symmetry in each hierarchy of CMA, not just asolution taken out by chance. We have also presented the correspondingASD Ricci-flat metric without Killing vectors. It has a rather complicatedform and further analysis is needed to study its other properties, such assingularities and asymptotic behavior.

Thus, at this stage we can by no means claim that our metric belongsto K3 instanton. However, we have demonstrated how invariance undernonlocal symmetries can be used in the attempts for constructing K3.

This method can serve as a direct approach for obtaining noninvariantsolutions of CMA from the invariance under other nonlocal flows in thehierarchy. We also point out that all our constructions and results havebeen obtained simultaneously for elliptic and hyperbolic CMA and so thecorresponding metrics have either Euclidean or neutral (ultra-hyperbolic)signature, respectively.

For the Euclidean signature, we have also explicitly calculated the Levi-Civita connection and Riemann curvature. These results are not restrictedto our solution only but more generally are applicable to any other solutionsof CMA. Further study of singularities and asymptotic behavior is neededto discover if our solution determines an instanton metric.

23

Appendix. Point symmetries and noninvariance

criterion

The full set of generators of point symmetries of CMA (8.1) has the form

X1 = z(t∂u + ∂v), X2 = (ty − xz)∂u + y∂v

X3 = t∂t + x∂x + u∂u, X4 = y∂y + z∂z + u∂u + v∂v

X5 = ∂t, X6 = ∂x, X7 = ∂y, X8 = ∂z

Xαβ = (α(t, x) + β(y, z))∂u + αt(t, x)∂v (A.1)

where α and β are arbitrary smooth solutions to the Laplace equations

αtt + αxx = 0, βyy + βzz = 0. (A.2)

The general symmetry generator is a linear combination of particular sym-metry generators (A.1)

X =8∑

k=1

ckXk +Xαβ

with constant coefficients ck. Invariance of the solution to CMA(

uv

)

=

(

F (t, x, y, z)Ft(t, x, y, z)

)

(A.3)

with respect to symmetry generator X has the general form

X

[(

FFt

)

−(

uv

)]∣

∣

∣

∣ u = Fv = Ft

=

(

00

)

. (A.4)

The explicit form of the equation in the first line of (A.4) reads

(c3t+ c5)Ft + (c3x+ c6)Fx + (c4y + c7)Fy + (c4z + c8)Fz

= (c3 + c4)F + c1tz + c2(ty − xz) + α+ β (A.5)

where we use for F the right-hand side of (1.5), whereas the equation in thesecond line of (A.4) is the time derivative of (A.5).

Applying criterion (A.4) to our solution with generic values of parameterswe find only trivial solution with all ck and α, β to be zero, hence our solutionis generically noninvariant which is enough for our purposes here. Nonzerosolutions for the constants may happen for special values of our solutionparameters which would mean that such special solutions are invariant. Thefull analysis of the situation is quite lengthy and complicated and deservesa separate study, which is clear from similar analysis of a much simplersolution of the Boyer-Finley equation in [14].

24

References

[1] J. F. Plebanski , Some solutions of complex Einstein equations, J. Math.

Phys. 16 (1975) 2395–2402.

[2] L.J. Mason and N. M. J. Woodhouse, Integrability, Self-Duality, andTwistor Theory, London Mathematical Society Monographs New Series,Vol. 15, Clarendon Press, Oxford, 1996.

[3] M. Dunajski, Solitons, Instantons, and Twistors, Oxford Graduate

Texts in Mathematics, Vol. 19, Oxford University Press, Oxford, 2010.

[4] T. Eguchi, P. B. Gilkey and A. J. Hanson, Gravitation, Gauge Theoriesand Differential Geometry, Phys. Rep. 66 No. 6 (1980), 213–393.

[5] M. F. Atiyah , N. J. Hitchin and I.M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. A 362 (1978), 425–461.

[6] N. J. Hitchin, Compact four-dimensional Einstein manifolds, J. Diff.

Geom. 9 (1974), 435–441.

[7] S.-T. Yau, Calabi’s conjecture and some new results in algebraic geom-etry, Proc. Natl. Acad. Sci. USA 74 (1977), 1798–1799.

[8] P. J. Olver, Applications of Lie Groups to Differential Equations, NewYork: Springer-Verlag (1986).

[9] Mikhail B. Sheftel and Andrei A. Malykh, Partner symmetries, groupfoliation and ASD Ricci-flat metrics without Killing vectors, SIGMA 9

(2013) 075.

[10] Y. Nutku, M. B. Sheftel, J. Kalayci, D. Yazıcı, Self-dual gravity is com-pletely integrable, J. Phys. A: Math. Theor. 41 (2008) 395206 (13pp);arXiv:0802.2203v4 [math-ph] (2008).

[11] F. Magri, A simple model of the integrable Hamiltonian equation, J.Math. Phys. 19 (1978) 1156–1162. ;F. Magri, A geometrical approach to the nonlinear solvable equations,in: Nonlinear Evolution Equations and Dynamical Systems (M. Boiti,F. Pempinelli, and G.Soliani, Eds.), Lecture Notes in Phys., 120,Springer: New York (1980) pp. 233–263.

25

[12] H. Stephani, Differential equations: their solution using symmetries,Cambridge: Cambridge University Press (1989).

[13] M. B. Sheftel and D. Yazıcı, Symmetries, integrals and hierarchies ofnew (3+1)-dimensional bi-Hamiltonian systems of Monge-Ampere type,J. Geom. Phys. Vol. 146 (2019) 103513 (19 pp.); arXiv:1904.11174b[math-ph].

[14] L. Martina, M. B. Sheftel and P. Winternitz, Group foliation and non-invariant solutions of the heavenly equation, J. Phys. A: Math. Gen 34

(2001) 9243–9263.

26