5.2. Case 2: = 0, »’1 = 0. The complete Lie algebra of symmetries

for the Dirac equations with nonvanishing rest mass is spanned by four-

teen generators, including ten in¬nitesimal generators of the Poincar´ group

e

X1 , . . . , X10 and the generators X19 , X20 , X23 , X16 . There is also a contin-

1 4

uous part generated by the functions F u , . . . , F v dependent on x1 , . . . , x4 ,

which satisfy Dirac equations (1.83) with nonvanishing rest mass.

5.3. Case 3: = 0, »’1 = 0. The complete Lie algebra in this situation

is spanned by fourteen generators. These generators are X1 , . . . , X10 , X19 ,

X20 , X23 , and X11 ’ X16 /2.

5.4. Case 4: = 0, »’1 = 0. The complete Lie algebra of symmetries

for the nonlinear Dirac equations with nonvanishing rest mass is spanned

by thirteen generators. The generators in this case are the ten in¬nitesimal

generators of the Poincar´ group, X1 , . . . , X10 , and X19 , X20 , X23 . This

e

result generalizes the result by Steeb [94] where X20 was found as additional

symmetry to the generators of the Poincar´ group.

e

6. Symmetries of the self-dual SU (2) Yang“Mills equations

We study here classical symmetries of the self-dual SU (2) Yang“Mills

equations. Two cases are considered: the general one and of the so-called

static gauge ¬elds. In the ¬rst case we obtain two instanton solutions (the

Belavin“Polyakov“Schwartz“Tyupkin [6] and ™t Hooft instantons [84]) as

invariant solutions for a special choice of symmetry subalgebras. In a similar

way, for the second case we derive a monopole solution [83].

We start with a concise description of the SU (2)-gauge theory referring

the reader to the survey paper by M. K. Prasad [83] for a more extensive

exposition.

6.1. Self-dual SU (2) Yang“Mills equations. Let M be a 4-dimen-

sional Euclidean space with the coordinates x1 , . . . , x4 . Due to nondegen-

erate metric in M , we make no distinction between contravariant and co-

variant indices, xµ = xµ . The basic object in the gauge theory is the Yang“

Mills gauge potential. The gauge potential is a set of ¬elds Aa ∈ C ∞ (M ),

µ

a = 1, . . . , 3, µ = 1, . . . , 4. It is convenient to introduce a matrix-valued

vector ¬eld Aµ (x), by setting

σa

a

Aa , a

Aµ = gT T= , a = 1, . . . , 3, µ = 1, . . . , 4, (1.93)

µ

2i

where σ a are the Pauli matrices

0 ’i

01 10

σ1 = σ2 = σ3 =

, , , (1.94)

0 ’1

10 i0

g being a constant, called the gauge coupling constant. Throughout this

section we shall use the Einstein summation convention when an index oc-

curs twice. From the matrix gauge potential Aµ dxµ one constructs the

44 1. CLASSICAL SYMMETRIES

matrix-valued ¬eld strength Fµν (x) by

‚ ‚

Aν ’

Fµν = Aµ + [Aµ , Aν ], µ, ν = 1, . . . , 4, (1.95)

‚xµ ‚xν

where [Aµ , Aν ] = Aµ Aν ’ Aν Aµ . If one de¬nes the covariant derivative

‚

Dµ = + Aµ , (1.96)

‚xµ

then (1.95) is rewritten as

Fµν = [Dµ , Dν ]. (1.97)

In explicit component form, one has

Fµν = gT a Fµν ,

a

(1.98)

where

‚a ‚a

a bc

Aν ’

Fµν = A +g abc Aµ Aν (1.99)

‚xν µ

‚xµ

and

±

+1 if abc is an even permutation of (1,2,3),

= ’1 if abc is an odd permutation of (1,2,3), (1.100)

abc

0 otherwise.

We shall use the expression static gauge ¬eld to refer to gauge potentials

that are independent of x4 (x4 to be considered as time), i.e.,

‚

Aµ (x) = 0, µ = 1, . . . , 4. (1.101)

‚x4

For gauge potentials that depend on all four coordinates x1 , . . . , x4 , the

action functional is de¬ned by

1

Fµν Fµν d4 x,

a a

S= (1.102)

4

the integral taken over R4 , while for static gauge ¬elds we de¬ne the energy

functional by

1

Fµν Fµν d3 x,

a a

E= (1.103)

4

whereas in (1.103) the integral is taken over R3 .

The extremals of the action S (or of the energy E for static gauge ¬elds)

are found by standard calculus of variations techniques leading to the Euler“

Lagrange equations

‚

Fµν + [Aµ , Fµν ] ≡ [Dµ , Fµν ] = 0, (1.104)

‚xµ

or in components

‚a

Fµν + g abc Ab Fµν = 0.

c

(1.105)

µ

‚xµ

6. SYMMETRIES OF THE SELF-DUAL SU (2) YANG“MILLS EQUATIONS 45

Equations (1.105) is a system of second order nonlinear partial dif-

ferential equations for the twelve unknown functions Aa , a = 1, . . . , 3,

µ

µ = 1, . . . , 4, that seems hard to solve.

Then one introduces the dual gauge ¬eld strength — Fµν as

1

—

Fµν = µν»ρ F»ρ , (1.106)

2

where µν»ρ is the completely antisymmetric tensor on M de¬ned by

±

+1 if µν»ρ is an even permutation of (1,2,3,4),

µν»ρ = (1.107)

’1 if µν»ρ is an odd permutation of (1,2,3,4),

0 otherwise.

Since the ¬elds Dµ (1.96) satisfy the Jacobi identity

[D» , [Dµ , Dν ]] + [Dµ , [Dν , D» ]] + [Dν , [D» , Dµ ]] = 0, (1.108)

multiplication of (1.108) by and summation result in

µν»ρ

[Dµ , — Fµν ] = 0. (1.109)

If we compare (1.104) with (1.109), we see that any gauge ¬eld which is

self-dual , i.e., for which

—

Fµν = Fµν , (1.110)

automatically satis¬es (1.101). Consequently, the only equations to solve are

(1.110) with — Fµν given by (1.106). This is a system of ¬rst order nonlinear

partial di¬erential equations.

Instanton solutions for general Yang“Mills equations and monopole so-

lutions for static gauge ¬elds satisfy (1.110) under the condition that S

(1.102) or E (1.103) are ¬nite.

Written in components, (1.110) takes the form

F13 = ’F24 ,

F12 = F34 , F14 = F23 . (1.111)

So in components, the self-dual Yang“Mills equations are described as a

system of nine nonlinear partial di¬erential equations,

’A1 + A1 ’ A1 + A1 ’ g(A2 A3 ’ A2 A3 + A2 A3 ’ A2 A3 ) = 0,

4,1 3,2 2,3 1,4 14 23 32 41

’A2 + A2 ’ A2 + A2 + g(A1 A3 ’ A1 A3 + A1 A3 ’ A1 A3 ) = 0,

4,1 3,2 2,3 1,4 14 23 32 41

’A3 + A3 ’ A3 + A3 ’ g(A1 A2 ’ A1 A2 + A1 A2 ’ A1 A2 ) = 0,

4,1 3,2 2,3 1,4 14 23 33 41

A1 + A1 ’ A1 ’ A1 + g(A2 A3 + A2 A3 ’ A2 A3 ’ A2 A3 ) = 0,

3,1 4,2 1,3 2,4 13 24 31 42

A2 + A2 ’ A2 ’ A2 ’ g(A1 A3 ’ A1 A3 ’ A1 A3 ’ A1 A3 ) = 0,

3,1 4,2 1,3 2,4 13 24 31 42

A3 + A3 ’ A3 ’ A3 + g(A1 A2 + A1 A2 ’ A1 A2 ’ A1 A2 ) = 0,

3,1 4,2 1,3 2,4 13 24 31 42

A1 ’ A1 ’ A1 + A1 + g(A2 A3 ’ A2 A3 ’ A2 A3 + A2 A3 ) = 0,

2,1 1,2 4,3 3,4 12 21 34 43

A2 ’ A2 ’ A2 + A2 ’ g(A1 A3 ’ A1 A3 ’ A1 A3 + A1 A3 ) = 0,

2,1 1,2 4,3 3,4 12 21 34 43

A3 ’ A3 ’ A3 + A3 + g(A1 A2 ’ A1 A2 ’ A1 A2 + A1 A2 ) = 0,

2,1 1,2 4,3 3,4 12 21 34 43

(1.112)

46 1. CLASSICAL SYMMETRIES

whereas in (1.112)

‚a

Aa = A, a = 1, . . . , 3, µ, ν = 1, . . . , 4. (1.113)

µ,ν

‚xν µ

Thus, we obtain a system E ‚ J 1 (π) for π : R12 — R4 ’ R4 .

6.2. Classical symmetries of self-dual Yang“Mills equations. In

order to construct the Lie algebra of classical symmetries of (1.112), we start

at a vector ¬eld V given by

‚ ‚ 1‚ 3‚

V = V x1 + · · · + V x4 + V A1 + · · · + V A4 . (1.114)

‚A1 ‚A3

‚x1 ‚x4 1 4

The condition for V to be a symmetry of equations (1.112) now leads to an

overdetermined system of partial di¬erential equations for the components

1 3

V x1 , . . . , V x4 , V A1 , . . . , V A4 , which are functions dependent of the variables

x1 , . . . , x4 , A 1 , . . . , A 3 .

1 4

The general solution of this overdetermined system of partial di¬erential

equations constitutes a Lie algebra of symmetries, generated by the vector

¬elds

1‚ 1‚ 1‚ 1‚

1

V1f = fx1 + f x2 + f x3 + f x4

‚A1 ‚A1 ‚A1 ‚A1

1 2 3 4

‚ ‚ ‚ ‚

+ f 1 gA3 2 + f 1 gA3 2 + f 1 gA3 2 + f 1 gA3 2

1 2 3 4

‚A1 ‚A2 ‚A3 ‚A4

‚ ‚ ‚ ‚

’ f 1 gA2 3 ’ f 1 gA2 3 ’ f 1 gA2 3 ’ f 1 gA2 3 ,

1 2 3 4

‚A1 ‚A2 ‚A3 ‚A4

‚ ‚ ‚ ‚

2

V2f = ’f 2 gA2 1 ’ f 2 gA2 1 ’ f 2 gA2 1 ’ f 2 gA2 1

1 2 3 4

‚A1 ‚A2 ‚A3 ‚A4

‚ ‚ ‚ ‚

+ f 2 gA1 2 + f 2 gA1 2 + f 2 gA1 2 + f 2 gA1 2

1 2 3 4

‚A1 ‚A2 ‚A3 ‚A4

2‚ 2‚ 2‚ 2‚

’ f x1 ’ f x2 ’ f x3 ’ f x4 ,

‚A3 ‚A3 ‚A3 ‚A3

1 2 3 4

‚ ‚ ‚ ‚

3

V3f = f 3 gA3 1 + f 3 gA3 1 + f 3 gA3 1 + f 3 gA3 1

1 2 3 4

‚A1 ‚A2 ‚A3 ‚A4

3‚ 3‚ 3‚ 3‚

’ f x1 ’ f x2 ’ f x3 ’ f x4

‚A2 ‚A2 ‚A2 ‚A2

1 2 3 4

‚ ‚ ‚ ‚

’ f 3 gA1 3 ’ f 3 gA1 3 ’ f 3 gA1 3 ’ f 3 gA1 3 ,

1 2 3 4

‚A1 ‚A2 ‚A3 ‚A4

‚ ‚ ‚ ‚

V4 = , V5 = , V6 = , V7 = ,

‚x1 ‚x2 ‚x3 ‚x4

‚ ‚ ‚ ‚

+ A1 1 ’ A1 1

’ x1

V8 = x 2 2 1

‚x1 ‚x2 ‚A1 ‚A2

‚ ‚ ‚ ‚

+ A2 2 ’ A2 2 + A3 3 ’ A3 3 ,

2 1 2 1

‚A1 ‚A2 ‚A1 ‚A2

6. SYMMETRIES OF THE SELF-DUAL SU (2) YANG“MILLS EQUATIONS 47

‚ ‚ ‚ ‚

’ A1 1 + A1 1

V9 = ’x3 + x1 3 1

‚x1 ‚x3 ‚A1 ‚A3

‚ ‚ ‚ ‚

’ A2 2 + A2 2 ’ A3 3 + A3 3 ,

3 1 3 1

‚A1 ‚A3 ‚A1 ‚A3

‚ ‚ ‚ ‚

’ A1 1 + A1 1

V10 = ’x4 + x1 4 1

‚x1 ‚x4 ‚A1 ‚A4

‚ ‚ ‚ ‚

’ A2 2 + A2 2 ’ A3 3 + A3 3 ,

4 1 4 1

‚A1 ‚A4 ‚A1 ‚A4

‚ ‚ ‚ ‚

’ A1 1 + A1 1

V11 = ’x3 + x2 3 2

‚x2 ‚x3 ‚A2 ‚A3

‚ ‚ ‚ ‚

’ A2 2 + A2 2 ’ A3 3 + A3 3 ,

3 2 3 2

‚A2 ‚A3 ‚A2 ‚A3

‚ ‚ ‚ ‚

+ A1 1 ’ A1 1

’ x2

V12 = x4 4 2

‚x2 ‚x4 ‚A2 ‚A4

‚ ‚ ‚ ‚

+ A2 2 ’ A2 2 + A3 3 ’ A3 3 ,

4 2 4 2

‚A2 ‚A4 ‚A2 ‚A4

‚ ‚ ‚ ‚

’ A1 1 + A1 1

V13 = ’x4 + x3 4 3

‚x3 ‚x4 ‚A3 ‚A4

‚ ‚ ‚ ‚

’ A2 2 + A2 2 ’ A3 3 + A3 3 ,

4 3 4 3

‚A3 ‚A4 ‚A3 ‚A4

‚ ‚ ‚ ‚

V14 = x1 + x2 + x3 + x4

‚x1 ‚x2 ‚x3 ‚x4

‚ ‚ ‚ ‚

’ A1 1 ’ A1 1 ’ A1 1 ’ A1 1

1 2 3 4

‚A1 ‚A2 ‚A3 ‚A4

‚ ‚ ‚ ‚

’ A2 2 ’ A2 2 ’ A2 2 ’ A2 2

1 2 3 4

‚A1 ‚A2 ‚A3 ‚A4

‚ ‚ ‚ ‚

’ A3 3 ’ A3 3 ’ A3 3 ’ A3 3 ,

1 2 3 4

‚A1 ‚A2 ‚A3 ‚A4

‚ ‚ ‚ ‚

V15 = (’x2 + x2 + x2 + x2 ) ’ 2x1 x2 ’ 2x1 x3 ’ 2x1 x4

1 2 3 4

‚x1 ‚x2 ‚x3 ‚x4

‚ ‚

+ 2(x1 A1 + x2 A1 + x3 A1 + x4 A1 ) 1 + 2(x1 A1 ’ x2 A1 ) 1

1 2 3 4 2 1

‚A1 ‚A2

‚ ‚

+ 2(x1 A1 ’ x3 A1 ) 1 + 2(x1 A1 ’ x4 A1 ) 1

3 1 4 1

‚A3 ‚A4

‚ ‚

+ 2(x1 A2 + x2 A2 + x3 A2 + x4 A2 ) 2 + 2(x1 A2 ’ x2 A2 ) 2

1 2 3 4 2 1

‚A1 ‚A2

‚ ‚

+ 2(x1 A2 ’ x3 A2 ) 2 + 2(x1 A2 ’ x4 A2 ) 2

3 1 4 1

‚A3 ‚A4

48 1. CLASSICAL SYMMETRIES

‚ ‚

+ 2(x1 A3 + x2 A3 + x3 A3 + x4 A3 ) + 2(x1 A3 ’ x2 A3 ) 3

1 2 3 4 2 1

‚A3 ‚A2

1

‚ ‚

+ 2(x1 A3 ’ x3 A3 ) 3 + 2(x1 A3 ’ x4 A3 ) 3 ,

3 1 4 1

‚A3 ‚A4

‚ ‚ ‚ ‚

+ (x2 ’ x2 + x2 + x2 )

V16 = ’2x2 x1 ’ 2x2 x3 ’ 2x2 x4

1 2 3 4

‚x1 ‚x2 ‚x3 ‚x4

‚ ‚

+ 2(’x1 A1 + x2 A1 ) 1 + 2(x1 A1 + x2 A1 + x3 A1 + x4 A1 ) 1

2 1 1 2 3 4

‚A1 ‚A2

‚ ‚

+ 2(x2 A1 ’ x3 A1 ) 1 + 2(x2 A1 ’ x4 A1 ) 1

3 2 4 2

‚A3 ‚A4

‚ ‚

+ 2(’x1 A2 + x2 A2 ) 2 + 2(x1 A2 + x2 A2 + x3 A2 + x4 A2 ) 2

2 1 1 2 3 4

‚A1 ‚A2

‚ ‚

+ 2(x2 A2 ’ x3 A2 ) 2 + 2(x2 A2 ’ x4 A2 ) 2

3 2 4 2

‚A3 ‚A4

‚ ‚

+ 2(’x1 A3 + x2 A3 ) 3 + 2(x1 A3 + x2 A3 + x3 A3 + x4 A3 ) 3

2 1 1 2 3 4

‚A1 ‚A2

‚ ‚

+ 2(x2 A3 ’ x3 A3 ) 3 + 2(x2 A3 ’ x4 A3 ) 3 ,

3 2 4 2

‚A3 ‚A4

‚ ‚ ‚ ‚

+ (x2 + x2 ’ x2 + x2 )

V17 = ’2x3 x1 ’ 2x3 x2 ’ 2x3 x4

1 2 3 4

‚x1 ‚x2 ‚x3 ‚x4

‚ ‚

+ 2(’x1 A1 + x3 A1 ) 1 + 2(’x2 A1 + x3 A1 ) 1

3 1 3 2

‚A1 ‚A2

‚ ‚

+ 2(x1 A1 + x2 A1 + x3 A1 + x4 A1 ) 1 + 2(x3 A1 ’ x4 A1 ) 1

1 2 3 4 4 3

‚A3 ‚A4

‚ ‚

+ 2(’x1 A2 + x3 A2 ) 2 + 2(’x2 A2 + x3 A2 ) 2

3 1 3 2

‚A1 ‚A2

‚ ‚

+ 2(x1 A2 + x2 A2 + x3 A2 + x4 A2 ) 2 + 2(x3 A2 ’ x4 A2 ) 2

1 2 3 4 4 3

‚A3 ‚A4

‚ ‚

+ 2(’x1 A3 + x3 A3 ) 3 + 2(’x2 A3 + x3 A3 ) 3

3 1 3 2

‚A1 ‚A2

‚ ‚

+ 2(x1 A3 + x2 A3 + x3 A3 + x4 A3 ) 3 + 2(x3 A3 ’ x4 A3 ) 3 ,

1 2 3 4 4 3

‚A3 ‚A4

‚ ‚ ‚ ‚

+ (x2 + x2 + x2 ’ x2 )

V18 = ’2x4 x1 ’ 2x4 x2 ’ 2x4 x3 1 2 3 4

‚x1 ‚x2 ‚x3 ‚x4

‚ ‚

+ 2(’x1 A1 + x4 A1 ) 1 + 2(’x2 A1 + x4 A1 ) 1

4 1 4 2

‚A1 ‚A2

‚ ‚

+ 2(’x3 A1 + x4 A1 ) 1 + 2(x1 A1 + x2 A1 + x3 A1 + x4 A1 ) 1

4 3 1 2 3 4

‚A3 ‚A4

6. SYMMETRIES OF THE SELF-DUAL SU (2) YANG“MILLS EQUATIONS 49

‚ ‚

+ 2(’x1 A2 + x4 A2 ) + 2(’x2 A2 + x4 A2 )

4 1 4 2

‚A2 ‚A2

1 2

‚ ‚

+ 2(’x3 A2 + x4 A2 ) 2 + 2(x1 A2 + x2 A2 + x3 A2 + x4 A2 )

4 3 1 2 3 4

‚A2

‚A3 4

‚ ‚

+ 2(’x1 A3 + x4 A3 ) 3 + 2(’x2 A3 + x4 A3 )

4 1 4 2

‚A3

‚A1 2

‚ ‚

+ 2(’x3 A3 + x4 A3 ) 3 + 2(x1 A3 + x2 A3 + x3 A3 + x4 A3 ) .

4 3 1 2 3 4

‚A3

‚A3 4

(1.115)

The functions F 1 , F 2 , F 3 in the symmetries V1 , V2 , V3 are arbitrary,

depending on the variables x1 , x2 , x3 , x4 . The vector ¬elds V1 , V2 , V3 are

just the generators of the gauge transformations.

The vector ¬elds V4 , V5 , V6 , V7 are generators of translations while the

¬elds V8 , . . . , V13 refer to in¬nitesimal rotations in R4 , X4 , . . . , X18 being the

in¬nitesimal generators of the conformal group.

6.3. Instanton solutions. In order to construct invariant solutions

associated to symmetries of the self-dual Yang“Mills equations (1.112), we

start from the vector ¬elds X1 , X2 , X3 de¬ned by

1 2 3

f1 f1 f1

X1 = V 8 + V1 + V2 + V3 ,

f1 f2 f3

X2 = V 9 + V 1 2 + V 2 2 + V 3 2 ,

f1 f2 f3

X3 = V10 + V1 3 + V2 3 + V3 3 , (1.116)

i.e., we take a combination of a rotation and a special choice for the

gauge transformations choosing particular values fij of arbitrary func-

tions f j . We also construct commutators of the vector ¬elds X1 , X2 , X3 ,

[X1 , X2 ], [X1 , X3 ], [X2 , X3 ] (1.117)

and make the following choice for the gauge transformations

1 2 3

f1 = ’1,

f1 = 0, f1 = 0,

1 2 3

f2 = ’1,

f2 = 0, f2 = 0,

1 2 3

f3 = ’1, f3 = 0, f3 = 0. (1.118)

In order to derive invariant solutions (see equations (1.40) on p. 28), we

impose the additional conditions. Namely, we compute generating functions

(•i )j = Yi ωA j , j = 1, . . . , 3, µ = 1, . . . , 4, (1.119)

µ µ

whereas in (1.119) ωAj is the contact form associated to Aj , i.e.,

µ

µ

ωAj = dAj ’ Aj dxν ,

µ µ,ν

µ

50 1. CLASSICAL SYMMETRIES

while Yi refers to the ¬elds X1 , X2 , X3 , [X1 , X2 ], [X1 , X3 ], [X2 , X3 ]. Then

we impose additional equations

j

(x1 , . . . , x4 , . . . , Aj , . . . , Aj , . . . )

•i =0 (1.120)

µ µν

µ

and solve them together with the initial system. From conditions (1.119)

we arrive at a system of 6 — 12 = 72 equations.

The resulting system can be solved in a straightforward way, leading to

the following intermediate presentation

A1 = x4 F (r), A1 = x3 F (r), A1 = ’x2 F (r), A1 = ’x1 F (r),

1 2 3 4

A2 = ’x3 F (r), A2 = x4 F (r), A2 = x1 F (r), A2 = ’x2 F (r),

1 2 3 4

A3 = x2 F (r), A3 = ’x1 F (r), A3 = x4 F (r), A3 = ’x3 F (r),

1 2 3 4

(1.121)

where

1

r = (x2 + x2 + x2 + x2 ) 2 . (1.122)

1 2 3 4

When obtaining the monopole solution (see below), we shall discuss in some

more detail how to solve a system of partial di¬erential equations like (1.120).

Substitution of (1.122) in (1.95) yields an ordinary di¬erential equation for

the function F (r), i.e.,

dF (r)

+ grF (r)2 = 0, (1.123)

dr

the solution of which is given by

2g ’1

F (r) = 2 , (1.124)

r +C

C being a constant. The result (1.124) is just the Belavin“Polyakov“

Schwartz“Tyupkin instanton solution!

More general, if we choose

2 3 1

f1 = ±1, f2 = ±1, f3 = ±1, (1.125)

and

231

f1 f2 f3 = ’1, (1.126)

or equivalently

1 23

f3 = ’f1 f2 , (1.127)

we arrive at

A1 = x4 F (r), A1 = x3 F (r), A1 = ’x2 F (r), A1 = ’x1 F (r),

1 2 3 4

A2 = x3 F (r)f1 ,

2

A2 = ’x4 F (r)f1 , A2 = ’x1 F (r)f1 , A2 = ’x2 F (r)f1 ,

2 2 2

1 2 3 4

A3 = ’x2 F (r)f3 , A3 = x1 F (r)f3 ,

2 2

A3 = ’x4 F (r)f3 , A3 = ’x3 F (r)f3 ,

2 2

1 2 3 4

(1.128)

6. SYMMETRIES OF THE SELF-DUAL SU (2) YANG“MILLS EQUATIONS 51

while (1.128) with F (r) has to satisfy (1.112), which results in

‚F (r)

+ grf1 f3 F (r)2 = 0.

22

(1.129)

‚r

2 3 1

Choosing f1 , f2 , f3 as in (1.125) but with

231

f1 f2 f3 = +1, (1.130)

then the result is

A1 = x4 F (r), A1 = ’ x3 F (r), A1 = x2 F (r), A1 = ’ x1 F (r),

1 2 3 4

A2 = ’x3 F (r)f1 , A2 = ’ x4 F (r)f1 , A2 = x1 F (r)f1 ,

2 2 2

A2 =x2 F (r)f1 ,

2

1 2 3 4

A3 = x2 F (r)f3 ,

2

A3 = ’ x1 F (r)f3 , A3 = ’x4 F (r)f3 , A3 =x3 F (r)f3 ,

2 2 2

1 2 3 4

(1.131)

while F (r) has to satisfy

dF (r)

+ 4F (r) + gr 2 f1 f2 F (r)2 = 0.

23

r (1.132)

dr

The solution of (1.132)

a2

2 2 3 ’1

F (r) = ’ (f1 f2 ) (1.133)

(r2 + a2 )r2

g

together with (1.131) is just the ™t Hooft instanton solution with instanton

number k = 1. This solution can be obtained from (1.124) by a gauge

transformation.

6.4. Classical symmetries for static gauge ¬elds. The equations

for the static SU (2) gauge ¬eld are described by (1.109) and (1.101). The

symmetries for the static gauge ¬eld are obtained from those for the time-

dependent case or straightforwardly in the following way . The respective

computations then results in the following Lie algebra of symmetries for the

static self-dual SU (2) Yang“Mills equations

‚ ‚ ‚

1

V1C = Cx1

1 1 1

+ C x2 + C x3

‚A1 ‚A1 ‚A1

1 2 3

‚ ‚ ‚ ‚

+ C 1 gA3 2 + C 1 gA3 2 + C 1 gA3 2 + C 1 gA3 2

1 2 3 4

‚A1 ‚A2 ‚A3 ‚A4

‚ ‚ ‚ ‚

’ C 1 gA2 3 ’ C 1 gA2 3 ’ C 1 gA2 3 ’ C 1 gA2 3 ,

1 2 3 4

‚A1 ‚A2 ‚A3 ‚A4

‚ ‚ ‚ ‚

2

V2C = ’C 2 gA2 1 ’ C 2 gA2 1 ’ C 2 gA2 1 ’ C 2 gA2 1

1 2 3 4

‚A1 ‚A2 ‚A3 ‚A4

‚ ‚ ‚ ‚

+ C 2 gA1 2 + C 2 gA1 2 + C 2 gA1 2 + C 2 gA1 2

1 2 3 4

‚A1 ‚A2 ‚A3 ‚A4

‚ ‚ ‚

2 2 2

’ C x1 ’ C x2 ’ C x3 ,

‚A3 ‚A3 ‚A3

1 2 3

52 1. CLASSICAL SYMMETRIES

‚ ‚ ‚ ‚

3

V3C = C 3 gA3 + C 3 gA3 1 + C 3 gA3 1 + C 3 gA3 1

1 2 3 4

‚A1 ‚A2 ‚A3 ‚A4

1

‚ ‚ ‚

3 3 3

’ C x1 ’ C x2 ’ C x3

‚A2 ‚A2 ‚A2

1 2 3

‚ ‚ ‚ ‚

’ C 3 gA1 3 ’ C 3 gA1 3 ’ C 3 gA1 3 ’ C 3 gA1 3 ,

1 2 3 4

‚A1 ‚A2 ‚A3 ‚A4

‚ ‚ ‚

V4 = , V5 = , V6 = ,

‚x1 ‚x2 ‚x3

‚ ‚ ‚ ‚

+ A1 1 ’ A1 1

’ x1

V7 = x 2 2 1

‚x1 ‚x2 ‚A1 ‚A2

‚ ‚ ‚ ‚

+ A2 2 ’ A2 2 + A3 3 ’ A3 3 ,

2 1 2 1

‚A1 ‚A2 ‚A1 ‚A2

‚ ‚ ‚ ‚

+ A1 1 ’ A1 1

V8 = ’x3 + x1 1 3

‚x1 ‚x3 ‚A3 ‚A1

‚ ‚ ‚ ‚

’ A2 2 + A2 2 ’ A3 3 + A3 3 ,

3 1 3 1

‚A1 ‚A3 ‚A1 ‚A3

‚ ‚ ‚ ‚

’ A1 1 + A1 1

V9 = ’x3 + x2 3 2

‚x2 ‚x3 ‚A2 ‚A3

‚ ‚ ‚ ‚

’ A2 2 + A2 2 ’ A3 3 + A3 3 ,

3 2 3 2

‚A2 ‚A3 ‚A2 ‚A3

‚ ‚ ‚ ‚

V10 = x1 + x2 + x3 + x4

‚x1 ‚x2 ‚x3 ‚x4

‚ ‚ ‚ ‚

’ A1 1 ’ A1 1 ’ A1 1 ’ A1 1

1 2 3 4

‚A1 ‚A2 ‚A3 ‚A4

‚ ‚ ‚

’ A2 2 ’ A2 2 ’ A2 2

1 2 3

‚A1 ‚A2 ‚A3

‚ ‚ ‚ ‚ ‚

’ A2 2 ’ A3 3 ’ A3 3 ’ A3 3 ’ A3 3 . (1.134)

4 1 2 3 4

‚A4 ‚A1 ‚A2 ‚A3 ‚A4

In (1.134) C 1 , C 2 , C 3 are arbitrary functions of x1 , . . . , x3 , while V1 ,

V2 , V3 themselves are just the generators of the gauge transformations. The

¬elds V7 , V8 , V9 generate rotations, while V10 is the generator of the scale

change of variables.

6.5. Monopole solution. In order to construct invariant solutions to

the static SU (2) gauge ¬eld, we proceed in a way analogously to the one for

the time-dependent ¬eld setting. We de¬ne the vector ¬elds Y1 , Y2 , Y3 by

Y1 = V7 ’ V21 ,

Y2 = V8 ’ V31 , (1.135)

Y3 = V9 ’ V11 ,

6. SYMMETRIES OF THE SELF-DUAL SU (2) YANG“MILLS EQUATIONS 53

i.e., put C 1 , C 2 , and C 3 equal to g ’1 . It results in 36 equations for the

functions Aa :

µ

A2 + A1 ’ x2 A1 + x1 A1 = 0, nonumber

1: (1.136)

1 2 1,1 1,2

’A1 + A2 ’ x2 A2 + x1 A2 = 0,

2: 1 2 1,1 1,2

A3 ’ x2 A3 + x1 A3 = 0,

3: 2 1,1 1,2

’A1 + A2 ’ x2 A1 + x1 A1 = 0,

4: 1 2 2,1 2,2

’A2 ’ A1 ’ x2 A2 + x1 A2 = 0,

5: 1 2 2,1 2,2

’A3 ’ x2 A3 + x1 A3 = 0,

6: 1 2,1 2,2

A2 ’ x2 A1 + x1 A1 = 0,

7: 3 3,1 3,2

’A1 ’ x2 A2 + x1 A2 = 0,

8: 3 3,1 3,2

’x2 A3 + x1 A3 = 0,

9: 3,1 3,2

A2 ’ x2 A1 + x1 A1 = 0,

10 : 4 4,1 4,2

’A1 ’ x2 A2 + x1 A2 = 0,

11 : 4 4,1 4,2

’x2 A3 + x1 A3 = 0,

12 : (1.137)

4,1 4,2

’A3 ’ A1 ’ x1 A1 + x3 A1 = 0,

13 : 1 3 1,3 1,1

’A2 ’ x1 A2 + x3 A2 = 0,

14 : 3 1,3 1,1

A1 ’ A3 ’ x1 A3 + x3 A3 = 0,

15 : 1 3 1,3 1,1

’A3 ’ x1 A1 + x3 A1 = 0,

16 : 2 2,3 2,1

’x1 A2 + x3 A2 = 0,

17 : 2,3 2,1

A1 ’ A3 ’ x1 A3 + x3 A3 = 0,

18 : 2 3 2,3 2,1

A1 ’ x1 A1 + x3 A1 = 0,

19 : 1 3,3 3,1

A2 ’ x1 A2 + x3 A2 = 0,

20 : 1 3,3 3,1

A3 + A1 ’ x1 A3 + x3 A3 = 0,

21 : 1 3 3,3 3,1

’A3 ’ x1 A1 + x3 A1 = 0,

22 : (1.138)

4 4,3 4,1

’x1 A2 + x3 A2 = 0,

23 : 4,3 4,1

A1 ’ x1 A3 + x3 A3 = 0,

24 : (1.139)

4 4,3 4,1

’x2 A1 + x3 A1 = 0,

25 : 1,3 1,2

’A3 ’ x2 A2 + x3 A2 = 0,

26 : 1 1,3 1,2

A2 ’ x2 A3 + x3 A3 = 0,

27 : 1 1,3 1,2

’A1 ’ x2 A1 + x3 A1 = 0,

28 : 3 2,3 2,2

’A3 ’ A2 ’ x2 A2 + x3 A2 = 0,

29 : 2 3 2,3 2,2

A2 ’ A3 ’ x2 A3 + x3 A3 = 0,

30 : 2 3 2,3 2,2

54 1. CLASSICAL SYMMETRIES

A1 ’ x2 A1 + x3 A1 = 0,

31 : 2 3,3 3,2

A2 ’ A3 ’ x2 A2 + x3 A2 = 0.

32 : 2 3 3,3 3,2

A3 + A2 ’ x2 A3 + x3 A3 = 0.

33 : 2 3 3,3 3,2

’x2 A1 + x3 A1 = 0.

34 : 4,3 4,2

’A3 ’ x2 A2 + x3 A2 = 0.

35 : 4 4,3 4,2

A2 ’ x2 A3 + x3 A3 = 0.

36 : (1.140)

4 4,3 4,2

We shall now indicate in more detail how to solve (1.140).

Note, that due to (1.137)

A3 = F 1 (r1,2 , x3 ), (1.141)

4

where

1

r1,2 = (x2 + x2 ) 2 , (1.142)

1 2

and due to (1.139)

‚F 1 (r1,2 , x3 ) x3 ‚F 1 (r1,2 , x3 )

A1 ’

= x1 . (1.143)

4

‚x3 r1,2 ‚r1,2

Now let

‚F 1 (r1,2 , x3 ) x3 ‚F 1 (r1,2 , x3 ) def

’ = H(r1,2 , x3 ). (1.144)

‚x3 r1,2 ‚r1,2

Substitution of (1.141) and (1.144) into (1.138) results in

x2 ‚H(r1,2 , x3 ) ‚H(r1,2 , x3 )

1

’ x2

F (r1,2 , x3 ) = x3 H(r1,2 , x3 ) + 1 , (1.145)

1

r1,2 ‚r1,2 ‚x3

or

1 ‚H(r1,2 , x3 ) ‚H(r1,2 , x3 )

F 1 (r1,2 , x3 ) ’ x3 H(r1,2 , x3 ) = x2 ’ .

1

r1,2 ‚r1,2 ‚x3

(1.146)

Di¬erentiation of (1.146) with respect to x1 , x2 yields

1 ‚H(r1,2 , x3 ) ‚H(r1,2 , x3 )

’ = 0,

r1,2 ‚r1,2 ‚x3

F 1 (r1,2 , x3 ) = x3 H(r1,2 , x3 ). (1.147)

From the second equation in (1.147) and equation (1.144) we obtain

H(r1,2 , x3 ) = l(r), (1.148)

where

1

r = (x2 + x2 + x2 ) 2 , (1.149)

1 2 3

and ¬nally, due to (1.147) and (1.123), one has

A2 = x1 l(r), A2 = x2 l(r), A3 = x3 l(r). (1.150)

4 4 4

6. SYMMETRIES OF THE SELF-DUAL SU (2) YANG“MILLS EQUATIONS 55

Handling the remaining system in a similar way, a straightforward but te-

dious computation leads to the general solution of (1.123), i.e.,

1 1

A1 = x2 f (r) + k(r), A1 = x1 x2 f (r) ’ x3 u(r),

1

21 2

2

1 1

A1 = x1 x3 f (r) + x2 u(r), A2 = x1 x2 l(r) + x3 u(r),

3 1

2 2

1 1

A2 = x2 f (r) + k(r), A2 = x2 x3 f (r) ’ x1 u(r),

2

22 3

2

1 1

A3 = x1 x3 f (r) ’ x2 u(r), A3 = x2 x3 f (r) + x1 u(r),

1 2

2 2

1

A3 = x2 f (r) + k(r), A2 = x1 l(r),

3 3 4

2

A2 = x2 l(r), A3 = x3 l(r), (1.151)

4 4

where u, l, k, f are functions of r.

Substitution of (1.151) into (1.95) and (1.95) yields a system of three

ordinary di¬erential equations for the functions u, l, k, f :

1

l + u ’ gru2 ’ grul + grf k = 0,

2

1

r2 u + 2ru ’ rl ’ gr 3 ul + grk 2 + gr3 f k = 0,

2

1 1

k ’ rf ’ grku ’ grlk ’ gr3 f u = 0. (1.152)

2 2

If we choose

h(r) a(r)

, u(r) = ’

f (r) = k(r) = 0, l(r) = , (1.153)

r r

we are led by (1.151), (1.153) to the monopole solution obtained by Prasad

and Sommerfeld [84] by imposing the ansatz (1.151), (1.153).

56 1. CLASSICAL SYMMETRIES

CHAPTER 2

Higher symmetries and conservation laws

In this chapter, we specify general constructions described for in¬nite jets

to in¬nitely prolonged di¬erential equations. We describe basic structures

existing on these objects, give an outline of di¬erential calculus over them

and introduce the notions of a higher symmetry and of a conservation law.

We also compute higher symmetries and conservation laws for some

equations of mathematical physics.

1. Basic structures

Now we introduce the main object of our interest:

Definition 2.1. The inverse limit proj liml’∞ E l with respect to pro-

jections πl+1,l is called the in¬nite prolongation of the equation E and is

denoted by E ∞ ‚ J ∞ (π).

In the sequel, we shall mostly deal with formally integrable equations

E ‚ J k (π) (see De¬nition 1.20 on p. 30), which means that all E l are smooth

manifolds while the mappings πk+l+1,k+l : E l+1 ’ E l are smooth locally

trivial ¬ber bundles.

In¬nite prolongations are objects of the category M∞ (see Example 1.5

on p. 10). Hence, general approach exposed in Subsection 1.3 of Chapter 1

can be applied to them just in the same manner as it was done for manifolds

of in¬nite jets. In this section, we give a brief outline of calculus over E ∞ and

describe essential structures speci¬c for in¬nite prolongations of di¬erential

equations.

1.1. Calculus. Let π : E ’ M be a vector bundle and E ‚ J k (π) be

a k-th order di¬erential equation. Then we have the embeddings µl : E l ‚

J k+l (π) for all l ≥ 0. We de¬ne a smooth function on E l as the restriction

f |E l of a smooth function f ∈ Fk+l (π). The set Fl (E) of all functions

on E l forms an R-algebra in a natural way and µ— : Fk+l (π) ’ Fl (E) is a

l

homomorphism. In the case of formally integrable equations, the algebra

def

Fl (E) coincides with C ∞ (E l ). Let Il = ker µ— .

l

57

58 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

Due to commutativity of the diagram

µ—

l

Fk+l (π) ’ Fl (E)

— —

πk+l+1,k+l πk+l+1,k+l

“ “

µ—

l+1

Fk+l+1 (π) ’ Fl+1 (E)

one has Il (E) ‚ Il+1 (E). Then I(E) = l≥0 Il (E) is an ideal in F(π) which

is called the ideal of the equation E. The function algebra on E ∞ is the quo-

tient F(E) = F(π)/I(E) and coincides with inj liml’∞ Fl (E) with respect

—

to the system of homomorphisms πk+l+1,k+l . For all l ≥ 0, we have the

homomorphisms µ— : Fl (E) ’ F(E). When E is formally integrable, they

l

are monomorphic, but in any case the algebra F(E) is ¬ltered by the images

of µ— .

l

Now, to construct di¬erential calculus on E ∞ , one needs the general

algebraic scheme exposed in Chapter 4 and applied to the ¬ltered algebra

F(E). However, in the case of formally integrable equations, due to the

fact that all E l are smooth manifolds, this scheme may be simpli¬ed and

combined with a purely geometrical approach (cf. similar constructions of

Subsection 1.3 of Chapter 1).

Namely, di¬erential forms in this case are de¬ned as elements of the

def

module Λ— (E) = l≥0 Λ— (E l ), where Λ— (E l ) is considered to be embedded

into Λ— (E l+1 ) by πk+l+1,k+l . A vector ¬eld on E ∞ is a derivation X : F(E) ’

—

F(E) agreed with ¬ltration, i.e., such that X(Fl (E)) ‚ Fl+± (E) for some

integer ± = ±(X) ∈ Z. Just like in the case J ∞ (π), we de¬ne the de Rham

complex over E ∞ and obtain “usual” relations between standard operations

(contractions, de Rham di¬erential and Lie derivatives).

In special coordinates the in¬nite prolongation of the equation E is de-

termined by the system similar to (1.41) on p. 29 with the only di¬erence

that |σ| is unlimited now. Thus, the ideal I(E) is generated by the functions

Dσ F j , |σ| ≥ 0, j = 1, . . . , m. From these remarks we obtain the following

important fact.

Example 2.1. Let E be a formally integrable equation. Then from the

above said it follows that the ideal I(E) is stable with respect to the action

of the total derivatives Di , i = 1, . . . , n = dim M . Consequently, the action

E def E

Di = Di F (E) : F(E) ’ F(E) is well de¬ned and Di are ¬ltered deriva-

tions. We can reformulate it in other words by saying that the vector ¬elds

Di are tangent to any in¬nite prolongation and thus determine vector ¬elds

on E ∞ . We shall often skip the superscript E in the notation of the above

de¬ned restrictions.

The fact established in the last example plays a crucial role in the theory

of in¬nite prolongations. We continue to discuss it in the next section.

1. BASIC STRUCTURES 59

To ¬nish this one, let us make a remark concerning local coordinates.

Let E be locally represented with equations (1.41). Assume that the latter

is resolved in the form

±

uσj = f j (x1 , . . . , xn , . . . , uβ , . . . ),

j

j = 1, . . . , r,

σ

in such a way that

(i) the set of functions u±1 , . . . , u±r has at the left-hand side the empty

1 r

σ

σ

intersection with the set of functions uβ at the left-hand side and

σ

±j

±i

(ii) uσi +„ = uσj +„ for no „, „ unless i = j.

In this case, all coordinate functions in the system under consideration may

±j

be partitioned into two parts: those of the form uσj +„ , |„ | ≥ 0, j = 1, . . . , r,

and all others. We call the latter ones internal coordinates on E ∞ . Note

that all constructions of di¬erential calculus over E ∞ can be expressed in

terms of internal coordinates.

Example 2.2. Consider a system of evolution equations of the form

(1.22) (see p. 16). Then the functions x1 , . . . , xn , t, . . . , uj 1 ,...,σn ,0 , σi ≥ 0,

σ

j = 1, . . . , m, where t = xn+1 , may be taken for internal coordinates on E ∞ .

The total derivatives restricted onto E ∞ are expressed as

n

‚ ‚

uj i

Di = + , i = 1, . . . , n,

σ+1

‚xi ‚uσ

j=1 |σ|≥0

n

‚ ‚

Dσ (f j )

Dt = + (2.1)

‚t ‚uσ

j=1 |σ|≥0

in these coordinates, while the Cartan forms are written down as

n

uj i dxi ’ Dσ (f j ) dt,

j

duj ’

ωσ = (2.2)

σ σ+1

i=1

where all multi-indices σ are of the form σ = (σ1 , . . . , σn , 0).

1.2. Cartan distribution. Let π : E ’ M be a vector bundle and

E ‚ J k (π) be a formally integrable equation.

Definition 2.2. Let θ ∈ J ∞ (π). Then

(i) The Cartan plane Cθ = Cθ (π) ‚ Tθ J ∞ (π) at θ is the linear envelope of

tangent planes to all manifolds j∞ (•)(M ), • ∈ “(π), passing through

the point θ.

def

(ii) If θ ∈ E ∞ , the intersection Cθ (E) = Cθ (π) © Tθ E ∞ is called Cartan

plane of E ∞ at θ.

The correspondence θ ’ Cθ (π), θ ∈ J ∞ (π) (respectively, θ ’ Cθ (E ∞ ),

θ ∈ E ∞ ) is called the Cartan distribution on J ∞ (π) (respectively, on E ∞ ).

The following result shows the crucial di¬erence between the Cartan

distributions on ¬nite and in¬nite jets (or between those on ¬nite and in¬nite

prolongations).

60 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

Proposition 2.1. For any vector bundle π : E ’ M and a formally

integrable equation E ‚ J k (π) one has:

(i) The Cartan plane Cθ (π) is n-dimensional at any point θ ∈ J ∞ (π).

(ii) Any point θ ∈ E ∞ is generic, i.e., Cθ (π) ‚ Tθ E ∞ and thus Cθ (E ∞ ) =

Cθ (J ∞ ).

(iii) Both distributions, C(J ∞ ) and C(E ∞ ), are integrable.

Proof. Let θ ∈ J ∞ (π) and π∞ (θ) = x ∈ M . Then the point θ com-

pletely determines all partial derivatives of any section • ∈ “loc (π) such that

its graph passes through θ. Consequently, all such graphs have a common

tangent plane at this point which coincides with Cθ (π). This proves the ¬rst

statement.

To prove the second one, recall Example 2.1: locally, any vector ¬eld Di

j

is tangent to E ∞ . But as it follows from (1.27) on p. 18, one has iDi ωσ = 0

j

for any Di and any Cartan form ωσ . Hence, linear independent vector ¬elds

D1 , . . . , Dn locally lie both in C(π) and in C(E ∞ ) which gives the result.

Finally, as it follows from the above said, the module

def

CD(π) = {X ∈ D(π) | X lies in C(π)} (2.3)

is locally generated by the ¬elds D1 , . . . , Dn . But it is easily seen that

[D± , Dβ ] = 0 for all ±, β = 1, . . . , n and consequently [CD(π), CD(π)] ‚

CD(π). The same reasoning is valid for

def

CD(E) = {X ∈ D(E ∞ ) | X lies in C(E ∞ )}. (2.4)

This ¬nishes the proof of the proposition.

We shall describe now maximal integral manifolds of the Cartan distri-

butions on J ∞ (π) and E ∞ .

Proposition 2.2. Maximal integral manifolds of the Cartan distribu-

tion C(π) are graph of j∞ (•), • ∈ “loc (π).

Proof. Note ¬rst that graphs of in¬nite jets are integral manifolds of

the Cartan distribution of maximal dimension (equaling to n) and that any

integral manifold projects onto J k (π) and M without singularities.

def

Let now N ‚ J ∞ (π) be an integral manifold and N k = π∞,k N ‚ J k (π),

def

N = π∞ N ‚ M . Hence, there exists a di¬eomorphism • : N ’ N 0 such

that π —¦ • = idN . Then by the Whitney theorem on extension for smooth

functions [71], there exists a local section • : M ’ E satisfying • |N = •

and jk (•)(M ) ⊃ N k for all k > 0. Consequently, j∞ (•)(M ) ⊃ N .

Corollary 2.3. Maximal integral manifolds of the Cartan distribution

on E ∞ coincide locally with graphs of in¬nite jets of solutions.

We use the results obtained here in the next subsection.

1. BASIC STRUCTURES 61

1.3. Cartan connection. Consider a point θ ∈ J ∞ (π) and let x =

π∞ (θ) ∈ M . Let v be a tangent vector to M at the point x. Then, since

the Cartan plane Cθ isomorphically projects onto Tx M , there exists a unique

tangent vector Cv ∈ Tθ J ∞ (π) such that π∞,— (Cv) = v. Hence, for any vector

def

¬eld X ∈ D(M ) we can de¬ne a vector ¬eld CX ∈ D(π) by setting (CX)θ =

C(Xπ∞ (θ) ). Then, by construction, the ¬eld CX is projected by π∞,— to X

while the correspondence C : D(M ) ’ D(π) is C ∞ (M )-linear. In other

words, this correspondence is a connection in the bundle π∞ : J ∞ (π) ’ M .

Definition 2.3. The connection C : D(M ) ’ D(π) de¬ned above is

called the Cartan connection in J ∞ (π).

Let now E ‚ J k (π) be a formally integrable equation. Then, due to

the fact that Cθ (E ∞ ) = Cθ (π) at any point θ ∈ E ∞ , we see that the ¬elds

CX are tangent to E ∞ for all vector ¬elds X ∈ D(M ). Thus we obtain the

Cartan connection in the bundle π∞ : E ∞ ’ M which is denoted by the

same symbol C.

Let x1 , . . . , xn , . . . , uj , . . . be special coordinates in J ∞ (π) and X =

σ

X1 ‚/‚x1 + · · · + Xn ‚/‚xn be a vector ¬eld on M represented in this co-

ordinate system. Then the ¬eld CX is to be of the form CX = X + X v ,

where X v = j,σ Xσ ‚/‚uj is a π∞ -vertical ¬eld. The de¬ning conditions

j

σ

j j

iCX ωσ = 0, where ωσ are the Cartan forms on J ∞ (π), imply

«

n n

‚ ‚

j

Xi

CX = + uσ+1i j = Xi Di . (2.5)

‚xi ‚uσ

i=1 j,σ i=1

In particular, we see that C(‚/‚xi ) = Di , i.e., total derivatives are just

liftings to J ∞ (π) of the corresponding partial derivatives by the Cartan

connection.

To obtain a similar expression for the Cartan connection on E ∞ , it needs

only to obtain coordinate representation for total derivatives in internal

coordinates. For example, in the case of equations (1.22) (see p. 16) we have

n n

‚ ‚

C Xi +T = Xi Di + T D t ,

‚xi ‚t

i=1 i=1

where D1 , . . . , Dn , Dt are given by formulas (2.1) and Xi , T ∈ C ∞ (M ) are

the coe¬cients of the ¬eld X ∈ D(M ).

Consider the following construction now. Let V be a vector ¬eld on E ∞

and θ ∈ E ∞ be a point. Then the vector Vθ can be projected parallel to

the Cartan plane Cθ onto the ¬ber of the projection π∞ : E ∞ ’ M passing

through θ. Thus we get a vertical vector ¬eld V v . Hence, for any f ∈ F(E)

a di¬erential one-form UE (f ) ∈ Λ1 (E) is de¬ned by

def

iV (UE (f )) = V v (f ), V ∈ D(E). (2.6)

62 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

The correspondence f ’ UE (f ) is a derivation of the algebra F(E) with the

values in the F(E)-module Λ1 (E), i.e.,

UE (f g) = f UE (g) + gUE (f )

for all f, g ∈ F(E). This correspondence contains all essential data about

the equation E.

Definition 2.4. The derivation UE : F(E) ’ Λ1 (E) is called the struc-

tural element of the equation E.

For the “empty” equation, i.e., in the case E ∞ = J ∞ (π), the structural

element Uπ is locally represented in the form

‚

j

ωσ — j ,

Uπ = (2.7)

‚uσ

j,σ

j

where ωσ are the Cartan forms on J ∞ (π). To obtain the expression in the

general case, one needs to rewrite (2.7) in local coordinates. For example,

in the case of evolution equations we get the same expression with σ =

j

(σ1 , . . . , σn , 0) and the forms ωσ given by (2.2). Contrary to the Cartan

forms, the structural element is independent of local coordinates.

We shall now give a “more algebraic” version of the Cartan connection

de¬nition.

Proposition 2.4. For any vector ¬eld X ∈ D(M ), the equality

j∞ (•)— (CX(f )) = X(j∞ (•)— (f )) (2.8)

takes place, where f ∈ F(π) and • ∈ “loc (π). Equality (2.8) uniquely deter-

mines the Cartan connection in J ∞ (π).

Proof. Both statements follow from the fact that in special coordinates

the right-hand side of (2.8) is of the form

‚ |σ| •j

‚f

X .

j ‚xσ

j,σ ‚uσ j∞ (•)(M )

Corollary 2.5. The Cartan connection in E ∞ is ¬‚at, i.e.,

C[X, Y ] = [CX, CY ]

for any X, Y ∈ D(M ).

Proof. Consider the case E ∞ = J ∞ (π). Then from Proposition 2.4 we

have

j∞ (•)— (C[X, Y ](f )) = [X, Y ](j∞ (•)— (f ))

= X(Y (j∞ (•)— (f ))) ’ Y (X(j∞ (•)— (f )))

for any • ∈ “loc (π), f ∈ F(π). On the other hand,

1. BASIC STRUCTURES 63

j∞ (•)— ([CX, CY ](f )) = j∞ (•)— (CX(CY (f )) ’ CY (CX(f )))

= X(j∞ (•)— (Y (f ))) ’ Y (j∞ (•)— (CX(f )))

= X(Y (j∞ (•)— (f ))) ’ Y (X(j∞ (•)— (f )))

To prove the statement for an arbitrary formally integrable equation E, it

su¬ces to note that the Cartan connection in E ∞ is obtained by restricting

the ¬elds CX onto in¬nite prolongation of E.

The construction of Proposition 2.4 can be generalized.

1.4. C-di¬erential operators. Let π : E ’ M be a vector bundle and

ξ1 : E1 ’ M , ξ2 : E2 ’ M be another two vector bundles.

Definition 2.5. Let ∆ : “(ξ1 ) ’ “(ξ2 ) be a linear di¬erential operator.

The lifting C∆ : F(π, ξ1 ) ’ F(π, ξ2 ) of the operator ∆ is de¬ned by

j∞ (•)— (C∆(f )) = ∆(j∞ (•)— (f )), (2.9)

where • ∈ “loc (π) and f ∈ F(π, ξ1 ) are arbitrary sections.

Immediately from the de¬nition, we obtain the following properties of

operators C∆:

Proposition 2.6. Let π : E ’ M , ξi : Ei ’ M , i = 1, 2, 3, be vector

bundles. Then

(i) For any C ∞ (M )-linear di¬erential operator ∆ : “(ξ1 ) ’ “(ξ2 ), the

operator C∆ is an F(π)-linear di¬erential operator of the same order.

(ii) For any ∆, : “(ξ1 ) ’ “(ξ2 ) and f, g ∈ F(π), one has

C(f ∆ + g ) = f C∆ + gC .

(iii) For ∆1 : “(ξ1 ) ’ “(ξ2 ), ∆2 : “(ξ2 ) ’ “(ξ3 ), one has

C(∆2 —¦ ∆1 ) = C∆2 —¦ C∆1 .

From this proposition and from Proposition 2.4 it follows that if ∆ is a

scalar di¬erential operator C ∞ (M ) ’ C ∞ (M ) locally represented as ∆ =

|σ| ∞

σ aσ ‚ /‚xσ , aσ ∈ C (M ), then

C∆ = a σ Dσ (2.10)

σ

in the corresponding special coordinates. If ∆ = ∆ij is a matrix operator,

then C∆ = C∆ij .

From Proposition 2.6 it follows that C∆ may be understood as a dif-

ferential operator acting from sections of the bundle π to linear di¬erential

operators from “(ξ1 ) to “(ξ2 ). This observation is generalized as follows.

Definition 2.6. An F(π)-linear di¬erential operator ∆ : F(π, ξ1 ) ’

F(π, ξ2 ) is called a C-di¬erential operator, if it admits restriction onto graphs

of in¬nite jets, i.e., if for any section • ∈ “(π) there exists an operator

∆• : “(ξ1 ) ’ “(ξ2 ) such that

j∞ (•)— (∆(f )) = ∆• (j∞ (•)— (f )) (2.11)

64 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

for all f ∈ F(π, ξ1 ).

Thus, C-di¬erential operators are nonlinear di¬erential operators taking

their values in C ∞ (M )-modules of linear di¬erential operators. The follow-

ing proposition gives a complete description of such operators.

Proposition 2.7. Let π, ξ1 , ξ2 be vector bundles over M . Then any C-

di¬erential operator ∆ : F(π, ξ1 ) ’ F(π, ξ2 ) can be presented in the form

a± C∆± , a± ∈ F(π),

∆=

±

where ∆± are linear di¬erential operators acting from “(ξ1 ) to “(ξ2 ).

Proof. Recall ¬rst that we consider the ¬ltered theory; in particular,

there exists an integer l such that ∆(Fk (π, ξ1 )) ‚ Fk+l (π, ξ2 ) for all k.

Consequently, since “(ξ1 ) is embedded into F0 (π, ξ1 ), we have ∆(“(ξ1 )) ‚

¯

Fl (π, ξ2 ) and the restriction ∆ = ∆ “(ξ1 ) is a C ∞ (M )-di¬erential operator

taking its values in Fl (π, ξ2 ). Then one can easily see that the equality

¯

∆• = j∞ (•)— —¦ ∆ holds, where • ∈ “loc (π) and ∆• is the operator from

(2.11). It means that any C-di¬erential ∆ operator is completely determined

¯

by its restriction ∆.

¯ ¯

On the other hand, the operator ∆ is represented in the form ∆ =

∞

± a± ∆± , a± ∈ Fl (π) and ∆± : “(ξ1 ) ’ “(ξ2 ) being C (M )-linear dif-

¯ def

ferential operators. Let us de¬ne C ∆ = ± a± C∆± . Then the di¬erence

¯

∆ ’ C ∆ is a C-di¬erential operator such that its restriction onto “(ξ1 ) van-

¯

ishes. Therefore ∆ = C ∆.

Remark 2.1. From the result obtained it follows that C-di¬erential op-

erators are operators “in total derivatives”. By this reason, they are called

total di¬erential operators sometimes.

Corollary 2.8. C-di¬erential operators admit restrictions onto in¬nite

prolongations: if ∆ : F(π, ξ1 ) ’ F(π, ξ2 ) is a C-di¬erential operator and

E ‚ J k (π) is a k-th order equation, then there exists a linear di¬erential

operator ∆E : F(E, ξ1 ) ’ F(E, ξ2 ) such that

µ— —¦ ∆ = ∆ E —¦ µ — ,

where µ : E ∞ ’ J ∞ (π) is the natural embedding.

Proof. The result immediately follows from Example 2.1 and Proposi-

tion 2.7.

We shall now consider an example which will play a very important role

in the sequel.

Example 2.3. Let ξ1 = „i— , ξ2 = „i+1 , where „p : p T — M ’ M (see

— —

Example 1.2 on p. 6), and ∆ = d : Λi (M ) ’ Λi+1 (M ) be the de Rham

def

di¬erential. Then we obtain the ¬rst-order operator dh = Cd : Λi (π) ’

h

p

i+1 — ). Due Corollary 2.8, the

Λh (π), where Λh (π) denotes the module F(π, „p

operators d : Λi (E) ’ Λi+1 (E) are also de¬ned, where Λp (E) = F(E, „p ).

—

h h h

1. BASIC STRUCTURES 65

Definition 2.7. Let E ‚ J k (π) be an equation.

(i) Elements of the module Λi (E) are called horizontal i-forms on E ∞ .

h

i (E) ’ Λi+1 (E) is called the horizontal de Rham

(ii) The operator dh : Λh h

di¬erential on E ∞ .

(iii) The sequence

d d

0 ’ F(E) ’ Λ1 (E) ’ · · · ’ Λi (E) ’ Λi+1 (E) ’ · · ·

’h ’h

h

is called the horizontal de Rham sequence of the equation E.

From Proposition 2.6 (iii) it follows that d—¦d = 0 and hence the de Rham

sequence is a complex. It cohomologies are called the horizontal de Rham

— i

cohomologies of E and are denoted by Hh (E) = i≥0 Hh (E).

In local coordinates, horizontal forms of degree p on E ∞ are represented

as ω = i1 <···<ip ai1 ...ip dxi1 § · · · § dxip , where ai1 ...ip ∈ F(E), while the

horizontal de Rham di¬erential acts as

n

Di (ai1 ...ip ) dxi § dxi1 § · · · § dxip .

dh (ω) = (2.12)

i=1 i1 <···<ip

In particular, we see that both Λi (E) and Hh (E) vanish for i > dim M .

i

h

Remark 2.2. In fact, the above introduced cohomologies are horizontal

cohomologies with trivial coe¬cients. The case of more general coe¬cients

will be considered in Chapter 4 (see also [98, 52]). Below we make the ¬rst

step to deal with a nontrivial case.

Consider the algebra Λ— (E) of all di¬erential forms on E ∞ and let us note

that one has the embedding Λ— (E) ’ Λ— (E). Let us extend the horizontal

h

de Rham di¬erential onto this algebra as follows:

(i) dh (dω) = ’d(dh (ω)),

(ii) dh (ω § θ) = dh (ω) § θ + (’1)ω ω § dh (θ).

Obviously, conditions (i), (ii) de¬ne the di¬erential dh : Λi (E) ’ Λi+1 (E) and

its restriction onto Λ— (E) coincides with the horizontal de Rham di¬erential.

h

def

Let us also set dC = d ’ dh : Λ— (E) ’ Λ— (E). Then, by de¬nition,

h h

dh —¦ dh = dC —¦ dC = 0, dC —¦ dh + dh —¦ dC = 0.

d = d h + dC ,

In other words, the pair (dh , dC ) forms a bicomplex in Λ— (E) with the total

di¬erential d. Hence, the corresponding spectral sequence converges to the

de Rham cohomology of E ∞ .

Remark 2.3. We shall rede¬ne this bicomplex in a more general alge-

braic situation in Chapter 4. On the other hand, it should be noted that

the above mentioned spectral sequence (in the case, when dh is taken for the

¬rst di¬erential and dC for the second one) is a particular case of the Vino-

gradov C-spectral sequence (or the so-called variational bicomplex) which is

essential to the theory of conservation laws and Lagrangian formalism with

constraints; cf. Subsection 2.2 below.

66 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

To conclude this section, let us write down the coordinate representation

for the di¬erential dC and the extended dh . First note that by de¬nition and

due to (2.12), one has

n

uj i dxi ,

dC (uj ) d(uj ) dh (uj ) duj

’ ’

= =

σ σ σ σ σ+1

i=1

i.e., dC takes coordinate functions uj to the corresponding Cartan forms.

σ

Since obviously dC (xi ) = 0 for any coordinate function on the base, we

obtain

‚f j

f ∈ F(π).

dC (f ) = ωσ , (2.13)

j

j,σ ‚uσ

The same representation, written in internal coordinates, is valid on E ∞ .

Therefore, the image of dC spans the Cartan submodule CΛ1 (E) in Λ1 (E).

By this reason, we call dC the Cartan di¬erential on E ∞ . From the equality

d = dh + dC it follows that the direct sum decomposition

Λ1 (E) = Λ1 (E) • CΛ1 (E)

h

takes place which extends to the decomposition

Λq (E) — C p Λ(E).

Λi (E) = (2.14)

h

p+q=i

Here the notation

def

C p Λ(E) = CΛ1 (E) § · · · § CΛ1 (E)

p times

j

is used. Consequently, to ¬nish computations, it su¬ces to compute dh (ωσ ).

But we have

dh (ωσ ) = dh dC (uj ) = ’dC dh (uj )

j

σ σ

and thus

n

j

j

dh (ωσ ) = ’ ωσ+1i § dxi . (2.15)

i=1

Note that from the results obtained it follows, that

dh (Λq (E) — C p Λ(E)) ‚ Λq+1 (E) — C p Λ(E),

h h

dC (Λq (E) — C p Λ(E)) ‚ Λq (E) — C p+1 Λ(E).

h h

Remark 2.4. Note that the sequence dh : Λq (E) — C — (E) ’ Λq+1 (E) —

h h

— (E) can be considered as the horizontal de Rham complex with coe¬cients

C

in Cartan forms

2. HIGHER SYMMETRIES AND CONSERVATION LAWS 67

Remark 2.5. From (2.14) it follows that to any form ω ∈ Λ— (E) we

can put into correspondence its “purely horizontal” component ωh ∈ Λ— (E).

h

k (π), then, due to the equality duj =

Moreover, if the form ω “lives” on J σ

j

n — k+1 (π)). This correspon-

i=1 uσ+1i dxi + ωσ , the form ωh belongs to Λ (J

dence coincides with the one used in Example 1.7 on p. 14 to construct

Monge“Ampere equations.

2. Higher symmetries and conservation laws

In this section, we brie¬‚y expose the theory of higher (or Lie“B¨cklund)

a

symmetries and conservation laws for nonlinear partial di¬erential equations

(for more details and examples see [60, 12]).

2.1. Symmetries. Let π : E ’ M be a vector bundle and E ‚ J k (π)

be a di¬erential equation. We shall still assume E to be formally integrable,

though is it not restrictive in this context.

Consider a symmetry F of the equation E and let θk+1 be a point of E 1

such that πk+1,k (θk+1 ) = θk ∈ E. Then the R-plane Lθk+1 is taken to an

R-plane F— (Lθk+1 ) by F , since F is a Lie transformation, and F— (Lθk+1 ) ‚

TF (θk ) , since F is a symmetry. Consequently, the lifting F (1) : J k+1 (π) ’

J k+1 (π) is a symmetry of E 1 . By the same reasons, F (l) is a symmetry of the

l-th prolongation of E. From here it also follows that for any in¬nitesimal

symmetry X of the equation E, its l-th lifting is a symmetry of E l as well.

In fact, the following result is valid:

Proposition 2.9. Symmetries of a formally integrable equation E ‚

J k (π)

coincide with symmetries of any prolongation of this equation. The

same is valid for in¬nitesimal symmetries.

Proof. We have shown already that to any (in¬nitesimal) symmetry

of E there corresponds an (in¬nitesimal) symmetry of E l . Consider now an

(in¬nitesimal) symmetry of E l . Then, due to Theorems 1.12 and 1.13 (see

pp. 24 and 26), it is πk+l,k -¬berwise and therefore generates an (in¬nitesimal)

symmetry of E.

The result proved means that a symmetry of E generates a symmetry of

E∞ which preserves every prolongation up to ¬nite order. A natural step to

generalize the concept of symmetry is to consider “all symmetries” of E ∞ .

Let us clarify such a generalization.

First of all note that only in¬nitesimal point of view may be e¬cient

in the setting under consideration. Otherwise we would have to deal with

di¬eomorphisms of in¬nite-dimensional manifolds with all natural di¬cul-

ties arising as a consequence. Keeping this in mind, we proceed with the

following de¬nition. Recall the notation

def

CD(π) = {X ∈ D(π) | X lies in C(π)},

cf. (1.31) on p. 25.

68 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

Definition 2.8. Let π be a vector bundle. A vector ¬eld X ∈ D(π) is

called a symmetry of the Cartan distribution C(π) on J ∞ (π), if [X, CD(π)] ‚

CD(π).

Thus, the set of symmetries coincides with DC (π) (see (1.32) on p. 25)

and forms a Lie algebra over R and a module over F(π). Note that since the

Cartan distribution on J ∞ (π) is integrable, one has CD(π) ‚ DC (π) and,

moreover, CD(π) is an ideal in the Lie algebra DC (π).

Note also that symmetries belonging to CD(π) are of a special type:

they are tangent to any integral manifold of the Cartan distribution. By

this reason, we call such symmetries trivial. Respectively, the elements of

the quotient Lie algebra

def

sym(π) = DC (π)/CD(π)

are called nontrivial symmetries of the Cartan distribution on J ∞ (π).

Let now E ∞ be the in¬nite prolongation of an equation E ‚ J k (π).

Then, since CD(π) is spanned by the ¬elds of the form CY , Y ∈ D(M ) (see

Example 2.1), any vector ¬eld from CD(π) is tangent to E ∞ . Consequently,

either all elements of the coset [X] = X mod CD(π), X ∈ D(π), are tangent

to E ∞ or neither of them is. In the ¬rst case we say that the coset [X] is

tangent to E ∞ .

Definition 2.9. An element [X] = X mod CD(π), X ∈ D(π), is called

a higher symmetry of E, if it is tangent to E ∞ .

The set of all higher symmetries forms a Lie algebra over R and is de-

noted by sym(E). We shall usually omit the adjective higher in the sequel.

Let us describe the algebra sym(E) in e¬cient terms. We start with