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‚ ‚ ‚ ‚
’ 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

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