Dunajski M. Solitons, Instantons, and Twistors
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116 6: Gauge field theory
distances the theory is however non-abelian which guarantees finite energy and
smoothness.
The non-vanishing part of the gauge field at infinity is the U(1) magnetic
field:
B
k
(m)
=
1
2
ε
ijk
F
a
ij
ˆ
a
.
To find the magnetic charge of B
(m)
note that asymptotically D
i
ˆ
= 0 with
|
ˆ
| = 1 which yields
A
a
i
= −ε
abc
∂
i
ˆ
b
ˆ
c
+ k
i
ˆ
a
(6.3.13)
for some vector k
i
. The last term is globally well defined and gives no net flux.
We shall calculate the corresponding field
F
a
ij
= ∂
i
A
a
j
− ∂
j
A
a
i
− ε
abc
A
b
i
A
c
j
=2ε
abc
∂
i
ˆ
b
∂
j
ˆ
c
− (ε
pqr
∂
i
ˆ
p
∂
j
ˆ
q
ˆ
r
)
ˆ
a
.
The corresponding magnetic charge is
Q =
S
2
∞
1
2
ε
ijk
(F
a
jk
ˆ
a
)n
i
d
2
S
=
S
2
∞
1
2
ε
ijk
(ε
pqr
∂
i
ˆ
p
∂
j
ˆ
q
)
ˆ
r
n
k
d
2
S
=4π N, (6.3.14)
where n is a unit normal to S
2
∞
since the integrand is the area form on S
2
vac
pulled back to S
2
∞
,4π is the area of the unit sphere and
N = deg(
ˆ
∞
)
is the topological degree (A7). The integer N is called the monopole number,
and 4π is the unit of magnetic charge.
6.3.2 Bogomolny–Prasad–Sommerfeld (BPS) limit
Consider the limit c =0, |
∞
| = 1 of the field equations (6.3.12), and define
the non-abelian magnetic field by
B
i
=
1
2
ε
ijk
F
jk
.
Theorem 6.3.1 The energy of a non-abelian magnetic monopole is bounded
from below:
E ≥ 4π|N|, (6.3.15)
6.3 Non-abelian monopoles 117
where N is the topological degree of the asymptotic electromagnetic field at
infinity. For positive N the bound is saturated if
∗
3
F = D, (6.3.16)
where ∗
3
:
k
−→
3−k
is the three-dimensional Hodge operator given by
∗
3
dx
k
=
1
2
ε
ijk
dx
j
∧ dx
k
.
Proof The energy functional is given by
E =
R
3
1
4
F
a
ij
F
a
ij
+
1
2
(D
k
)
a
(D
k
)
a
d
3
x
=
1
2
R
3
%
B
a
k
B
a
k
+(D
k
)
a
(D
k
)
a
&
d
3
x
=
1
2
R
3
(B
k
− D
k
)
a
(B
k
− D
k
)
a
d
3
x +
R
3
B
a
k
(D
k
)
a
d
3
x
= E
1
+ E
2
, (6.3.17)
where E
1
is non-negative (recall that E =0 as A
0
= 0). The spatial Bianchi
identity D
[i
F
jk]
= 0 is equivalent to D
k
B
k
= 0. We use this identity, Stoke’s
theorem, and the cyclic property of trace to rewrite the second term on the
RHS as
E
2
= −2
R
3
Tr( B
k
D
k
)d
3
x = −2
R
3
Tr[ D
k
(B
k
)]d
3
x
= −2
R
3
∂
k
Tr( B
k
)d
3
x =
S
2
∞
B
a
k
a
n
k
d
2
S =4π N,
where n
k
is the unit outward normal to the sphere at infinity and we have
identified the flux of the asymptotic electromagnetic field (6.3.14). This yields
the Bogomolny bound (6.3.15) which (for a positive N) is saturated if B
k
=
D
k
or, in terms of differential forms, if (6.3.16) holds.
Equations (6.3.16) are nine coupled non-linear PDEs known as the Bogomolny
equations. They imply the static field equations as (A,) is a critical point of
the energy functional. For a given monopole number N the space of finite-
energy solutions (6.3.16) modulo the gauge transformations (6.1.1) is 4N − 4
dimensional up to an overall rotation [180]. In Section 8.1 we shall see that
the system (6.3.16) is integrable but explicit formulae can only be found for
N = 1 and (to some extend) for N =2.
118 6: Gauge field theory
In practice the monopole number can be read off from the asymptotic
behaviour of the Higgs field as
|| =1−
N
r
+ O(r
−2
), where r −→ ∞. (6.3.18)
To verify this calculate the topological term E
2
in (6.3.17) on solutions to the
Bogomolny equations:
E
2
=
R
3
B
a
k
(D
k
)
a
d
3
x =
R
3
(D
k
)
a
(D
k
)
a
d
3
x
=
1
2
R
3
||
2
d
3
x =
1
2
S
2
∞
∇
1 −
2N
r
+ O(r
−2
)
· dS =4π N,
where we have used the static field equations D
k
D
k
= 0 and the diver-
gence theorem. This is in agreement with the calculation leading to
(6.3.15).
r
Example. To find explicit solutions we make the spherically symmetric
ansatz:
a
= h(r)
x
a
r
and A
i
a
= −ε
aij
x
j
r
2
[1 − k(r)].
This makes use of the isomorphism su(2)
∼
=
R
3
as the ansatz replaces the Lie
algebra indices by the space-time indices. The Bogomolny equations reduce
toapairofODEs:
dh
dr
= r
−2
(1 − k
2
) and
dk
dr
= −kh.
Using the change of variables H = h + r
−1
and K = k/r one finds the Prasad–
Sommerfield solution [138]:
a
=
x
a
r
coth(r) −
1
r
and A
a
i
= −ε
aij
x
j
r
2
1 −
r
sinh (r)
. (6.3.19)
This solution has N = 1 which follows from (6.3.18) as
1 −||
2
=
1
r
− 2e
−r
+ O(e
−3r
).
Asymptotically the solution approaches the Dirac one-monopole (6.2.8) with
g =4π and mass equal to 4π. It is the lowest energy one–monopole configu-
ration, therefore the solution is stable.
The corresponding energy density is given by a spherically symmetric func-
tion concentrated around the origin r = 0, thus supporting the interpretation
of the monopole as a particle located at the origin.
6.4 Yang–Mills equations and instantons 119
6.4 Yang–Mills equations and instantons
Let us start with the following:
Definition 6.4.1 Instantons are non-singular solutions of classical equations of
motion in Euclidean space whose action is finite.
In this section we shall study instantons in pure YM theory. Our interest in
such configurations is motivated by QM, where in the WKB approximation
the tunnelling amplitudes are controlled by the exponentially small factor
e
−S/
, where S is the minimal Euclidean action to pass form initial to final
state. In this section we use the Euclidean metric η
µν
= diag(1, 1, 1, 1), and
the coordinates x
µ
, where µ =1,...,4. The term ‘instanton’ is used because
a solution localized in R
4
with a Euclidean metric dx
2
+ dτ
2
is simultaneously
localized in space and in an instant of Euclidean time.
The Euclidean YM action
S = −
R
4
Tr( F ∧∗F )
yields the YM equations
D ∗ F =0. (6.4.20)
Finiteness of the action is ensured by
F
µν
(x) ∼ O(1/r
3
) and A
µ
(x) ∼−∂
µ
gg
−1
+ O(1/r
2
), as r →∞,
(6.4.21)
the important point being that the gauge transformation g(x) needs only
to be defined asymptotically, so that g : S
3
∞
→ SU(2). This function can be
continuously extended to R
4
if its degree (A8) vanishes. Making another gauge
transformation of A
µ
at infinity will change g, but not its homotopy class.
The boundary conditions can be understood in terms of the one-point com-
pactification S
4
= R
4
∪{∞}, which has a metric conformally equivalent to the
flat metric on R
4
. The YM equations are conformally invariant and solutions
extend from R
4
to S
4
. Any smooth solution of YM equations on S
4
project
stereographically to a connection on R
4
with a curvature which vanishes at
infinity with the rate (6.4.21). Uhlenbeck [165] established a converse of this
result: For any finite action smooth solution A to the YM equations on R
4
there exists a bundle over S
4
which stereographic projects to A. The proof
uses the conformal invariance of the YM equations and of the Hodge operator
in four dimensions. In this approach the base space S
4
is not contractible, so
the principal YM bundles need not be topologically trivial (in fact they are
classified by the same integer which classified the gauge equivalence classes of
A at ∞ in R
4
). Let ω be a connection one-form on a principal bundle P −→ S
4
,
120 6: Gauge field theory
and let
f
N
: S
4
− (N = {0, 0, 0, 1}) and f
S
: S
4
− (S = {0, 0, 0, −1})
be stereographic projections from the north pole and south pole, respectively.
The connection projects down to (d + A
N
) and (d + A
S
)onR
4
, but the pull
back ( f
N
)
∗
(A
N
)toS
4
does not extend smoothly to N unless P is trivial, and
the similar statement can be made about the pull–back of A
S
by f
S
. The two
one-forms ( f
N
)
∗
(A
N
) and ( f
S
)
∗
(A
S
) are defined on S
4
−{N ∪ S}, and related
by the gauge transformation g : S
3
× R → SU(2) = S
3
:
( f
N
)
∗
(A
N
)=g( f
S
)
∗
(A
S
)g
−1
− (dg)g
−1
.
Let F
N
and F
S
be the gauge fields of A
N
and A
S
on R
4
. The gauge invariance
implies that the four forms
( f
N
)
∗
Tr( F
N
∧∗F
N
)=(f
S
)
∗
Tr( F
S
∧∗F
S
)
agree on S
4
−{N ∪ S}, and so they are equal and well defined everywhere on
S
4
.
6.4.1 Chern and Chern–Simons forms
Consider a pure gauge theory on R
D
.IfF takes values in su(n), then the Chern
class [62, 175] is given by
C(F ) = det
1 +
i
2π
F
=1+C
1
(F )+C
2
(F )+···,
where C
p
(F )isa2p-form (a polynomial in Tr(F
k
)). The pth Chern form C
p
(F )
is gauge invariant (which is why we can work with F and not defined in
Section 6.1.2) and closed because the Bianchi identity implies that Tr(F
k
)is
closed for all k. From now on take G = SU(2). We have
C
1
(F )=
i
2π
Tr( F )=0 and
C
2
(F )=
1
8π
2
[
Tr( F ∧ F ) − Tr( F ) ∧ Tr( F )
]
=
1
8π
2
Tr( F ∧ F ).
Explicitly and in any dimension
dC
2
=
1
4π
2
Tr(dF ∧ F )=
1
4π
2
Tr( DF ∧ F − A ∧ F ∧ F + F ∧ A ∧ F )=0,
where we used the Bianchi identity (6.1.2) and the cyclic property of the trace.
Therefore, since R
D
is contractible, C
2
= dY
3
, where Y
3
is the so-called Chern–
Simons three-form given by
Y
3
=
1
8π
2
Tr(dA∧ A +
2
3
A
3
).
6.4 Yang–Mills equations and instantons 121
This can be verified using Tr(A
4
)=0.
Integrating the Chern classes over manifolds of appropriate dimensions gives
integer Chern numbers. If D =4
c
2
=
R
4
C
2
=
1
8π
2
R
4
dY
3
=
1
8π
2
R
4
dTr
F ∧ A −
1
3
A
3
= −
1
24π
2
S
3
∞
Tr( A
3
) ∈ Z, (6.4.22)
since
A
∞
= −(dg) g
−1
, F
∞
=0, and
1
24π
2
S
3
∞
Tr
'
%
(dg) g
−1
&
3
(
= deg(g) ∈ Z,
where we used (A8).
We shall now explain how to understand this result from the point of view
of bundles over S
4
with no boundary conditions. Cover S
4
by two hemispheres
U
+
and U
−
with the overlap being a cylinder U
+
∩ U
−
= S
3
× [−, ], where S
3
is the equatorial three-sphere and [−, ] is a line segment. The connection and
the curvature (6.1.7) are given by
ω =
γ
+
−1
A
+
γ
+
+ γ
+
−1
dγ
+
on U
+
γ
−
−1
A
−
γ
−
+ γ
−
−1
dγ
−
on U
−
and
=
γ
+
−1
F
+
γ
+
on U
+
γ
−
−1
F
−
γ
−
on U
−
,
where γ
−
= gγ
+
and (A
+
, A
−
), (F
+
, F
−
) are related by the gauge transforma-
tions:
A
−
= gA
+
g
−1
− (dg)g
−1
and F
−
= gF
+
g
−1
, where g = g(x
µ
) ∈ G.
We claim that the second Chern number characterizing the bundle is given by
(6.4.22), albeit with a different interpretation of g. We shall use the fact that
both hemispheres U
±
are separately contractible, so Chern–Simons three-form
can be introduced on each of them:
k = −
1
8π
2
S
4
Tr( ∧ )=−
1
8π
2
U
+
Tr( F
+
∧ F
+
)+
U
−
Tr( F
−
∧ F
−
)
= −
1
8π
2
S
3
Tr
F
+
∧ A
+
−
1
3
(A
+
)
3
− Tr
F
−
∧ A
−
−
1
3
(A
−
)
3
122 6: Gauge field theory
= −
1
8π
2
S
3
Tr
1
3
%
(dg) g
−1
&
3
− d
%
A
+
∧ (dg) g
−1
&
= −
1
24π
2
S
3
Tr
'
%
(dg) g
−1
&
3
(
in agreement with (6.4.22). In this calculation we interpret g as the transition
function of the bundle P → S
4
. The bundle is trivial when restricted to U
+
or
U
−
, and the non-triviality arises when the hemispheres are patched together
using g : U
+
∩ U
−
→ SU(2). In the case of the one instanton solution k =1
the total space of P is S
7
[163]. The argument we have just given readily
generalizes to show that principal G bundles over S
n
are classified by elements
of the homotopy group π
n−1
(G).
6.4.2 Minimal action solutions and the anti-self-duality condition
Theorem 6.4.2 The YM action S within a given topological sector
c
2
=
1
8π
2
R
4
Tr(F ∧ F ) > 0
is bounded from below by 8π
2
c
2
. The bound is saturated if the anti-self-dual
Yang–Mills (ASDYM) equations
F = −∗ForF
12
= −F
34
, F
13
= −F
42
, F
14
= −F
23
(6.4.23)
hold.
Proof Note that F ∧ F = ∗F ∧∗F and calculate
S = −
1
2
R
4
Tr[(F + ∗F ) ∧ (F + ∗F )] +
R
4
Tr( F ∧ F )
= −
1
2
R
4
Tr[(F + ∗F ) ∧∗(F + ∗F )] + 8π
2
c
2
≥ 8π
2
c
2
, (6.4.24)
as the first integral is non-negative. This gives the Bogomolny bound for S,
with the equality iff the field tensor is ASD and the ASDYM equations (6.4.23)
hold.
A similar calculation with c
2
≤ 0 would lead to the SD equations F = ∗F .If
the YM connection satisfies the SD or ASD conditions, then the YM equations
(6.4.20) are satisfied by virtue of the Bianchi identity (6.1.2). Changing the
sign of the volume form (reversing the orientation) interchanges ASD and SD
fields.
6.4 Yang–Mills equations and instantons 123
The ASDYM fields in R
4
which satisfy the boundary conditions (6.4.21)
are called instantons, and k = −c
2
is called the instanton number. There is an
8|k|−3-dimensional manifold of instantons of charge k [7].
6.4.3 Ansatz for ASD fields
In this section we shall present an ansatz, originally due to Corrigan and Fairlie
[34], which reduces the non-linear ASDYM equations to the Laplace equation
in R
4
. The ansatz will lead to explicit instantons with arbitrarily large instanton
number.
Introduce the antisymmetric objects σ
µν
on R
4
by
σ
ab
= ε
abc
T
c
,σ
a4
= −σ
4a
= T
a
,
where T
a
, a =1, 2, 3, form a basis of su(2), that is, [T
a
, T
b
]=−ε
abc
T
c
. Note
σ
12
= σ
34
etc. One can check the relation
[σ
µκ
,σ
νλ
]=−δ
µν
σ
κλ
+ δ
µλ
σ
κν
+ δ
κν
σ
µλ
− δ
κλ
σ
µν
which implies the identities
σ
µν
σ
µκ
= −
3
4
1δ
νκ
− σ
νκ
,σ
µν
σ
µν
= −31. (6.4.25)
Note that
1
2
ε
µνκλ
σ
κλ
= σ
µν
so the objects σ
µν
are SD. This has the following interpretation. The Lie algebra
so(4) = su(2) ⊕ su(2) regarded as a six-dimensional vector space is isomorphic
to the space
2
of two-forms on R
4
. The forms σ
µν
select the three-dimensional
space of SD two-forms
2
+
from the six-dimensional space
2
=
2
+
⊕
2
−
and
project it onto su(2). The three-dimensional vector spaces su(2) and
2
+
are
isomorphic.
Proposition 6.4.3 Let ρ : R
4
→ R be a function. The potential
A = σ
µν
∂
ν
ρ
ρ
dx
µ
(6.4.26)
satisfies the ASDYM equations (6.4.23) iff the Laplace equation
ρ = 0 (6.4.27)
holds, where = η
µν
∂
µ
∂
ν
.
Proof The YM field corresponding to (6.4.26) will be ASD iff
1
2
σ
µν
F
µν
= σ
µν
(∂
µ
A
ν
+ A
µ
A
ν
)=0,
124 6: Gauge field theory
which follows from
σ
12
F
12
+ ···+ σ
34
F
34
+ ···= σ
12
(F
12
+ F
34
)+··· .
Now compute
σ
µν
(∂
µ
A
ν
+ A
µ
A
ν
)
= σ
µν
σ
νλ
∂
µ
∂
λ
ρ
ρ
+ σ
µλ
σ
νκ
∂
λ
ρ∂
κ
ρ
ρ
2
=
3
4
δ
µλ
+ σ
µλ
∂
µ
∂
λ
ρ
ρ
−
∂
λ
ρ∂
µ
ρ
ρ
2
+
−
3
4
δ
νλ
− σ
νλ
σ
νκ
∂
λ
ρ∂
κ
ρ
ρ
2
=
3
4
∂
µ
∂
µ
ρ
ρ
.
So F is ASD iff ∂
µ
∂
µ
ρ = 0 which is (6.4.27).
The basic solution ρ = r
−2
is a pure gauge and gives F = 0. The Jackiw–
Nohl–Rebbi (JNR) N-instanton solutions [91] are obtained by superposing
(N + 1) fundamental solutions to the Laplace equation:
ρ =
N
p=0
λ
p
|x − x
p
|
2
. (6.4.28)
This family of instantons depends on 5N + 4 parameters consisting of the
choice of N + 1 points in R
4
or S
4
as the overall scaling of ρ has no effect.
The instanton number is N, because each of the (N + 1) singularities can be
removed by a gauge transformation g
p
:(S
3
)
p
→ SU(2) of degree one defined
on a sphere S
3
p
surrounding the point x
p
and no other singularities, and one
unit is taken away because of the asymptotic behaviour of the gauge field.
The general N-instanton solution is known to depend on 8N − 3 param-
eters, and the JNR ansatz gives all solutions for N = 1 and 2. In the limiting
case, where one of the fixed points is at ∞ we recover the t’Hooft ansatz [156]:
ρ =1+
N
p=1
λ
p
|x − x
p
|
2
.
This depends on 5N parameters and, unlike (6.4.28), is not conformally
invariant, as we have selected a point in R
4
. Although the gauge potential
has singularities in both JNR and t’Hooft solutions, the field itself is regular.
6.4.4 Gradient flow and classical mechanics
This section is intended as a physical motivation for studying instanton solu-
tions of the YM equations in the Euclidean signature. We shall follow the
6.4 Yang–Mills equations and instantons 125
treatment of [40] and present a reformulation of Euclidean YM theory as
classical mechanics on an infinite-dimensional manifold of connections on R
3
.
Consider a trajectory q
k
(t) of a particle with unit mass moving in a potential
V(q)inR
D
, with coordinates q
k
, k =1,...,D. The equations of motion are
¨
q
k
= −
∂V(q)
∂q
k
. (6.4.29)
Assume that V = −|∇W|
2
/2, where W = W(q). Then any solution to the first-
order gradient-flow equations
˙
q
k
=
∂W
∂q
k
is automatically a solution to (6.4.29) as
¨
q
k
=
d
dt
∂W
∂q
k
=
∂
2
W
∂q
k
∂q
j
∂W
∂q
j
=
∂
∂q
k
1
2
|∇W|
2
.
The total energy of these special solutions is
1
2
|∇W |
2
+ V(q)=0,
and adding a constant E to the potential generalizes this to
1
2
|∇W |
2
+ V(q)=E. (6.4.30)
Now consider a corresponding QM problem governed by the Schrödinger
equation
−
2
2
∇
2
+ V = E,
where (q) is the complex wave function. Represent the wave function as
= a(q)e
iW(q)/
,
where a(q) and W(q) are real valued. In the WKB approximation one analyses
the leading terms in the expansion of the Schrödinger equation. The lowest
order coincides with the gradient flow (6.4.30). In the classically forbidden
region V > E the appropriate asymptotic form of the wave function is
= a(q)e
W(q)/
,
and one finds that the quantum system is approximated by a classical motion
along the gradient lines with a reversed potential, that is,
1
2
|∇W |
2
= V(q) − E. (6.4.31)