Chapter 3 Instanton Solutions in Non-Abelian Gauge Theory

26 Pages·2008·0.19 MB·English

Chapter 3
Instanton Solutions in
Non-Abelian Gauge Theory
Contents
3.1 Conventions
3.2 Decomposition SO(4) = SU(2)×SU(2) and Quater-
nions
3.3 (Anti-)Self-Dual Configurations as Classical Solu-
tions
3.4 Winding Number for Finite Action Configurations
3.5 One Instanton Solution for SU(2)
3.6 A Class of Multi-Instanton Solutions
3.7 ADHM Construction for Most General Multi-Instanton
Solutions
3.1-1
3.1 Conventions
Gauge fields:
A = gAaTa (3.1)
µ µ
Ta = anti-hermitian generator
(cid:163) (cid:164)
Ta,Tb = fabcTc
F = ∂ A − ∂ A + [A ,A ] = gFa Ta (3.2)
µν µ ν ν µ µ ν µν
Fa = ∂ Aa − ∂ Aa + gfabcAbAc (3.3)
µν µ ν ν µ µ ν
Normalization: We normalize the generator as
1
TrTaTb = − δab (3.4)
2
In the case of G = SU(2), this corresponds precisely to the
choice
τa
Ta = ta ≡ (3.5)
2i
Covariant derivative:
D ≡ ∂ + A (3.6)
µ µ µ
F = [D ,D ] (3.7)
µν µ ν
Inner product notation: Sometimes we use the following
inner product notation
(Ta,Tb) ≡ δab (= −2TrTaTb) (3.8)
Euclidean action:
(cid:90) (cid:90)
1 1
S = d4xFa Fa = − d4xTr(F F )
E 4 µν µν 2g2 µν µν
(cid:90)
1
= d4x(F ,F ) ≥ 0 (3.9)
µν µν
4g2
3.1-2
Indices and (cid:178)-tensor: To conform to some of the important
literatures, we use the convention
µ,ν = 0,1,2,3
(cid:178) = 1 (3.10)
0123
Remark: If one uses the convention µ = 1 ∼ 4 and (cid:178) = 1,
1234
self-dual and anti-self-dual solutions are switched.
3.1-3
3.2 Decomposition SO(4) = SU(2) × SU(2)
and Quaternions
In the following, the instanton for G = SU(2) will play a funda-
mental role. This solution intertwines the gauge group and the
spacetime symmetry group SO(4), which can be decomposed
as SU(2) × SU(2). This decomposition is intimately related
to the quaternion, which will play a basic role in the ADHM
construction.
3.2.1 Decomposition of SO(4)
(cid:50) SO(4) and its generators:
SO(4) rotation is expressed as
x(cid:48) = Λ x
µ µν ν
ΛTΛ = 1
Writing Λ = exp(ξ) and considering the infinitesimal transfor-
mation, we easily find that ξ is real 4×4 antisymmetric matrix.
The standard basis for such antisymmetric matrices can be taken
as L defined by (choosing a convenient overall sign)
µν
(L ) ≡ −(δ δ − δ δ ) (3.11)
µν ρσ µρ νσ µσ νρ
In other words, L has −1 at the position (µ,ν) and 1
µν
at (ν,µ) and 0 for all the other elements. The non-vanishing
commutator for these generators is of the form
[L ,L ] = L no sum over ν (3.12)
µν νρ ρµ
3.2-4
(For instance, [L ,L ] = L .) ξ can then be decomposed as
23 31 12
(watch for the order of indices)
1
ξ = − ξ L (3.13)
µν µν
2
L is a generator of rotation in the µ-ν plane. For example,
µν
Rotation in the 1-2 plane is
(cid:181) (cid:182) (cid:181) (cid:182)(cid:181) (cid:182)
x(cid:48) cosθ −sinθ x
1 = 1
x(cid:48) sinθ cosθ x
2 2
(cid:181) (cid:182)(cid:181) (cid:182)
0 −θ x
1
∼ = θL x (3.14)
12
θ 0 x
2
(cid:50) SU(2) × SU(2) decomposition:
If we define I and K (i = 1,2,3) as
i i
1
I ≡ (cid:178) L
i ijk jk
2
K ≡ L
i 0i
they satisfy the commutation relations
[I ,I ] = (cid:178) I
i j ijk k
[I ,K ] = (cid:178) K
i j ijk k
[K ,K ] = (cid:178) I
i j ijk k
Now define the following combinations
1
J± ≡ (I ± K ) (3.15)
i 2 i i
Then, we find that they generate separtely the algebra of SU(2):
(cid:163) (cid:164)
J±,J± = (cid:178) J±
i j ijk k
(cid:163) (cid:164)
J±,J∓ = 0
i j
3.2-5
To see that they generate really the group SU(2) × SU(2),
compute the exponent −1ξ L :
2 µν µν
1 1
− ξ L = −ξ K − ξ (cid:178) I
µν µν 0i i ij ijk k
2 2
(cid:181) (cid:182) (cid:181) (cid:182)
1 1
= − ξ (cid:178) − ξ J+ + − ξ (cid:178) + ξ J−
2 ij ijk 0k k 2 ij ijk 0k k
≡ θ+J+ + θ−J−
k k k k
Since θ± are real and independent, the decomposition is in-
k
deed SU(2)×SU(2). In this regard, recall that for the Lorentz
group, they are complex conjugate of each other.
(cid:50) Intertwiner:
We will need a more explicit relation between SO(4) and SU(2)×
SU(2).
Let M be an SU(2) transformation matrix and u be the
AB B
fundamental spinor representation:
u(cid:48) = M u , M†M = 1, A,B = 1,2 (3.16)
A AB B
˙ ˙
We will use dotted indices such as A,B for the second SU(2).
The above decomposition means that there must exist a 4×4
intertwining matrix T such that
AB˙,µ
Σ+
TJ+T−1 ≡ J+ = i ⊗ 1
i i 2i
Σ−
TJ−T−1 ≡ J− = 1 ⊗ i
i i 2i
or more explicitly
(cid:181) (cid:182)
Σ+
T (J+) = i ⊗ 1 T
AB˙,µ i µν 2i CD˙,ν
AB˙,CD˙
(cid:181) (cid:182)
Σ−
T (J−) = 1 ⊗ i T
AB˙,µ i µν 2i CD˙,ν
AB˙,CD˙
3.2-6
where Σ± are 2×2 SU(2) generators. A solution to these set of
i
equations is, regarding T as four 2 × 2 matrices,
AB˙,µ
T = 1, T = −iτ (3.17)
0 i i
Σ+ = τ , Σ− = −τT (3.18)
i i i i
where τ are the Pauli matricies. Indeed Σ± satisfy the same
i i
algebra. Hereafter, we shall write
σ ≡ T = (1,−iτ ) (3.19)
µ µ i
In this way, we obtain the following explicit decomposition
formula for general SO(4) transformation:
x(cid:48) = Λx = eθk+Jk+eθk−Jk−x (3.20)
(cid:104) (cid:105)(cid:104) (cid:105)
Tx(cid:48) = σµΛµνxν = Teθk+Jk+T−1 Teθk−Jk−T−1 Tx
(cid:104) (cid:105)
= eθk+Σ+/2i ⊗ eθk−Σ−/2i Tx
= eθk+Σ+/2iσνeθk−(Σ−)T/2ixν (3.21)
Removing x , this can be written as
ν
σ Λ = M σ MT (3.22)
µ µν + ν −
M = SU(2) (3.23)
± ±
Therefore, σ transforms under bifundamental (1, 1) represen-
µ 2 2
tation of SU(2) × SU(2).
3.2.2 Quaternions
σ defined above is deeply related to the quaternions, which
µ
forms an algebra (actually a field) denoted by H.
3.2-7
Quaternion q ∈ H is defined by
(cid:88)3 (cid:88)3
q = q e = q e + q e (3.24)
µ µ 0 0 i i
µ=0 i=1
where q are real numbers and e ’s satisfy the following closed
µ µ
algebra:
e e = e , e e = −e , (3.25)
0 0 0 i i 0
e e = e e = e , e e = (cid:178) e (i (cid:54)= j) (3.26)
0 i i 0 i i j ijk k
Since e e = −e e , quaternion algebra is in general non-commutative.
i j j i
This multiplication rule can be summarized as (µ,ν on the RHS
are not summed)
e e = −(−1)δµ0δ e + δ (1 − δ )e + δ (1 − δ )e
µ ν µν 0 µ0 ν0 ν ν0 µ0 µ
(cid:88)
+ (cid:178) e (3.27)
0µνρ ρ
ρ
When one sets e = e = 0, then it becomes a complex number,
2 3
with e being the imaginary unit i. Hereafter summation over
1
the repeated indices will be assumed, unless otherwise stated.
It is clear that H is closed under multiplication.
Remark: One can easily check that if we ignore the fact that
σ has the index structure (σ ) , namely that row and col-
µ µ AB˙
umn indices are acted on by different SU(2) groups, σ satisfies
µ
exactly the same algebra as quaternions defined above.
(cid:50) Conjugation, (anti-)self-duality and norm:
Quaternionic conjugate of q will be denoted by q† and is defined
by
q† ≡ q e† (3.28)
µ µ
† †
e = e , e = −e (3.29)
0 0 i i
3.2-8
Consider now the products e e† and e†e . Due to the following
µ ν µ ν
group theoretical reasons, they have very simple interpretations:
If we go back to σ interpretation of quaternions, e e† is a
µ µ ν
quantity which transforms solely by SU(2) , like M e e†M†.
+ + µ ν +
Thus, it is a singlet under SU(2) and is therefore a mixture of
−
(0,0) and (1,0):
1 1 1 1
( , ) ⊗ ( , ) = (1,1) ⊕ (1,0) ⊕ +(0,1) ⊕ (0,0)
2 2 2 2
The former must be represented by δ and the latter should
µν
be a self-dual antisymmetric tensor, which will be denoted by
2iσ .
µν
Similarly, e†e transforms under SU(2) like M∗e†e MT and
µ ν − − µ ν −
is a mixture of (0,0) and (0,1). The latter is the anti-self-dual
part of SO(4) tensor.
In fact, one easily verifies the following relations:
e e† = δ + 2iσ (3.30)
µ ν µν µν
e†e = δ + 2iσ¯ (3.31)
µ ν µν µν
1
σ = (e e† − e e†) (3.32)
µν 4i µ ν ν µ
1
σ¯ = (e†e − e†e ) (3.33)
µν 4i µ ν ν µ
where σ and σ¯ are antisymmetric, hermitian and satisfy the
µν µν
3.2-9
following duality properties:
1
∗σ ≡ (cid:178) σ = σ self-dual(SD)
µν µνρσ ρσ µν
2
1
∗σ¯ ≡ (cid:178) σ¯ = −σ¯ anti-self-dual(ASD)
µν µνρσ ρσ µν
2
where (cid:178) ≡ 1
0123
(For example, ∗(e e†) = e e† = −e = e e† satisfying self-
0 1 2 3 1 0 1
duality. )
’t Hooft tensor: σ and σ¯ are traceless as 2×2 matrices.
µν µν
Thus, they can be expanded in terms of e , with a = 1,2,3.
a
Explicitly,
1 τ
σ ≡ ηa (ie ) = ηa a (3.34)
µν 2 µν a µν 2
ηa = δ δ − δ δ + (cid:178) (3.35)
µν µ0 νa ν0 µa 0aµν
1 τ
σ¯ ≡ η¯a (ie ) = η¯a a (3.36)
µν 2 µν a µν 2
η¯a = −(δ δ − δ δ ) + (cid:178) (3.37)
µν µ0 νa ν0 µa 0aµν
ηa ,η¯a are often called ’t Hooft tensors. In the construction
µν µν
of instanton solution, e will be regarded as the basis for the
a
gauge group SU(2). Thus,’t Hooft tensors intertwine the
spacetime and internal groups.
Properties of the ’t Hooft tensors:
η = (cid:178) , if µ,ν = 1,2,3
aµν aµν
η = δ
a0ν aν
η = −δ
aµ0 aµ
η = 0
a00
η¯ = (−1)δµ0+δν0η
aµν aµν
3.2-10