24
3 Superfield Description of Three-Dimensional Supersymmetric Theories
we get
W α =
1
2
D
β D α A β −
i
2
[A
β
, D β A α ] −
1
6
[A
β
, {A β , A α }].
(3.69)
The W α defined in such a way is not invariant under transformation (3.64). Instead,
it is transformed covariantly by the rule
W α → e
−i K W α e
i K
,
(3.70)
or, in the infinitesimal form,
W α → W α + i[W α , K ].
(3.71)
Now, let us consider the Bianchi identity
{∇ α , [∇ β , ∇ γ δ ]} + {∇ β , [∇ γ δ , ∇ α ]} + [∇ γ δ , {∇ α , ∇ β }] = 0.
(3.72)
After contractions with the symbols C
αγ and C
βδ it gives
{∇
(α
, [∇
β)
, ∇ αβ ]} = −6{∇
α
, W α } = 0.
(3.73)
This is the non-Abelian generalization of the transversality condition (3.49).
The most natural definition of the non-Abelian gauge invariant action, i.e. the
action of the SYM theory is similar to the Abelian one:
S SY M =
1
2g 2 tr
d
5 zW
α W α .
(3.74)
Using the expression (3.69) and adding the gauge-fixing action
S g f = −
1
4g 2 ξ
tr
d
5 z(D
α A α )D
2
(D
β A β ),
(3.75)
one can write down the exact form of total action of A α superfield:
S total =
1
2g 2 tr
d
5 z
1
2
(1 +
1
ξ
)A
α
A α −
1
2
(1 −
1
ξ
)A
α i∂ αβ D
2 A
β
+
+
1
g 2 tr
d
5 z
−
i
4
D
γ D
α A γ [A
β
, D β A α ] −
1
12
D
γ D
α A γ [A
β
, {A β , A α }] −
−
1
8
[A
γ
, D γ A
α
][A
β
, D β A α ] +
i
12
[A
γ
, D γ A
α
][A
β
, {A β , A α }] +
+
1
72
[A
γ
, {A γ , A
α
}][A
β
, {A β , A α }]
.
(3.76)