22
3 Superfield Description of Three-Dimensional Supersymmetric Theories
sion of the action. However, in this case only the residual supersymmetry survives,
which makes the superfield description to be a bit senseless (for a discussion of
noncovariant gauges in superfield theories see e.g. [39]).
One more gauge invariant action in the three-dimensional superspace is the superfield analogue of the Chern-Simons term:
S C S =
m
2g 2
d
5 z A
α W α ,
(3.55)
which component structure is
S C S =
m
g 2
d
3 x(V
αβ f αβ − λ
α
λ α ).
(3.56)
As we noticed already, the f αβ is a vector dual to the stress tensor.
3. There is also an alternative free action for the spinor superfield initially introduced
in [40]. In this case, we have two spinor fields,
α and ¯
α whose component structure
is
α
= ψ
α
+ θ
α b + iθ β b
βα
− θ
2
φ
α
;
¯
α
= ¯
ψ
α
+ θ
α ¯
b + iθ β ¯
b
βα
− θ
2 ¯
φ
α
.
(3.57)
We can introduce the Dirac-like action for these fields:
S = −
d
5 z ¯
α
(i∂ αβ − MC αβ ))
β
.
(3.58)
The corresponding action for the component fields looks like
S M = S
(1/2)
M
+ S
(1)
M ,
(3.59)
where
S
(1/2)
M
=
d
3 x
φ
i γ
m
∂ m − M
ψ + ψ
i γ
m
∂ m − M
φ
,
(3.60)
S
(1)
M = −
d
3 x
1
2
ε
mnp b m ∂ n b p +
M
2
b
m b m + b∂
m b m + b∂
m b m − 2Mbb
,
(3.61)
We can eliminate the auxiliary field b using its equation of motion, thus
S
(1)
M = −
d
3 x
1
2
ε
mnp b m ∂ n b p +
M
2
b
m b m −
1
2M
∂
m b m
∂
m b m
, (3.62)