4.1 General Properties of the Four-Dimensional Superspace
53
d
4
θF(x, θ, ¯
θ) =
1
16
∂
2
∂θ 2
∂
2
∂ ¯
θ 2
F(x, θ, ¯
θ) =
1
16
F(x, θ, ¯
θ)| θ 2 ¯
θ 2 ;
d
2
θG(x, θ) = −
1
4
∂
2
∂θ 2 G(x, θ) = −
1
4
G(x, θ)| θ 2 .
(4.10)
Here | θ 2 , | θ 2 ¯
θ 2 denotes the corresponding component of the superfield. Of course,
differentiations with respect to Grassmannian coordinates anticommute.
The supersymmetry generators are required to satisfy the anticommutation relations (4.7) and possess several realizations in terms of
∂
∂x m and
∂
∂θ α
,
∂
∂ ¯
θ ˙
α
, e.g.
¯
Q ˙
α = i(
∂
∂ ¯
θ ˙
α
− iθ
α
(σ
m
) α ˙
α ∂ m ), Q α = i(
∂
∂θ α + i ¯
θ
˙
β
( ¯
σ
m
) ˙
βα ∂ m ).
(4.11)
This realization, or, as is the same, this representation of the supersymmetry algebra
is not unique, other forms of these generators can be introduced as well. However, we
emphasize again that all its possible representations must satisfy the relations (4.7).
The spinor supercovariant derivatives D A also must be constructed from
∂
∂x m and
∂
∂θ α
,
∂
∂ ¯
θ ˙
α
. They should anticommute with generators Q α , ¯
Q ˙
α which provides that
D A is transformed covariantly, i.e. according to (4.6), for any superfield we can
write
D A δ = δ(D A ) = (Q + ¯
¯
Q)D A ,
which implies that {D α , Q β } = { ¯
D ˙
α , Q β } = {D α , ¯
Q ˙
β } = { ¯
D ˙
α , ¯
Q ˙
β } = 0. For example, if generators of the supersymmetry are chosen in the form (4.11), the corresponding supercovariant derivatives are realized as
¯
D ˙
α = −i ¯
Q ˙
α + 2iθ
α
∂ α ˙
α =
∂
∂ ¯
θ ˙
α
+ iθ
α
(σ
m
) α ˙
α ∂ m ,
D α = −i Q α − 2i ¯
θ
˙
α
∂ α ˙
α =
∂
∂θ α − i ¯
θ
˙
β
( ¯
σ
m
) ˙
βα ∂ m .
(4.12)
The spinor supercovariant derivatives satisfy the following anticommutation relations
{D α , ¯
D ˙
α } = 2i∂ α ˙
α ; {D α , D β } = { ¯
D ˙
α , ¯
D ˙
β } = 0.
(4.13)
So we defined the operations of integration and differentiation in the superspace. Further we will use the integral measure for the complete superspace d
8 z = d
4 xd
2
θd
2 ¯
θ,
the integral measure for the chiral subspace d
6 z=d
4 xd
2
θ and the integral measure
for the antichiral one d
6
¯
z=d
4 xd
2 ¯
θ. We also can use the identities D
2
θ
2
= ¯
D
2 ¯
θ
2
= − 4.
Now, let us introduce the Grassmannian delta function. We suggest that it must
satisfy the condition analogous to that one for a standard delta function
d
4
θ
δ
4
(θ − θ
) f (θ
) = f (θ).
(4.14)
53
d
4
θF(x, θ, ¯
θ) =
1
16
∂
2
∂θ 2
∂
2
∂ ¯
θ 2
F(x, θ, ¯
θ) =
1
16
F(x, θ, ¯
θ)| θ 2 ¯
θ 2 ;
d
2
θG(x, θ) = −
1
4
∂
2
∂θ 2 G(x, θ) = −
1
4
G(x, θ)| θ 2 .
(4.10)
Here | θ 2 , | θ 2 ¯
θ 2 denotes the corresponding component of the superfield. Of course,
differentiations with respect to Grassmannian coordinates anticommute.
The supersymmetry generators are required to satisfy the anticommutation relations (4.7) and possess several realizations in terms of
∂
∂x m and
∂
∂θ α
,
∂
∂ ¯
θ ˙
α
, e.g.
¯
Q ˙
α = i(
∂
∂ ¯
θ ˙
α
− iθ
α
(σ
m
) α ˙
α ∂ m ), Q α = i(
∂
∂θ α + i ¯
θ
˙
β
( ¯
σ
m
) ˙
βα ∂ m ).
(4.11)
This realization, or, as is the same, this representation of the supersymmetry algebra
is not unique, other forms of these generators can be introduced as well. However, we
emphasize again that all its possible representations must satisfy the relations (4.7).
The spinor supercovariant derivatives D A also must be constructed from
∂
∂x m and
∂
∂θ α
,
∂
∂ ¯
θ ˙
α
. They should anticommute with generators Q α , ¯
Q ˙
α which provides that
D A is transformed covariantly, i.e. according to (4.6), for any superfield we can
write
D A δ = δ(D A ) = (Q + ¯
¯
Q)D A ,
which implies that {D α , Q β } = { ¯
D ˙
α , Q β } = {D α , ¯
Q ˙
β } = { ¯
D ˙
α , ¯
Q ˙
β } = 0. For example, if generators of the supersymmetry are chosen in the form (4.11), the corresponding supercovariant derivatives are realized as
¯
D ˙
α = −i ¯
Q ˙
α + 2iθ
α
∂ α ˙
α =
∂
∂ ¯
θ ˙
α
+ iθ
α
(σ
m
) α ˙
α ∂ m ,
D α = −i Q α − 2i ¯
θ
˙
α
∂ α ˙
α =
∂
∂θ α − i ¯
θ
˙
β
( ¯
σ
m
) ˙
βα ∂ m .
(4.12)
The spinor supercovariant derivatives satisfy the following anticommutation relations
{D α , ¯
D ˙
α } = 2i∂ α ˙
α ; {D α , D β } = { ¯
D ˙
α , ¯
D ˙
β } = 0.
(4.13)
So we defined the operations of integration and differentiation in the superspace. Further we will use the integral measure for the complete superspace d
8 z = d
4 xd
2
θd
2 ¯
θ,
the integral measure for the chiral subspace d
6 z=d
4 xd
2
θ and the integral measure
for the antichiral one d
6
¯
z=d
4 xd
2 ¯
θ. We also can use the identities D
2
θ
2
= ¯
D
2 ¯
θ
2
= − 4.
Now, let us introduce the Grassmannian delta function. We suggest that it must
satisfy the condition analogous to that one for a standard delta function
d
4
θ
δ
4
(θ − θ
) f (θ
) = f (θ).
(4.14)