14
3 Superfield Description of Three-Dimensional Supersymmetric Theories
Projecting this transformation to components, we can obtain the transformation laws
for components of any superfield. We note that there is only one set of generators
Q α , as it must be in the case of the N = 1 supersymmetry.
The (super)covariant derivative D α of any superfield must be consistent with the
supersymmetry, i.e. under supersymmetry transformations (3.10) it also should vary
as a superfield, δ D α = D α δδ. To satisfy this condition, the D α must anticommute
with the supersymmetry generators Q β , i.e. {D α , Q β } = 0, and commute with the
translation generators P αβ = i∂ αβ . Also, spinor supercovariant derivatives must be
linear in the simple derivatives ∂ α , ∂ αβ in order to satisfy the Leibnitz rule. All these
properties are satisfied if the D α looks like
D α = ∂ α + iθ
β
∂ βα .
(3.11)
The spinor supercovariant derivatives D α defined in such a way possess the following
properties:
{D α , D β } = 2i∂ αβ ; [D α , D β ] = −2C αβ D
2
.
(3.12)
After summation of two these expressions we arrive at the following very important
relation
D α D β = i∂ αβ − C αβ D
2
.
(3.13)
This identity is employed to carry out D-algebra transformations which are fundamental for simplifying the forms of the contributions of the supergraphs. Many
examples will be presented further. One can also notice that the first expression in
(3.12) can be interpreted in the sense that the superspace possesses a fundamental
torsion since the anticommutator of the covariant derivatives is proportional to the
supercovariant derivative as occurs in spacetimes with a torsion.
Let us obtain some other useful identities (some of details presented here can be
found also in [37]; note, however, that our conventions differ from those ones used
in [37]). First of all, let us note that the totally antisymmetric object with three spinor
indices should vanish. Indeed, such an object is equal to zero if any two of its indices
coincide, and since spinor indices can take only two values, that is, 1 and 2, this
object should have at least two coinciding indices. If such an object is constructed
through antisymmetrizing the product D α D β D γ , we get
D α D β D γ + D β D γ D α +D γ D α D β − D α D γ D β − D β D α D γ
−D γ D β D α = 0.
(3.14)
Contracting this expression to C
βγ we get
2(D α D β D
β
+ D β D
β D α + D
β D α D β ) = 0.
(3.15)
3 Superfield Description of Three-Dimensional Supersymmetric Theories
Projecting this transformation to components, we can obtain the transformation laws
for components of any superfield. We note that there is only one set of generators
Q α , as it must be in the case of the N = 1 supersymmetry.
The (super)covariant derivative D α of any superfield must be consistent with the
supersymmetry, i.e. under supersymmetry transformations (3.10) it also should vary
as a superfield, δ D α = D α δδ. To satisfy this condition, the D α must anticommute
with the supersymmetry generators Q β , i.e. {D α , Q β } = 0, and commute with the
translation generators P αβ = i∂ αβ . Also, spinor supercovariant derivatives must be
linear in the simple derivatives ∂ α , ∂ αβ in order to satisfy the Leibnitz rule. All these
properties are satisfied if the D α looks like
D α = ∂ α + iθ
β
∂ βα .
(3.11)
The spinor supercovariant derivatives D α defined in such a way possess the following
properties:
{D α , D β } = 2i∂ αβ ; [D α , D β ] = −2C αβ D
2
.
(3.12)
After summation of two these expressions we arrive at the following very important
relation
D α D β = i∂ αβ − C αβ D
2
.
(3.13)
This identity is employed to carry out D-algebra transformations which are fundamental for simplifying the forms of the contributions of the supergraphs. Many
examples will be presented further. One can also notice that the first expression in
(3.12) can be interpreted in the sense that the superspace possesses a fundamental
torsion since the anticommutator of the covariant derivatives is proportional to the
supercovariant derivative as occurs in spacetimes with a torsion.
Let us obtain some other useful identities (some of details presented here can be
found also in [37]; note, however, that our conventions differ from those ones used
in [37]). First of all, let us note that the totally antisymmetric object with three spinor
indices should vanish. Indeed, such an object is equal to zero if any two of its indices
coincide, and since spinor indices can take only two values, that is, 1 and 2, this
object should have at least two coinciding indices. If such an object is constructed
through antisymmetrizing the product D α D β D γ , we get
D α D β D γ + D β D γ D α +D γ D α D β − D α D γ D β − D β D α D γ
−D γ D β D α = 0.
(3.14)
Contracting this expression to C
βγ we get
2(D α D β D
β
+ D β D
β D α + D
β D α D β ) = 0.
(3.15)