12
3 Superfield Description of Three-Dimensional Supersymmetric Theories
the identity A [α B β] = −C αβ A
γ B γ following from the properties of the irreducible
representations of the Lorentz group, i.e. the (anti)symmetrization will be henceforth
defined without the 1/n! factor. The reader should note the presence of
1
2
factor in all
“square” contractions like D
2
=
1
2
D
α D α , W
2
=
1
2
W
α W α , etc. This is the difference
from the conventions for the three-dimensional superspace proposed by Ruiz Ruiz
and Nieuwenhuizen in [37].
To formulate the superspace, we start with introducing spinor coordinates θ
α with
α = 1, 2. These coordinates are transformed under the spinor representation of threedimensional Lorentz group, i.e. SO(1, 2) group (the spinor representation of the
Lorentz group in the three-dimensional space-time is given by the SL(2, R) group).
In a general case, there can be N sets of spinor coordinates θ
iα , with i = 1, . . . N .
However, throughout this book, both in three- and four-dimensional cases, we will
consider only the N = 1 superspace, i.e. we assume that there is only one set of
spinor coordinates and, correspondingly, only one set of supersymmetry generators.
The spinor coordinates θ
α , satisfying the Grassmannian anticommutation condition
{θ
α
, θ
β
} = 0, together with the usual bosonic coordinates x
m parametrize the superspace.
1
To develop field theory on superspace we must introduce integration and differentiation on superspace, i.e. with respect to Grassmannian coordinates [38]. We can
introduce left ∂ L and right ∂ R derivatives with respect to Grassmannian coordinates as
∂ L
∂θ α i
(θ
α 1 θ
α i−1 θ
α i θ
α i+1 . . . θ
α n ) = (−1)
i−1
(θ
α 1 θ
α i−1 θ
α i+1 . . . θ
α n );
∂ R
∂θ α i
(θ
α 1 θ
α i−1 θ
α i θ
α i+1 . . . θ
α n ) = (−1)
n−i+1
(θ
α 1 θ
α i−1 θ
α i+1 . . . θ
α n ).
(3.1)
Therefore these derivatives differ only by a sign factor. So, it is natural to choose
one of them, for example, the left one, and use it henceforth. Thus, we can define the
derivatives with respect to usual and Grassmannian coordinates as follows:
∂ α ≡
∂
∂θ α ; ∂ α θ
β
= δ
β
α ;
∂ αβ x
γ δ
= δ
γ
(α δ
δ
β) ; =
1
2
∂ αβ ∂
αβ
.
(3.2)
Here A (α B β) ≡
1
2
(A α B β + A β B α ) is a symmetrized product of spinors A α and B β .
The superfield is a (general) function of the superspace coordinates which can be
introduced in the form the Taylor series in θ which is finite because of the anticommuting nature of θ
α :
f (x, θ) = f 0 (x) + f
α
1 (x)θ α + f 2 (x)θ
2
,
(3.3)
1 In the Sect. 3.7 we will briefly discuss the simplest deformation of the anticommutation relation,
that is, {θ α , θ β } = C αβ , in that case the spinor coordinates form the Clifford algebra instead of the
Grassmann algebra; this methodology has been originally introduced in [31].
3 Superfield Description of Three-Dimensional Supersymmetric Theories
the identity A [α B β] = −C αβ A
γ B γ following from the properties of the irreducible
representations of the Lorentz group, i.e. the (anti)symmetrization will be henceforth
defined without the 1/n! factor. The reader should note the presence of
1
2
factor in all
“square” contractions like D
2
=
1
2
D
α D α , W
2
=
1
2
W
α W α , etc. This is the difference
from the conventions for the three-dimensional superspace proposed by Ruiz Ruiz
and Nieuwenhuizen in [37].
To formulate the superspace, we start with introducing spinor coordinates θ
α with
α = 1, 2. These coordinates are transformed under the spinor representation of threedimensional Lorentz group, i.e. SO(1, 2) group (the spinor representation of the
Lorentz group in the three-dimensional space-time is given by the SL(2, R) group).
In a general case, there can be N sets of spinor coordinates θ
iα , with i = 1, . . . N .
However, throughout this book, both in three- and four-dimensional cases, we will
consider only the N = 1 superspace, i.e. we assume that there is only one set of
spinor coordinates and, correspondingly, only one set of supersymmetry generators.
The spinor coordinates θ
α , satisfying the Grassmannian anticommutation condition
{θ
α
, θ
β
} = 0, together with the usual bosonic coordinates x
m parametrize the superspace.
1
To develop field theory on superspace we must introduce integration and differentiation on superspace, i.e. with respect to Grassmannian coordinates [38]. We can
introduce left ∂ L and right ∂ R derivatives with respect to Grassmannian coordinates as
∂ L
∂θ α i
(θ
α 1 θ
α i−1 θ
α i θ
α i+1 . . . θ
α n ) = (−1)
i−1
(θ
α 1 θ
α i−1 θ
α i+1 . . . θ
α n );
∂ R
∂θ α i
(θ
α 1 θ
α i−1 θ
α i θ
α i+1 . . . θ
α n ) = (−1)
n−i+1
(θ
α 1 θ
α i−1 θ
α i+1 . . . θ
α n ).
(3.1)
Therefore these derivatives differ only by a sign factor. So, it is natural to choose
one of them, for example, the left one, and use it henceforth. Thus, we can define the
derivatives with respect to usual and Grassmannian coordinates as follows:
∂ α ≡
∂
∂θ α ; ∂ α θ
β
= δ
β
α ;
∂ αβ x
γ δ
= δ
γ
(α δ
δ
β) ; =
1
2
∂ αβ ∂
αβ
.
(3.2)
Here A (α B β) ≡
1
2
(A α B β + A β B α ) is a symmetrized product of spinors A α and B β .
The superfield is a (general) function of the superspace coordinates which can be
introduced in the form the Taylor series in θ which is finite because of the anticommuting nature of θ
α :
f (x, θ) = f 0 (x) + f
α
1 (x)θ α + f 2 (x)θ
2
,
(3.3)
1 In the Sect. 3.7 we will briefly discuss the simplest deformation of the anticommutation relation,
that is, {θ α , θ β } = C αβ , in that case the spinor coordinates form the Clifford algebra instead of the
Grassmann algebra; this methodology has been originally introduced in [31].