Chapter 3
Superfield Description
of Three-Dimensional Supersymmetric
Theories
In this chapter we describe the superfield formalism for three-dimensional supersymmetric field theory models. Afterwards, we describe various approaches to calculate
quantum corrections in three-dimensional superfield theories, including the noncommutative ones. These approaches are illustrated with different examples discussed in
details.
3.1 Definitions and Conventions
The basic concept of supersymmetric field theory consists in the existence of some
essentially new symmetry transformations with a fermionic parameter which mix
fermionic and bosonic dynamical variables of the theory. To provide a nontrivial
connection of the supersymmetry transformations with usual Poincaré transformations, we also suggest that the anticommutator of two supersymmetry generators
differs from zero. In the most used, and simplest, versions of the supersymmetry
algebra, including the versions considered in this book, this anticommutator is proportional to a space-time translation.
In this chapter we follow conventions of [23]: in the three-dimensional space-time
we choose the Minkowski metric of the form η mn = diag(− + +), and the Dirac
matrices are the 2 × 2 matrices whose explicit form is: (γ
0
)
α
β = −iσ
2
, (γ
1
)
α
β =
σ
1
, (γ
2
)
α
β = σ
3 , with {γ
m
, γ
n
} = 2η
mn . We use bispinor notations based on converting any vector index into two spinor indices by the rule A
m
→ A
αβ
= A
m
(γ m )
αβ . We
note that the Dirac matrices with two lower indices are (γ
m
) αβ = (−1 2 , −σ
3
, σ
1
)
are symmetric, hence all vectors (in particular, coordinates x
m and corresponding
derivatives ∂ m ) can be mapped into symmetric bispinors. To raise and lower the
spinor indices we use the Hermitian Levi-Civita-like C symbol: C αβ = −i αβ =
0 −i
i 0
= −C
αβ , with ψ
α
= C
αβ
ψ β , ψ α = ψ
β C βα (“north-western” convention).
Also, one has C
αβ C βγ = −δ
α
γ , and C
αβ C αβ = 2. We define ψ
2
=
1
2
ψ
α
ψ α , and use
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. Petrov, Quantum Superfield Supersymmetry, Fundamental Theories of Physics 202,
https://doi.org/10.1007/978-3-030-68136-4_3
11
Superfield Description
of Three-Dimensional Supersymmetric
Theories
In this chapter we describe the superfield formalism for three-dimensional supersymmetric field theory models. Afterwards, we describe various approaches to calculate
quantum corrections in three-dimensional superfield theories, including the noncommutative ones. These approaches are illustrated with different examples discussed in
details.
3.1 Definitions and Conventions
The basic concept of supersymmetric field theory consists in the existence of some
essentially new symmetry transformations with a fermionic parameter which mix
fermionic and bosonic dynamical variables of the theory. To provide a nontrivial
connection of the supersymmetry transformations with usual Poincaré transformations, we also suggest that the anticommutator of two supersymmetry generators
differs from zero. In the most used, and simplest, versions of the supersymmetry
algebra, including the versions considered in this book, this anticommutator is proportional to a space-time translation.
In this chapter we follow conventions of [23]: in the three-dimensional space-time
we choose the Minkowski metric of the form η mn = diag(− + +), and the Dirac
matrices are the 2 × 2 matrices whose explicit form is: (γ
0
)
α
β = −iσ
2
, (γ
1
)
α
β =
σ
1
, (γ
2
)
α
β = σ
3 , with {γ
m
, γ
n
} = 2η
mn . We use bispinor notations based on converting any vector index into two spinor indices by the rule A
m
→ A
αβ
= A
m
(γ m )
αβ . We
note that the Dirac matrices with two lower indices are (γ
m
) αβ = (−1 2 , −σ
3
, σ
1
)
are symmetric, hence all vectors (in particular, coordinates x
m and corresponding
derivatives ∂ m ) can be mapped into symmetric bispinors. To raise and lower the
spinor indices we use the Hermitian Levi-Civita-like C symbol: C αβ = −i αβ =
0 −i
i 0
= −C
αβ , with ψ
α
= C
αβ
ψ β , ψ α = ψ
β C βα (“north-western” convention).
Also, one has C
αβ C βγ = −δ
α
γ , and C
αβ C αβ = 2. We define ψ
2
=
1
2
ψ
α
ψ α , and use
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. Petrov, Quantum Superfield Supersymmetry, Fundamental Theories of Physics 202,
https://doi.org/10.1007/978-3-030-68136-4_3
11