Perturbative and non-perturbative infrared behavior of ...

2. Nonanticommutative superspace
where A, B, C, D are functions of the superspace variables, Ā, ¯ C are the hermitian conjugates of
A and C respectively and ¯ B, ¯ D are hermitian operators.
While the covariance under Lorentz transformation is manifest, we require (2.1.2) to be covariant
under (super)translations
θ ′α = θ α + ǫ α
¯θ ′ ˙α = ¯ θ ˙α + ¯ǫ ˙α
x ′α ˙α = x α ˙α + a α ˙α − i
2
ǫ α¯ θ ˙α + ¯ǫ ˙α θ α
(2.1.3)
which constraints the functional dependence of the functions A, B, C, D upon the coordinates.
By performing a supertranslation on the coordinates (2.1.2) and computing the new algebra,
one finds that A and B are independent of the coordinates and C and D are independent of the
bosonic coordinates. More precisely, we get the algebra
where
θ α ,θ β
x α ˙α ,θ β
x α ˙α , ¯ θ ˙ β
x α ˙α ,x β ˙ β
= A αβ
= iC α ˙αβ (θ, ¯ θ)
= i ¯ C α ˙α ˙ β (θ, ¯ θ)
,
¯θ ˙α , ¯ θ ˙ β
= Ā ˙α ˙ β
, θ α , ¯ θ ˙α α ˙α
= B
= iD α ˙αβ ˙ β (θ, ¯ θ) (2.1.4)
C α ˙αβ (θ, ¯ θ) = C α ˙αβ − 1
2 θαB β ˙α − 1
2 ¯ θ ˙α A αβ
D α ˙αβ ˙ β
(θ, θ) ¯ α ˙αβ
= D ˙ β i
−
− i
4
2
θ β ¯ C α ˙α ˙ β − ¯ θ ˙α C β ˙ βα − θ α ¯ C β ˙ β ˙α + ¯ θ ˙ βC α ˙αβ
θ α Ā ˙α ˙ β θ β + θ α B β ˙α¯ θ ˙ β + ¯ θ ˙α B α ˙ β θ β + ¯ θ ˙α A αβ ¯ θ ˙ β
and A, B, C and D are constant functions.
2.2 Euclidean
(2.1.5)
In the Minkowski case it is impossible to break only half of the supersymmetries due to the
conjugation rules of the spinorial variables. It is known that relaxing these constraints amounts
to deal with extended supersymmetry or consider a space with euclidean signature. The main
difference with respect to the Minkowski one relies on the reality conditions satisfied by the
fermionic variables. In euclidean signature a reality condition on spinors is applicable only in the
case of extended supersymmetry. The simplest case is then the N = 2 superspace.
The spinors of the N = 2 euclidean superspace satisfy the pseudo-Majorana condition
10
(θ α i )∗ = θ i α ≡ Cij θ β
j Cβα , ( ¯ θ ˙α,i ) ∗ = ¯ θ ˙α,i ≡ ¯ θ ˙ β,j C ˙ β ˙α Cji (2.2.6)