3.7 On the Noncommutativity in the Fermionic Sector of the Superspace
45
Let us suppose that the spinor coordinates of the superspace, instead of the Grassmann algebra, form the Clifford algebra, thus obeying the following anticommutation
relations:
{θ
α
, θ
β
} =
αβ
.
(3.163)
It is clear that, to maintain the key relation of the supersymmetry algebra, that is,
(3.9), we should deform the supersymmetry generators, which will imply in a need
to deform the spinor supercovariant derivatives via introducing second-derivative
terms into them, therefore the Leibnitz rule will not be satisfied. Therefore, we must
try to extend the supersymmetry (indeed, in the four-dimensional case the similar
deformation of the superspace had implied in a partial breaking of the supersymmetry [31]).
In the first way, we introduce an additional set of the supersymmetry generators
thus considering the extended, N = 2 supersymmetry, with the generators are
Q
i
α = i∂
i
α + θ
iβ
∂ βα ,
(3.164)
with i = 1, 2 is a number of the set of spinor coordinates (and, hence—of the supersymmetry generators). Then, we suppose that only in one of the sets of the spinor coordinates, say i = 2, the anticommutation relations are deformed in a manner (3.163)
while for i = 1 they persist to be the same. Thus, only the unbroken generators Q
1
α
satisfy the usual anticommutation relation {Q
1
α , Q
1
β } = 2i∂ αβ . The supercovariant
derivatives anticommuting with them are
D
i
α = ∂
i
α + iθ
iβ
∂ βα ,
(3.165)
with the only deformed anticommutation relation
{D
2
α , D
2
β } = 2i∂ αβ −
γ δ
∂ αγ ∂ βδ ,
(3.166)
while all other anticommutation relations between the supercovariant derivatives
persist to be the same.
The Moyal product compatible with this definition of the supersymmetry is defined
as
1 (z) ∗ 2 (z) = exp(−
1
2
αβ D
2
α1 D
2
β2 )) 1 (z 1 )) 2 (z 2 )| z 1 =z 2 =z .
(3.167)
It is easy to see that in this case already the quadratic action will suffer noncommutative deformation which is a highly unusual situation (one should remind that for
the usual bosonic Moyal product (3.153), the quadratic action does not suffer any
deformation, while the interaction vertices are deformed).
45
Let us suppose that the spinor coordinates of the superspace, instead of the Grassmann algebra, form the Clifford algebra, thus obeying the following anticommutation
relations:
{θ
α
, θ
β
} =
αβ
.
(3.163)
It is clear that, to maintain the key relation of the supersymmetry algebra, that is,
(3.9), we should deform the supersymmetry generators, which will imply in a need
to deform the spinor supercovariant derivatives via introducing second-derivative
terms into them, therefore the Leibnitz rule will not be satisfied. Therefore, we must
try to extend the supersymmetry (indeed, in the four-dimensional case the similar
deformation of the superspace had implied in a partial breaking of the supersymmetry [31]).
In the first way, we introduce an additional set of the supersymmetry generators
thus considering the extended, N = 2 supersymmetry, with the generators are
Q
i
α = i∂
i
α + θ
iβ
∂ βα ,
(3.164)
with i = 1, 2 is a number of the set of spinor coordinates (and, hence—of the supersymmetry generators). Then, we suppose that only in one of the sets of the spinor coordinates, say i = 2, the anticommutation relations are deformed in a manner (3.163)
while for i = 1 they persist to be the same. Thus, only the unbroken generators Q
1
α
satisfy the usual anticommutation relation {Q
1
α , Q
1
β } = 2i∂ αβ . The supercovariant
derivatives anticommuting with them are
D
i
α = ∂
i
α + iθ
iβ
∂ βα ,
(3.165)
with the only deformed anticommutation relation
{D
2
α , D
2
β } = 2i∂ αβ −
γ δ
∂ αγ ∂ βδ ,
(3.166)
while all other anticommutation relations between the supercovariant derivatives
persist to be the same.
The Moyal product compatible with this definition of the supersymmetry is defined
as
1 (z) ∗ 2 (z) = exp(−
1
2
αβ D
2
α1 D
2
β2 )) 1 (z 1 )) 2 (z 2 )| z 1 =z 2 =z .
(3.167)
It is easy to see that in this case already the quadratic action will suffer noncommutative deformation which is a highly unusual situation (one should remind that for
the usual bosonic Moyal product (3.153), the quadratic action does not suffer any
deformation, while the interaction vertices are deformed).