3.8 Conclusion
47
tions of the noncommutative analogue of the supergraphs studied in Sect. 3.4, differ
only by multiplying of all contributions by the common phase factor, which is equal
to 1 in the case of the coupling in the fundamental representation and
1
4
sin
2
(k ∧ p),
where k ∧ p = k m
mn p n , and
mn is a noncommutativity matrix, in the adjoint
representation case (see [15] for more details).
We should note that neither the supersymmetric multiplets (scalar and spinor
one) nor the actions presented in this section are unique possible ones. The important example of the action for the spinor supersymmetric multiplet is the Dirac-like
action whose properties were studied in [40]. Another important example of the multiplet and the corresponding action is the three-dimensional superfield supergravity
discussed in [23]. However, detailed perturbative study of this theory was not yet
carried out.
Let us briefly discuss the most important actual lines of studies in the threedimensional supersymmetric theories. Presently, the main line of interest in the
three-dimensional supersymmetry is related to the extended supersymmetric ChernSimons theories, especially, N = 6 and N = 8 ones, which are known to be superconformally invariant and thus finite [4]. Another interesting line of study of these
theories is based on the essentially N = 2 supersymmetric description of such theories, whose properties are very similar to the usual N = 1 supersymmetry in fourdimensional space-time [57]. Different aspects of the N -extended supersymmetric
Chern-Simons theories are presented in [58]. Besides of this, one more very important
line of studies in these theories could consist in a detailed investigation of possible
applications of three-dimensional supersymmetry within the condensed matter context where one of the first steps has been done in [24].
We close this chapter by mentioning of the noncommutative superspace where
the anticommutation relations between fermionic superspace coordinates are the
Clifford ones instead of the Grassmann ones. However, such a formulation seems
to be realized only by paying a price of a partial supersymmetry breaking, i.e. to
proceed with this formulation in the three-dimensional space one should start with
the N -extended supersymmetric theories, with further some of the supersymmetries
are broken. A first attempt of developing such a formulation is presented in [55],
however, this issue certainly requires more studies.
47
tions of the noncommutative analogue of the supergraphs studied in Sect. 3.4, differ
only by multiplying of all contributions by the common phase factor, which is equal
to 1 in the case of the coupling in the fundamental representation and
1
4
sin
2
(k ∧ p),
where k ∧ p = k m
mn p n , and
mn is a noncommutativity matrix, in the adjoint
representation case (see [15] for more details).
We should note that neither the supersymmetric multiplets (scalar and spinor
one) nor the actions presented in this section are unique possible ones. The important example of the action for the spinor supersymmetric multiplet is the Dirac-like
action whose properties were studied in [40]. Another important example of the multiplet and the corresponding action is the three-dimensional superfield supergravity
discussed in [23]. However, detailed perturbative study of this theory was not yet
carried out.
Let us briefly discuss the most important actual lines of studies in the threedimensional supersymmetric theories. Presently, the main line of interest in the
three-dimensional supersymmetry is related to the extended supersymmetric ChernSimons theories, especially, N = 6 and N = 8 ones, which are known to be superconformally invariant and thus finite [4]. Another interesting line of study of these
theories is based on the essentially N = 2 supersymmetric description of such theories, whose properties are very similar to the usual N = 1 supersymmetry in fourdimensional space-time [57]. Different aspects of the N -extended supersymmetric
Chern-Simons theories are presented in [58]. Besides of this, one more very important
line of studies in these theories could consist in a detailed investigation of possible
applications of three-dimensional supersymmetry within the condensed matter context where one of the first steps has been done in [24].
We close this chapter by mentioning of the noncommutative superspace where
the anticommutation relations between fermionic superspace coordinates are the
Clifford ones instead of the Grassmann ones. However, such a formulation seems
to be realized only by paying a price of a partial supersymmetry breaking, i.e. to
proceed with this formulation in the three-dimensional space one should start with
the N -extended supersymmetric theories, with further some of the supersymmetries
are broken. A first attempt of developing such a formulation is presented in [55],
however, this issue certainly requires more studies.