46
3 Superfield Description of Three-Dimensional Supersymmetric Theories
In the second way, we propose the following generators:
Q
1
α = i∂
1
α + θ
2β
∂ βα ,
Q
2
α = i∂
2
α + θ
1β
∂ βα .
(3.168)
Then, we again impose the nontrivial anticommutation relation {θ
2α
, θ
2β
} =
αβ .
The Moyal product is again chosen in the form (3.167). It is easy to see that in
this case the quadratic action is not deformed, and the only impact of the nontrivial
anticommutation relations will present in arising of additive vertices of interaction,
just as in [31]. This approach has received further development in the case of N =
2 supersymmetry where the resulting supersymmetry algebra is rather similar to
the four-dimensional N = 1 supersymmetry algebra [56]. However, studying of
quantum effects within both these approaches is a completely open problem.
3.8 Conclusion
In this chapter we gave a brief introduction to the properties of three-dimensional
supersymmetric field theories within the superfield approach. We described the structure of the most important three-dimensional supermultiplets, that is, scalar and
spinor ones. We developed a detailed description of properties of the supercovariant derivatives and of the superfield approach for calculating quantum corrections,
including the prescriptions for obtaining the effective action in one-loop and higherloop orders. The examples we presented show that supersymmetry indeed allows
to improve essentially the renormalization properties of the field theories. Really,
almost all supersymmetric field theories formulated in three-dimensional space-time
are one-loop finite except of the exotic theories with an effective dynamics providing
a nontrivial asymptotic behavior of the effective propagators, such as, for example,
the three-dimensional nonlinear sigma model [42] and C P
N −1 model [15]. Moreover, the three-dimensional supersymmetric QED is explicitly finite in all loop orders
(it was argued in [37] and explicitly proved in [41]).
This improvement of the renormalization properties plays an important role in
the context of the noncommutative field theories. Indeed, it is well known that the
noncommutative theories suffer from arising of nonintegrable infrared divergences
due to the UV/IR mixing mechanism which converts part of ultraviolet divergences
to infrared singularities whose presence in a general case can break the perturbative
expansion [30]. At the same time, we have shown that the supersymmetry improves
convergence of the Feynman diagrams, leaving, for many theories, only logarithmic
UV divergences which in the noncommutative case are just converted to harmless
logarithmic IR singularities whose presence does not break the perturbative expansion. This is the key result shown in the papers [15]. At the same time, we have
observed that the formalism of the supercovariant D-algebra is applicable in the
noncommutative case as well as in the commutative one, for example, the calcula-
3 Superfield Description of Three-Dimensional Supersymmetric Theories
In the second way, we propose the following generators:
Q
1
α = i∂
1
α + θ
2β
∂ βα ,
Q
2
α = i∂
2
α + θ
1β
∂ βα .
(3.168)
Then, we again impose the nontrivial anticommutation relation {θ
2α
, θ
2β
} =
αβ .
The Moyal product is again chosen in the form (3.167). It is easy to see that in
this case the quadratic action is not deformed, and the only impact of the nontrivial
anticommutation relations will present in arising of additive vertices of interaction,
just as in [31]. This approach has received further development in the case of N =
2 supersymmetry where the resulting supersymmetry algebra is rather similar to
the four-dimensional N = 1 supersymmetry algebra [56]. However, studying of
quantum effects within both these approaches is a completely open problem.
3.8 Conclusion
In this chapter we gave a brief introduction to the properties of three-dimensional
supersymmetric field theories within the superfield approach. We described the structure of the most important three-dimensional supermultiplets, that is, scalar and
spinor ones. We developed a detailed description of properties of the supercovariant derivatives and of the superfield approach for calculating quantum corrections,
including the prescriptions for obtaining the effective action in one-loop and higherloop orders. The examples we presented show that supersymmetry indeed allows
to improve essentially the renormalization properties of the field theories. Really,
almost all supersymmetric field theories formulated in three-dimensional space-time
are one-loop finite except of the exotic theories with an effective dynamics providing
a nontrivial asymptotic behavior of the effective propagators, such as, for example,
the three-dimensional nonlinear sigma model [42] and C P
N −1 model [15]. Moreover, the three-dimensional supersymmetric QED is explicitly finite in all loop orders
(it was argued in [37] and explicitly proved in [41]).
This improvement of the renormalization properties plays an important role in
the context of the noncommutative field theories. Indeed, it is well known that the
noncommutative theories suffer from arising of nonintegrable infrared divergences
due to the UV/IR mixing mechanism which converts part of ultraviolet divergences
to infrared singularities whose presence in a general case can break the perturbative
expansion [30]. At the same time, we have shown that the supersymmetry improves
convergence of the Feynman diagrams, leaving, for many theories, only logarithmic
UV divergences which in the noncommutative case are just converted to harmless
logarithmic IR singularities whose presence does not break the perturbative expansion. This is the key result shown in the papers [15]. At the same time, we have
observed that the formalism of the supercovariant D-algebra is applicable in the
noncommutative case as well as in the commutative one, for example, the calcula-