2
1 Introduction
metry allows for the cancellation not only of ultraviolet divergences, as usual, but
also of dangerous infrared divergences arising due to the UV/IR mixing mechanism
(see e.g. [15]). Further, the concept of the extended supersymmetry has been introduced, and various versions of an extended supersymmetric formalism have been
elaborated. Within this lecture course, nevertheless, we concentrate on the N = 1
superfield formalism which is known as one of the most universal tools for studying
the supersymmetric field theories.
Now, let us briefly describe the main steps in development of the superfield
methodology. The first example of the successful application of the superfield concept was the model proposed by Wess and Zumino in their seminal paper [11] where
the simplest superfield model, further called the Wess-Zumino model, has been formulated. Further, in [9] they introduced the superfield gauge model, that is, the SYM
theory. Intensive studies of different issues related quantum aspects of these theories began to be carried out. One of the most important results is the finiteness of
N = 4 SYM theory that was proved in [16] (for discussions of finiteness of the
supersymmetric gauge theories see also [17]), which implied in a strongest interest
to the supergauge theories. Then, the superfield supergravity was formulated [18].
In 1984, the first consistent methodology possessing an explicit extended (N = 2)
supersymmetry has been developed, that is, the harmonic superspace [19, 20]. At the
same year, the superfield approach has been successfully applied to the superstring
theory [21], which emphasized the importance of this methodology within the string
context. A bit earlier, the superfield formulation has been developed for the supersymmetric theories in a three-dimensional space-time [22, 23], which further manifested
itself as a very convenient laboratory for studies of different issues related to the
supersymmetry. It should be noted that actually, the interest to three-dimensional
field theories, besides of their simplicity and better renormalization properties, is
driven by studies of graphene. In our context, it worth mentioning that the superfield
supersymmetric model for graphene has been formulated as well [24].
A new epoch for studies of the superfield theories began in 1991 when, in the
papers [25, 26], the chiral quantum contributions to the effective action were discussed for the first time. These papers called an attention to the superfield methodology for evaluating the effective potential whose development has been carried out
in a series of works initiated by the paper [27], where this methodology has been
successfully applied to the Wess-Zumino model. Further, the success of the paper
[28] strongly increased the interest to various issues, both perturbative and nonperturbative ones, related to supergauge theories, especially those ones possessing
the extended supersymmetry.
Among the most successful applications of the superfield methodology, also its use
for the noncommutative supersymmetric field theories deserves to be mentioned. The
famous paper [29] established the fact that, since the space-time noncommutativity
does not affect the anticommuting (Grassmannian) coordinates of superfields, the
superfield methodology can be used as a very powerful instrument to deal with the
famous problem of the UV/IR mixing [30] known for generating the new infrared
divergences which are able to break the perturbative expansion. It was shown first
in [29] that the supersymmetric extension of field theories, implying in improving
1 Introduction
metry allows for the cancellation not only of ultraviolet divergences, as usual, but
also of dangerous infrared divergences arising due to the UV/IR mixing mechanism
(see e.g. [15]). Further, the concept of the extended supersymmetry has been introduced, and various versions of an extended supersymmetric formalism have been
elaborated. Within this lecture course, nevertheless, we concentrate on the N = 1
superfield formalism which is known as one of the most universal tools for studying
the supersymmetric field theories.
Now, let us briefly describe the main steps in development of the superfield
methodology. The first example of the successful application of the superfield concept was the model proposed by Wess and Zumino in their seminal paper [11] where
the simplest superfield model, further called the Wess-Zumino model, has been formulated. Further, in [9] they introduced the superfield gauge model, that is, the SYM
theory. Intensive studies of different issues related quantum aspects of these theories began to be carried out. One of the most important results is the finiteness of
N = 4 SYM theory that was proved in [16] (for discussions of finiteness of the
supersymmetric gauge theories see also [17]), which implied in a strongest interest
to the supergauge theories. Then, the superfield supergravity was formulated [18].
In 1984, the first consistent methodology possessing an explicit extended (N = 2)
supersymmetry has been developed, that is, the harmonic superspace [19, 20]. At the
same year, the superfield approach has been successfully applied to the superstring
theory [21], which emphasized the importance of this methodology within the string
context. A bit earlier, the superfield formulation has been developed for the supersymmetric theories in a three-dimensional space-time [22, 23], which further manifested
itself as a very convenient laboratory for studies of different issues related to the
supersymmetry. It should be noted that actually, the interest to three-dimensional
field theories, besides of their simplicity and better renormalization properties, is
driven by studies of graphene. In our context, it worth mentioning that the superfield
supersymmetric model for graphene has been formulated as well [24].
A new epoch for studies of the superfield theories began in 1991 when, in the
papers [25, 26], the chiral quantum contributions to the effective action were discussed for the first time. These papers called an attention to the superfield methodology for evaluating the effective potential whose development has been carried out
in a series of works initiated by the paper [27], where this methodology has been
successfully applied to the Wess-Zumino model. Further, the success of the paper
[28] strongly increased the interest to various issues, both perturbative and nonperturbative ones, related to supergauge theories, especially those ones possessing
the extended supersymmetry.
Among the most successful applications of the superfield methodology, also its use
for the noncommutative supersymmetric field theories deserves to be mentioned. The
famous paper [29] established the fact that, since the space-time noncommutativity
does not affect the anticommuting (Grassmannian) coordinates of superfields, the
superfield methodology can be used as a very powerful instrument to deal with the
famous problem of the UV/IR mixing [30] known for generating the new infrared
divergences which are able to break the perturbative expansion. It was shown first
in [29] that the supersymmetric extension of field theories, implying in improving