148
6 Conclusions
approach developed in [19, 20]. This method is based on defining superfields
as functions of bosonic space-time coordinates x
a , two sets of Grassmannian
coordinates θ
iα
, ¯
θ
i ˙
α with i = 1, 2 and spherical harmonics u
±i . Introducing of
the analytic superfield [19] allows to develop formulation of various theories in
terms of unconstrained N = 2 superfields and to avoid arising of component
fields with higher spins. The N = 2 and N = 4 SYM theories in the harmonic
superspace were formulated in [19, 20], the background field method for these
theories was introduced in [121–123], and examples of quantum calculations
are given in [111, 120, 124–126]. The most important results presented in these
papers are obtaining of the holomorphic action of N = 2 matter hypermultiplets
in the external N = 2 gauge superfield, the calculation of the one-loop nonholomorphic effective potential in the N = 4 SYM theory and proof of its absence
in higher loops, finding of the one-loop effective action for N = 4 SYM theory
for a constant strength tensor F ab , and computing of a superconformal anomaly
of N = 2 matter interacting with N = 2 supergravity. During last years, other
important results of these investigations were the calculations of two-loop contributions depending on derivatives of N = 2 SYM strength W [127, 128] and
of the one-loop effective action in the matter hypermultiplet sector [129], and the
development of a quantum approach for N = 3 SYM theory [130] (the N = 3
harmonic superspace technique was introduced in the paper [60]). Also, it is
necessary to mention the intensive studies of three-dimensional extended supersymmetric theories, especially N = 6 and N = 8 Chern-Simons theories (see
e.g. [4]), where also the harmonic superspace approach has been successfully
developed and applied (see e.g. [131]).
2. Noncommutative supersymmetric theories. Noncommutative theories have been
intensively studied during recent years. The concept of the space-time noncommutativity was introduced to quantum field theory being motivated by some
consequences of the D-branes theory [132] and by the interest to behaviour of
quantum theories at very small distances where quantum fluctuations of geometry are essential. The consideration of supersymmetric noncommutative theories is quite natural. During last years some interesting results in studying of
noncommutative supersymmetric theories were obtained but they were mostly
based on component approach. The first superfield results were the calculation
of leading (∼F
4 ) correction to the one-loop effective action for N = 4 SYM
theory [108] and the formulation of a supergraph technique for the noncommutative Wess-Zumino model [29]. Further, the quantum superfield studies for noncommutative extensions of the Wess-Zumino model [133] and four-dimensional
superfield QED [103] and SYM theories [104] were carried out. There are also
various examples of calculations in three-dimensional supersymmetric field theories presented in [15, 54], and some of them have been considered in this
book. These theories were shown to be consistent in the sense of the absence of
the nonintegrable UV/IR infrared divergences. Thus, we can speak about constructing of consistent noncommutative generalizations of the supersymmetric
theories of electromagnetic, strong and weak interactions. Therefore, the next
most important problem could consist in the development of the noncommu-
6 Conclusions
approach developed in [19, 20]. This method is based on defining superfields
as functions of bosonic space-time coordinates x
a , two sets of Grassmannian
coordinates θ
iα
, ¯
θ
i ˙
α with i = 1, 2 and spherical harmonics u
±i . Introducing of
the analytic superfield [19] allows to develop formulation of various theories in
terms of unconstrained N = 2 superfields and to avoid arising of component
fields with higher spins. The N = 2 and N = 4 SYM theories in the harmonic
superspace were formulated in [19, 20], the background field method for these
theories was introduced in [121–123], and examples of quantum calculations
are given in [111, 120, 124–126]. The most important results presented in these
papers are obtaining of the holomorphic action of N = 2 matter hypermultiplets
in the external N = 2 gauge superfield, the calculation of the one-loop nonholomorphic effective potential in the N = 4 SYM theory and proof of its absence
in higher loops, finding of the one-loop effective action for N = 4 SYM theory
for a constant strength tensor F ab , and computing of a superconformal anomaly
of N = 2 matter interacting with N = 2 supergravity. During last years, other
important results of these investigations were the calculations of two-loop contributions depending on derivatives of N = 2 SYM strength W [127, 128] and
of the one-loop effective action in the matter hypermultiplet sector [129], and the
development of a quantum approach for N = 3 SYM theory [130] (the N = 3
harmonic superspace technique was introduced in the paper [60]). Also, it is
necessary to mention the intensive studies of three-dimensional extended supersymmetric theories, especially N = 6 and N = 8 Chern-Simons theories (see
e.g. [4]), where also the harmonic superspace approach has been successfully
developed and applied (see e.g. [131]).
2. Noncommutative supersymmetric theories. Noncommutative theories have been
intensively studied during recent years. The concept of the space-time noncommutativity was introduced to quantum field theory being motivated by some
consequences of the D-branes theory [132] and by the interest to behaviour of
quantum theories at very small distances where quantum fluctuations of geometry are essential. The consideration of supersymmetric noncommutative theories is quite natural. During last years some interesting results in studying of
noncommutative supersymmetric theories were obtained but they were mostly
based on component approach. The first superfield results were the calculation
of leading (∼F
4 ) correction to the one-loop effective action for N = 4 SYM
theory [108] and the formulation of a supergraph technique for the noncommutative Wess-Zumino model [29]. Further, the quantum superfield studies for noncommutative extensions of the Wess-Zumino model [133] and four-dimensional
superfield QED [103] and SYM theories [104] were carried out. There are also
various examples of calculations in three-dimensional supersymmetric field theories presented in [15, 54], and some of them have been considered in this
book. These theories were shown to be consistent in the sense of the absence of
the nonintegrable UV/IR infrared divergences. Thus, we can speak about constructing of consistent noncommutative generalizations of the supersymmetric
theories of electromagnetic, strong and weak interactions. Therefore, the next
most important problem could consist in the development of the noncommu-