6 Conclusions
149
tative supersymmetric generalization for the last fundamental interaction—the
gravitational one. However, this problem is certainly extremely difficult (see
discussion of the problem e.g. in [134, 135]).
3. Noncommutative superspace One more approach in the superfield quantum
theory is based on using of the noncommutative superspace [31]. Within it,
the fermionic superspace coordinates form the Clifford algebra instead of the
Grassmann one, which, in particular, leads to the modified construction of the
Moyal product. Within the framework of this approach, the generalizations of
the Wess-Zumino (see e.g. [136] and references therein), gauge (see e.g. [137]
and references therein) and general chiral superfield models [138] were studied.
A three-dimensional version of the noncommutative superspace [55] has been
presented in the Sect. 3.7 of this review.
4. Lorentz-breaking supersymmetric theories. As it is known (see e.g. [139]), the
breaking of the Lorentz symmetry is introduced through additive terms proportional to constant vectors, or, in general, constant tensors, which introduce
privileged directions in space-time. Therefore, the natural question is—how
one can implement Lorentz symmetry breaking in supersymmetric theories?
There are three known manners to do it. Within one of them, we introduce the
Lorentz-breaking operator at the level of the superfield action, without modifying structures of superfields or a supersymmetry algebra. This approach,
motivated by the idea to construct a supersymmetric generalization of HoravaLifshitz-like theories displaying a space-time anisotropy, is presented in [140].
Within another manner, one introduces a new superfield so that some its components are proportional to Lorentz-breaking vectors (tensors) which allows to
construct supersymmetric extensions of known Lorentz-breaking terms, such as
the Carroll-Field-Jackiw term and the aether term [141]. Finally, the third manner
is based on the Kostelecky-Berger deformation of the supersymmetry algebra
through the replacement ∂ m → ∂ m + k mn ∂
n within supersymmetry generators
(4.11), with k mn are constant tensors defined in such a manner that |k mn | | 1 for
any m, n to ensure smallness of the Lorentz symmetry breaking. It was explicitly
demonstrated that in superfield theories based on this approach, all perturbative
superfield calculations can be performed with no more difficulties that in usual
superfield theories, see e.g. [143].
Besides of all this, there are a lot of applications of the superfield approach to various problems of supersymmetric quantum field theory, e.g. to studying of AdS/CFT
correspondence which was carried out mostly on the base of a component approach,
and of course to considering of many problems originated from strings/M-theory.
As a final conclusion, we can suppose that superfield approach in quantum field
theory is a very perspective methodology, and there are a lot of ways for its development and more applications, including contexts of phenomenology and even condensed matter.
149
tative supersymmetric generalization for the last fundamental interaction—the
gravitational one. However, this problem is certainly extremely difficult (see
discussion of the problem e.g. in [134, 135]).
3. Noncommutative superspace One more approach in the superfield quantum
theory is based on using of the noncommutative superspace [31]. Within it,
the fermionic superspace coordinates form the Clifford algebra instead of the
Grassmann one, which, in particular, leads to the modified construction of the
Moyal product. Within the framework of this approach, the generalizations of
the Wess-Zumino (see e.g. [136] and references therein), gauge (see e.g. [137]
and references therein) and general chiral superfield models [138] were studied.
A three-dimensional version of the noncommutative superspace [55] has been
presented in the Sect. 3.7 of this review.
4. Lorentz-breaking supersymmetric theories. As it is known (see e.g. [139]), the
breaking of the Lorentz symmetry is introduced through additive terms proportional to constant vectors, or, in general, constant tensors, which introduce
privileged directions in space-time. Therefore, the natural question is—how
one can implement Lorentz symmetry breaking in supersymmetric theories?
There are three known manners to do it. Within one of them, we introduce the
Lorentz-breaking operator at the level of the superfield action, without modifying structures of superfields or a supersymmetry algebra. This approach,
motivated by the idea to construct a supersymmetric generalization of HoravaLifshitz-like theories displaying a space-time anisotropy, is presented in [140].
Within another manner, one introduces a new superfield so that some its components are proportional to Lorentz-breaking vectors (tensors) which allows to
construct supersymmetric extensions of known Lorentz-breaking terms, such as
the Carroll-Field-Jackiw term and the aether term [141]. Finally, the third manner
is based on the Kostelecky-Berger deformation of the supersymmetry algebra
through the replacement ∂ m → ∂ m + k mn ∂
n within supersymmetry generators
(4.11), with k mn are constant tensors defined in such a manner that |k mn | | 1 for
any m, n to ensure smallness of the Lorentz symmetry breaking. It was explicitly
demonstrated that in superfield theories based on this approach, all perturbative
superfield calculations can be performed with no more difficulties that in usual
superfield theories, see e.g. [143].
Besides of all this, there are a lot of applications of the superfield approach to various problems of supersymmetric quantum field theory, e.g. to studying of AdS/CFT
correspondence which was carried out mostly on the base of a component approach,
and of course to considering of many problems originated from strings/M-theory.
As a final conclusion, we can suppose that superfield approach in quantum field
theory is a very perspective methodology, and there are a lot of ways for its development and more applications, including contexts of phenomenology and even condensed matter.