66
7 Summary
only logarithmic divergences). Moreover, there are known examples of completely
finite supersymmetric theories, the paradigmatic example is the N = 4 super-YangMills theory, where N is a number of supersymmetries (number of sets of generators
of supersymmetry). Clearly, this called interest to a possible supersymmetric extension of gravity, so, the supergravity (SUGRA) was introduced (see [127] for a review).
However, the N = 1 SUGRA is still non-renormalizable, therefore, the extensions
of SUGRA with larger values of N began to be introduced. The maximal N allowing for a consistent theory is 8, for SUGRA (for larger values of N , higher spin
fields arise, and they cannot be consistently coupled to gravity). It should be noted
also that the interest to SUGRA models with high N is motivated also by possible
applications of these theories to superstrings.
So, let us briefly review the most important results found within N = 8 SUGRA
obtained in series of papers by Bern, Dixon, Kosower and collaborators. In [128] it
was proved that the degree of divergence, at N = 8, in D dimensions and L loops,
is
ω = (D − 2)L − 10.
(7.1)
So we see that divergences in four dimensions can begin only from five-loop order!
It is interesting to note that the approach from the same paper allows to show that
the N = 4 super-Yang-Mills theory is all-loop finite.
Further, on the base of the unitarity cuts approach, in [129], it has been proved that
the four-point functions in N = 8 SUGRA satisfies the same finiteness condition in
the D-dimensional space-time
D <
6
L
+ 4,
(7.2)
which for D = 4 implies all-loop finiteness of these functions. Then, in [130], with
use of some identities applied for sets of more than 30 supergraphs, it was proved that
some extra cancellations occur, so, the finiteness of N = 8 SUGRA is achieved up to
four loops at D ≤ 5. Afterwards, in [131] it was proved that the five-loop correction
in this theory begins to diverge at D ≥ 24/5, so, in the four-dimensional space-time,
the theory is five-loop finite. Taking all together, we conclude that there is a natural
hope that N = 8 SUGRA is all-loop finite in D = 4. The next problem consists in
extracting some observable results for SUGRA (scattering amplitudes, corrections
to GR etc.) while, up to now, there are only some isolated conclusions.
We conclude out course with the ideas that, first, in study of gravity one still has
more questions that answers, second, apparently the most promising extensions of
gravity are the SUGRA, the nonlocal gravity and the HL gravity. However, each of
these modifications still has its difficulties which need to be solved. In principle,
there are some other approaches to gravity, for example, treating the gravity as
an emergent phenomenon caused by essentially quantum effects [132], asymptotic
safety also known as non-perturbative renormalizability, which allows to treat many
7 Summary
only logarithmic divergences). Moreover, there are known examples of completely
finite supersymmetric theories, the paradigmatic example is the N = 4 super-YangMills theory, where N is a number of supersymmetries (number of sets of generators
of supersymmetry). Clearly, this called interest to a possible supersymmetric extension of gravity, so, the supergravity (SUGRA) was introduced (see [127] for a review).
However, the N = 1 SUGRA is still non-renormalizable, therefore, the extensions
of SUGRA with larger values of N began to be introduced. The maximal N allowing for a consistent theory is 8, for SUGRA (for larger values of N , higher spin
fields arise, and they cannot be consistently coupled to gravity). It should be noted
also that the interest to SUGRA models with high N is motivated also by possible
applications of these theories to superstrings.
So, let us briefly review the most important results found within N = 8 SUGRA
obtained in series of papers by Bern, Dixon, Kosower and collaborators. In [128] it
was proved that the degree of divergence, at N = 8, in D dimensions and L loops,
is
ω = (D − 2)L − 10.
(7.1)
So we see that divergences in four dimensions can begin only from five-loop order!
It is interesting to note that the approach from the same paper allows to show that
the N = 4 super-Yang-Mills theory is all-loop finite.
Further, on the base of the unitarity cuts approach, in [129], it has been proved that
the four-point functions in N = 8 SUGRA satisfies the same finiteness condition in
the D-dimensional space-time
D <
6
L
+ 4,
(7.2)
which for D = 4 implies all-loop finiteness of these functions. Then, in [130], with
use of some identities applied for sets of more than 30 supergraphs, it was proved that
some extra cancellations occur, so, the finiteness of N = 8 SUGRA is achieved up to
four loops at D ≤ 5. Afterwards, in [131] it was proved that the five-loop correction
in this theory begins to diverge at D ≥ 24/5, so, in the four-dimensional space-time,
the theory is five-loop finite. Taking all together, we conclude that there is a natural
hope that N = 8 SUGRA is all-loop finite in D = 4. The next problem consists in
extracting some observable results for SUGRA (scattering amplitudes, corrections
to GR etc.) while, up to now, there are only some isolated conclusions.
We conclude out course with the ideas that, first, in study of gravity one still has
more questions that answers, second, apparently the most promising extensions of
gravity are the SUGRA, the nonlocal gravity and the HL gravity. However, each of
these modifications still has its difficulties which need to be solved. In principle,
there are some other approaches to gravity, for example, treating the gravity as
an emergent phenomenon caused by essentially quantum effects [132], asymptotic
safety also known as non-perturbative renormalizability, which allows to treat many