How old supergravity is: thirty five years or more?
These notes is aimed at comparing two different approaches to theory of gravity with a fermionic gauge field, which possesses the invariance under special transformations called supersymmetry. One of these approaches is the approach by Ferrara-Freedman-Nieuwenhuizen-Deser-Zumino (FFNDZ) of 1976; the...
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irk-123456789-1114642017-01-11T03:02:45Z How old supergravity is: thirty five years or more? Nurmagambetov, A.J. Ядерная физика и элементарные частицы These notes is aimed at comparing two different approaches to theory of gravity with a fermionic gauge field, which possesses the invariance under special transformations called supersymmetry. One of these approaches is the approach by Ferrara-Freedman-Nieuwenhuizen-Deser-Zumino (FFNDZ) of 1976; the other one is that of Volkov-Soroka (VS) of 1974. The analysis is based on the standard concept of realizing Supergravity as a gauge theory for the super- Poincare group. We deliberately sacrifice the rigor of the proposed consideration in compare to the pioneering papers on D=4 N=1 Supergravity to make our presentation simple as much as possible. In effect we emphasize the differences between the above mentioned approaches. We keep out of making rigorous conclusions throughout the paper, and suggest the reader to get the own answer to the question posed in the title. Метою цих нотаток є порівняння двох різних підходів до теорії гравітації з ферміонним калібровочним полем, що інваріантна щодо спеціальних перетворень, які одержали назву суперсиметрії. Одним з цих підходів є підхід Феррари, Фрідмана, Ньювенхойзена, Дезера і Зуміно 1976 року; іншим є підхід Волкова і Сороки, запропонований в 1974 році. Аналіз грунтується на стандартній реалізації супергравітації як калібровочної теорії для групи супер-Пуанкаре. Строгість розгляду, в порівнянні з піонерськими роботами з D=4 N=1 супергравітації, свідомо жертвується на користь максимального спрощення в поданні матеріалу. Наслідком цього є чітке висвітлення відмінностей між двома порівнюваними підходами. Нотатки не містять ніяких категоричних висновків, навпаки, читачеві пропонується отримати власну відповідь на питання, що фігурує в назві роботи. Целью данных заметок является сравнение двух различных подходов к теории гравитации с фермионным калибровочным полем, обладающей инвариантностью относительно специальных преобразований, получивших название суперсимметрии. Одним из этих подходов является подход Феррары, Фридмана, Ньювенхойзена, Дезера и Зумино 1976 года; другим является подход Волкова и Сороки, предложенный в 1974 году. Анализ основывается на стандартной реализации супергравитации как калибровочной теории для группы супер-Пуанкаре. Строгость рассмотрения, по сравнению с пионерскими работами по D=4 N=1 супергравитации, сознательно жертвуется в пользу максимального упрощения в представлении материала. Следствием этого является четкое обозначение различий между двумя сравниваемыми подходами. Заметки не содержат никаких категорических выводов; напротив, читателю предлагается получить собственный ответ на вопрос, фигурирующий в названии работы. 2011 Article How old supergravity is: thirty five years or more? / A.J. Nurmagambetov // Вопросы атомной науки и техники. — 2011. — № 5. — С. 3-6. — Бібліогр.: 17 назв. — англ. 1562-6016 PACS: 12.60.Jv; 04.65.+e http://dspace.nbuv.gov.ua/handle/123456789/111464 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Ядерная физика и элементарные частицы Ядерная физика и элементарные частицы Nurmagambetov, A.J. How old supergravity is: thirty five years or more? Вопросы атомной науки и техники |
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These notes is aimed at comparing two different approaches to theory of gravity with a fermionic gauge field, which possesses the invariance under special transformations called supersymmetry. One of these approaches is the approach by Ferrara-Freedman-Nieuwenhuizen-Deser-Zumino (FFNDZ) of 1976; the other one is that of Volkov-Soroka (VS) of 1974. The analysis is based on the standard concept of realizing Supergravity as a gauge theory for the super- Poincare group. We deliberately sacrifice the rigor of the proposed consideration in compare to the pioneering papers on D=4 N=1 Supergravity to make our presentation simple as much as possible. In effect we emphasize the differences between the above mentioned approaches. We keep out of making rigorous conclusions throughout the paper, and suggest the reader to get the own answer to the question posed in the title. |
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Nurmagambetov, A.J. |
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Nurmagambetov, A.J. |
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Nurmagambetov, A.J. |
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How old supergravity is: thirty five years or more? |
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How old supergravity is: thirty five years or more? |
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How old supergravity is: thirty five years or more? |
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How old supergravity is: thirty five years or more? |
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How old supergravity is: thirty five years or more? |
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how old supergravity is: thirty five years or more? |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Ядерная физика и элементарные частицы |
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How old supergravity is: thirty five years or more? / A.J. Nurmagambetov // Вопросы атомной науки и техники. — 2011. — № 5. — С. 3-6. — Бібліогр.: 17 назв. — англ. |
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| fulltext |
NUCLEAR PHYSICS AND ELEMENTARY PARTICLES
HOW OLD SUPERGRAVITY IS: THIRTY FIVE YEARS OR
MORE?
A.J. Nurmagambetov∗
National Science Center ”Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
(Received August 18, 2011)
These notes is aimed at comparing two different approaches to theory of gravity with a fermionic gauge field, which
possesses the invariance under special transformations called supersymmetry. One of these approaches is the approach
by Ferrara-Freedman-Nieuwenhuizen-Deser-Zumino (FFNDZ) of 1976; the other one is that of Volkov-Soroka (VS)
of 1974. The analysis is based on the standard concept of realizing Supergravity as a gauge theory for the super-
Poincare group. We deliberately sacrifice the rigor of the proposed consideration in compare to the pioneering papers
on D=4 N=1 Supergravity to make our presentation simple as much as possible. In effect we emphasize the differences
between the above mentioned approaches. We keep out of making rigorous conclusions throughout the paper, and
suggest the reader to get the own answer to the question posed in the title.
PACS: 12.60.Jv; 04.65.+e
FOREWORD
This paper is based on the authors’ talk at the Spe-
cial Research Scientific Council of Kharkov Institute
of Physics and Technology in July of 2005 to Com-
memorate the 80th Anniversary of Academician D.V.
Volkov. My notes would never have appeared in print
if it were not an accident this summer. Suddenly and
very unexpectedly for us Vyacheslav Aleksandrovich
Soroka passed away. He was a great man, good friend
and colleague. Involved in many activities, related
to studies of various aspects of modern theoretical
physics, he took an active and direct part in stud-
ies of supersymmetric models, which was resulted,
in particular, in the Supergravity invention. That is
why I took the liberty to publish these notes, which,
I hope, will be useful in dating the Supergravity age.
1. INTRODUCTION
There is a conventional wisdom that the foundation of
Supergravity, or more precisely the simple N=1 D=4
Supergravity, took place in 1976 and began with two
seminal papers by Ferrara-Freedman-Nieuwenhuizen
[6] and by Deser-Zumino [7] (FFNDZ). This point of
view is widespread [1], [2] and without no doubt is
true. However, the history of Science learns us that
every significant innovation has its own, perhaps dra-
matic, pre-history, when a little gap in insight does
not allow predecessors to make a final step towards
a new discovery. The history of Science also learns
us that sometimes a discovery of one person is er-
roneously ascribed to another one as it happened for
instance with the Kamerlingh Onnes superconductiv-
ity, which was in fact observed by Gilles Holst [3]. It
is not the unique example of course, so in the light of
the above it is natural to ask could it be something
like that with the Supergravity invention?
The latter question gains a new insight once I re-
call that in the Former Soviet Union the Supergrav-
ity foundation is ascribed to D.V. Volkov and V.A.
Soroka in view of [4], [5]. I make the special accent on
the dates of these papers: 1973 and 1974. Hence, if
the results of these two papers by Volkov-Soroka (VS)
are correct, one should at least get a little doubt on
the real date of the Supergravity foundation.
To figure out the (in)correctness of the VS ap-
proach we have to trace the way of reasoning by VS
and that of FFNDZ back and to compare the for-
mulations to each other. I should emphasize that
such a comparison of two formulations (more pre-
cisely, VS to FFNDZ) has been done, see e.g. [8],
[9, 10], [11]. However, it is not so easy to realize the
arguments of these papers since they appeal to the
clear understanding the VS construction. Roots of
misunderstanding are mainly twofold: The VS con-
struction is based on the non-linear realization of the
super-Poincare group with its subsequent gauging,
the approach which is not so common, popular and
well-known for the present days audience (see how-
ever [12]); and the notation in [8]-[10], borrowed in
part from the original papers [4, 5], is either far from
the modern notation accepted in Supergravity, or de-
notes different objects. Therefore, to make things
clear we have to reformulate old results in a modern
and commonly accepted fashion.
I would not stick to any “preferred” point of view
in the analysis below. Rather I suggest the reader
to be a referee and to get the own answer to the
question posed in the title.
∗ajn@kipt.kharkov.ua
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2011, N5.
Series: Nuclear Physics Investigations (56), p.3-6.
3
2. SUPERGRAVITY – WHAT’S THIS?
To realize the announced program let us get started
with recalling what Supergravity is? I will refer to the
standard definition of Supergravity (cf. e.g. [2]): Su-
pergravity is a theory of gauge fields, which is invari-
ant under the local (i.e. with space-time dependent
parameters) supersymmetry. Any theory of Super-
gravity contains a spin 2 gauge field or graviton, –
this field is responsible for the gravitational interac-
tion, – together with its spin 3/2 super(symmetric)
partner, the so-called gravitino. Supergravity may
also include scalars, vector and antisymmetric ten-
sor gauge fields of spin zero and one, together with
their superpartners with spin 1/2. A field content of
Supergravities is strongly dependent on the number
of space-time dimensions, features of the considered
theory (simple, i.e. with N = 1 local supersymmetry,
or extended, i.e. with N > 1, Supergravity), gauged
or un-gauged Supergravity and so on. However, in
all cases the field content should be supersymmetric,
that means it has to provide the precise balance of the
bosonic and fermionic degrees of freedom, at least on
the mass-shell. The off-shell matching the bosonic
and fermionic degrees of freedom is also desired, but
not necessary.
There are many ways to Supergravity construct-
ing. A general, but not always simple, method of the
construction of Supergravity is based on the corre-
sponding superalgebra gauging (see [13] for details).
In general, the superalgebra possesses a complicated
structure, but it always includes the super-Poincare
algebra as a subalgebra. To reach conclusions on VS
vs. FFNDZ model, no need to deal with a compli-
cated superalgebra. It is enough to gauge the N=1
D=4 super-Poincare algebra to this end.
3. GAUGING THE SIMPLE
SUPER-POINCARE ALGEBRA
We follow the way of N=1 D=4 Supergravity con-
structing in the spirit of the Yang-Mills theory, as it
was done (but in more complicated manner) in [4, 5],
and was recently discussed in [14]. The starting point
is the N=1 D=4 super-Poincare algebra
[Pa, Pb] = 0, [Jab, Pc] = ηac Pb − ηbc Pa,
[Jab, Jcd] = ηac Jbd − ηbc Jad + ηbd Jac − ηad Jbc,
[Jab, Q
α] = −1
2
(γab)α
β Qβ , [Pa, Qβ ] = 0,
{Qα, Qβ} =
1
2
(γa)α
β Pa. (1)
The next step is to construct the Yang-Mills-type
connections
A = AAtA = eaPa +
1
2
ωabJab + QΨ, (2)
and their field strengths (curvatures)
F = FAtA = dA + A ∧A =
= T aPa +
1
2
RabJab + RαQα,
T a = dea + ωa
b eb − 1
2
Ψ̄γaΨ,
Rab = dωab + ωa
b ωcb,
Rα = dΨα +
1
4
ωab(γab)α
βΨβ ≡ DΨα. (3)
Using the superalgebra (1) it is easy to verify that
the local transformations of the connections
δA = Dλ = dλ+[A, λ}, λ = ρaPa +
1
2
κabJab +Qε,
(4)
which transform curvatures in the covariant way
δF ∼ [F, λ}, (5)
can be split on
i) translations
δtranslations ea = Dρa, δtranslations ωab = 0,
δtranslations Ψα = 0; (6)
ii) local Lorentz transformations
δLorentz ea = κa
be
b, δLorentz ωab = −Dκa
b,
δLorentz Ψα =
1
4
κabγabΨα; (7)
iii) local supersymmetry transformations
δSUSY ea =
1
2
ε̄γaΨ, δSUSY ωab = 0,
δSUSY Ψα = Dεα. (8)
Clearly, the Poincare superalgebra gauging nat-
urally leads to δSUSYωab = 0. The same transfor-
mation property of the connection follows from the
VS papers [4, 5]. However, this result contradicts the
FFNDZ supergravity construction [6], [7], that can be
treated as a manifestation of the difference between
the Yang-Mills and gravitational theories, and calls
into question (see for instance [14]) the correctness of
the VS approach.
One could stop here, since it seems to be unrea-
sonable to proceed further after observing such a dis-
crepancy. But this is not the end of the game. We
have to turn to the corresponding actions, having in
mind what we have figured out above.
4. FFNDZ VS. VS: ACTIONS ANALYSIS
Now consider the action of D=4 N=1 supergravity,
proposed in [6, 7]
SFFNDZ =
∫
M4
εabcde
aebRcd + 4Ψ̄γ5e
aγaDΨ. (9)
This action is not invariant under translations (6)
without the additional requirement
T a = dea + ωa
b eb − 1
2
Ψ̄γaΨ = 0. (10)
In effect
4
1) the connection is not an independent variable
anymore
ωab = ωab(e, Ψ); (11)
2) transformations of the connection are different
from that of followed from the algebraic consid-
eration; in particular
δSUSY ωab 6= 0; (12)
3) the relative coefficient between two terms of the
action (9) is completely fixed by the require-
ment of the action invariance under the local
supersymmetry transformations;
4) the algebra of the gauge transformations is
closed off-shell only by use of auxiliary fields;
5) the structure constants of the gauge transfor-
mations algebra become the structure functions
of fields. That leads to additional drawbacks
upon the quantization of the model.
Now let us turn to the corresponding part of the
Volkov-Soroka action [4, 5]
SV S =
∫
M4
α1εabcdE
aEbRcd + 4α2ψ̄γ5E
aγaDψ
(13)
with arbitrary coefficients α1, α2. Aside from usual
vielbeins ea, (13) involves additional ‘coordinates’ ξa
entering the ‘generalized vielbeins’
Ea = ea + Dξa − 1
2
Ψ̄γaθ − 1
4
Dθ̄γaθ, (14)
together with Goldstone-type fermionic coordinates
θα, entering the combination
ψα = Ψα + Dθα. (15)
That is why the Volkov-Soroka action is invariant un-
der translations (6) and the invariance is guaranteed
by the following transformations
δtranslation Ea = 0 ⇐⇒ δtranslation ea = Dρa,
δtranslation ξa = −ρa, δtranslation θα = 0 (16)
The VS action is manifestly invariant under the local
SUSY transformations (8) as well, that is provided
by
δSUSY Ea = 0, δSUSY ψα = 0,
δSUSY ξa =
1
4
ε̄γaθ, δSUSY θα = −εα. (17)
As the result:
a) the connection and the vielbein are independent
variables;
b) the local supersymmetry transformations of the
connection are not different from that of coming
from the superalgebra gauging, i.e.
δSUSY ωab = 0;
c) the gauge transformation algebra is closed off-
shell and without introducing auxiliary fields;
d) the relative coefficients between different terms
of the action are not fixed since the action is
constructed out the manifestly invariant un-
der the local supersymmetry transformations
forms;
e) the transformation property of the ‘coordinates’
ξa, θα is the same as the transformation of the
Goldstone fields, and they have no impact on
physics described by the model;
f) upon eliminating the Goldstone fields, (13) is
reduced to (9). Then, arbitrary coefficients of
(13) are fixed by the requirement of the local su-
persymmetry of the action and δSUSY ωab 6= 0
is required.
Before turning to final remarks, let me make an
additional comment on the structure of the VS ac-
tion (13). At a first sight it seems non plausible
and unnatural that the local supersymmetry does
not uniquely fix the relative coefficient between two
manifestly invariant under the local supersymmetry
transformations terms of the action. However, the
requirement of local supersymmetry is not always
enough to fix all the parameters entering the Super-
gravity action. As a counter-example I recall that the
supergravity-scalar fields coupling requires introduc-
ing a real function of scalar fields, the precise form
of which is not uniquely fixed by the supersymmetry
invariance arguments (cf. e.g. [15]).
5. FINAL REMARKS
To summarize, we have compared two different ap-
proaches to Supergravity by Volkov-Soroka and by
Ferrara-Freedman-Nieuwenhuizen-Deser-Zumino. I
have emphasized the differences in these approaches,
and have established the source of the differences.
As I have mentioned at the beginning of the paper,
I would not stick to any “preferred” point of view,
suggesting the reader to make own conclusions on the
main question of the paper. Let me finally note that
the Goldstone-type variables ξa entering (14), which
were appeared in [4, 5], are called now the Poincare
coordinates, and the N=1 D=4 Supergravity formu-
lation, which is invariant under the Poincare group
translations, is presently known (notably due to the
authors of [16]) as the Grignani-Nardelli-Stelle-West
formulation [17, 16].
Acknowledgements. Discussions with V.A. Soroka
on the history of Supergravity were very useful and
are kindly acknowledged. The author is very in-
debted to N.P. Merenkov and G.I. Gakh for fruitful
discussions. Work is supported in part by the Joint
DFFD-RFBR Project # F40.2/040.
5
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НАСКОЛЬКО СУПЕРГРАВИТАЦИЯ СТАРА: ТРИДЦАТЬ ПЯТЬ ЛЕТ ИЛИ
БОЛЬШЕ?
А.Ю. Нурмагамбетов
Целью данных заметок является сравнение двух различных подходов к теории гравитации с фермионным ка-
либровочным полем, обладающей инвариантностью относительно специальных преобразований, получивших
название суперсимметрии. Одним из этих подходов является подход Феррары, Фридмана, Ньювенхойзена, Де-
зера и Зумино 1976 года; другим является подход Волкова и Сороки, предложенный в 1974 году. Анализ осно-
вывается на стандартной реализации супергравитации как калибровочной теории для группы супер-Пуанкаре.
Строгость рассмотрения, по сравнению с пионерскими работами по D=4 N=1 супергравитации, сознательно
жертвуется в пользу максимального упрощения в представлении материала. Следствием этого является четкое
обозначение различий между двумя сравниваемыми подходами. Заметки не содержат никаких категорических
выводов; напротив, читателю предлагается получить собственный ответ на вопрос, фигурирующий в названии
работы.
НАСКIЛЬКИ СУПЕРГРАВIТАЦIЯ СТАРА: ТРИДЦЯТЬ П’ЯТЬ РОКIВ ЧИ
БIЛЬШЕ?
О.Ю. Нурмагамбетов
Метою цих нотаток є порiвняння двох рiзних пiдходiв до теорiї гравiтацiї з фермiонним калiбровочним по-
лем, що iнварiантна щодо спецiальних перетворень, якi одержали назву суперсиметрiї. Одним з цих пiдходiв
є пiдхiд Феррари, Фрiдмана, Ньювенхойзена, Дезера i Зумiно 1976 року; iншим є пiдхiд Волкова i Сороки,
запропонований в 1974 роцi. Аналiз грунтується на стандартнiй реалiзацiї супергравiтацiї як калiбровочної
теорiї для групи супер-Пуанкаре. Строгiсть розгляду, в порiвняннi з пiонерськими роботами з D=4 N=1 супер-
гравiтацiї, свiдомо жертвується на користь максимального спрощення в поданнi матерiалу. Наслiдком цього є
чiтке висвiтлення вiдмiнностей мiж двома порiвнюваними пiдходами. Нотатки не мiстять нiяких категоричних
висновкiв, навпаки, читачевi пропонується отримати власну вiдповiдь на питання, що фiгурує в назвi роботи.
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