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World Scientific
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Published by
World Scientific Publishing Co. Pte. Ltd.
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Library of Congress Cataloging-in-Publication Data
Fioresi, Rita, 1966–
The Minkowski and conformal superspaces : the classical and quantum descriptions / by Rita
Fioresi (Università di Bologna, Italy), María Antonia Lledó (Universitat de València, Spain).
pages cm
Includes bibliographical references and index.
ISBN 978-9814566339 (hardcover : alk. paper)
1. Supersymmetry. 2. Generalized spaces. 3. Quantum groups. 4. Minkowski geometry.
I. Lledó, María Antonia. II. Title.
QC174.17.S9F56 2015
539.7'25--dc23
2015000733
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2015 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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Preface
This monograph originated from the desire to describe, in a mathemati-
cal rigorous way, the construction of the Minkowski and conformal super-
spacesashomogeneousspacesforthePoincar´eandconformalsupergroups,
respectively. Many authors, at different times, have contributed to the
development of the modern theory of the Minkowski superspace since the
pioneering works in supersymmetry [77,151,158].
Very little is novel in these notes. Our work has grown out from our
papers [26,27,66] on this subject and we have drawn heavily, especially
in the treatment of the infinitesimal setting, from the comprehensive and
beautiful monograph by V. S. Varadarajan[147].
Ideally these notes are aimed at graduate students in mathematics and
physics. We have made an effort to keep our treatment as elementary as
possible andself contained, providingthe reader with a terse, but intuitive
introduction to those advanced topics (like the theory of sheaves or the
functor of points approach) which are necessary for our purposes.
As for the physics part, we try to build the physical intuition from the
beginning, withintroductionstospecialrelativity,quantummechanicsand
even quantum field theory. This aims to describe the fundamentals, so
one cannotexpect a treatmentof supersymmetricfield theories here. Even
when our presentation greatly differs from the way in which supersymme-
try is usually introduced by physicists, we try to make contact with their
language.
WeareverygratefultoProf. V.S.Varadarajanforhisconstanthelpand
encouragement while this book was written. We have learnt this subject
following his 2000 seminar at UCLA and we are deeply indebted to him.
v
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vi The Minkowski and Conformal Superspaces
We also wish to thank Prof. A. Waldron, Prof. A. Schwarz and Dr. E.
Latini for their enthusiasm for this work; without them this book would
not have been written.
We finally wish to thank Prof. L. Andrianopoli, Prof. A. Brini, Prof.
C. Carmeli, Prof. A. Cattaneo, Prof. N. Ciccoli, Prof. F. Gavarini, Dr.
S. D. Kwok, Prof. L. Migliorini and Prof. J. Navarro-Salas for helpful
discussions andremarks. We also wantto thank the UCLA Departmentof
Mathematics, the Dipartimento di Matematica, Universit`a di Bologna and
theDepartamentdeF´ısicaTeo`rica,UniversitatdeVal`enciaforsupportand
hospitality during the realization of this work.
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Introduction
In the sixties and seventies particle physics underwent a very fast develop-
ment with many discoveries both, at the theoretical and the experimental
level. The Standard Model, the model for electromagnetic and nuclear
interactions, was taking its first steps.
There are two different nuclear interactions: the weak interaction and
the strong interaction.
The weak interaction is responsible for the radioactive decays of nuclei
of unstable isotopes of atoms by emission of particles and for the nuclear
fusion, in which two nuclei colliding at high speed form a new nucleus.
Nuclearfusionoftwohydrogennucleiintooneheliumnucleusistheprocess
that fuels the Sun. With the works of Glashow, Weinberg and Salam, the
electromagneticandtheweaknuclearinteractionswereunifiedintoasingle
theory called the theory of the electroweak interactions.
The strong nuclear interaction is the force that keeps together the pro-
tons and neutrons (nucleons) in the nucleus in spite of the electric repul-
sion of the positively chargedprotons. Not only nucleons, but other heavy
particles, generically called hadrons, were being discovered, and it seemed
that they could not all be fundamental particles. Instead, one could as-
sume the existence of only three fundamental particles, called quarks, that
would combine to produce all the different hadrons. The quarks are not
observed directly and their interactions are described by a theory that, in
its currentform, is called QuantumCromodynamics (QCD). QCD has not
been unified with the electroweak theory, at least not in the way in which
electromagnetic and weak interactions are unified, but they are both very
similar at the mathematical level. They are non abelian Yang-Mills the-
ories with gauge symmetry groups SU(2)×U(1) (electroweak) and SU(3)
(QCD). The word ‘non abelian’ simply means that the gauge symmetry
vii
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viii The Minkowski and Conformal Superspaces
groups are non abelian. Electroweaktheory and QCD form what is known
as the Standard Model (SM) of particle interactions. It is worth to point
out that quarks also undergo electroweak interactions.
Thisleavesoutthefourthinteraction,gravity,describedbyacompletely
different theory: Einstein’s General Relativity (GR). This is a theory in
which the geometry of the spacetime becomes the dynamical variable, and
itsmathematicaltreatmentgreatlydiffersfromtheSM.Nevertheless,since
itincorporatesthespecialtheoryofrelativity(onesaysata‘locallevel’)one
can think on the Poincar´e algebra (the Lorentz algebra plus translations)
as an algebra of infinitesimal symmetries.
In this state of things, physicists were looking for new constructions
using bigger symmetry groups to help with unresolved problems. Nowa-
days we know six different types of quarks: we say that quarks come in
six different ‘flavors’, up (u), down (d), strange (s), charm (c), bottom (b)
andtop(t), togetherwiththeir antiparticles,u¯,d¯,s¯,c¯,¯b,t¯. The firstquarks
discovered were the up and down quarks. They have similar masses and
there is an approximate SU(2) symmetry of the Standard Model in which
(u,d) conform a doublet of SU(2), while (u¯,d¯) are in the dual or contra-
gradient representation. This symmetry is called isospin: it is in fact an
oldacquaintance,sinceitismanifestedinthesimilarmassesandbehaviour
of the neutron (ddu) and the proton (uud), which also form a doublet of
SU(2). Wigner was the first to propose, back in 1936, a theory that mixed
the SU(2) isospin symmetry of nucleons together with the non relativistic
SU(2)spin,aclassicremnantofthespinofthePoincar´egroup. Wignerpro-
posedanSU(4)symmetrygroupthatwouldcontainbothSU(2)’s. Perhaps
wishfully, or ahead of his time, Wigner called the SU(4) representations or
multiplets of particles, supermultiplets.
The isospin group SU(2) was enhanced to SU(3) with the prediction of
the strange quark, s. Then, (u,d,s) would be a triplet of SU(3). Hadrons
then could be organized in other representations of SU(3) which are ob-
tained by tensoring the fundamental representation of SU(3) and its dual
several times. This successful classification of the hadrons was known as
theeightfold way,inreferencetothe8-dimensionaladjointrepresentationof
SU(3). Itwasproposedindependently byGell-MannandNe’emanin1961.
Soon after, Wigner’s supermultiplets were upgraded to representations of
an SU(6) symmetry group which would contain the SU(2) spin group and
the SU(3) flavor group [76,130].
The quarksu,d,s arethe lightquarks, being the c, b and t increasingly
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Introduction ix
heavier and behaving too differently as to suppose that there is an even
approximate symmetry among them.
Although this theory explained some facts, it presented problems of
interpretation regarding the spin-statistics connection. Nevertheless, this
lineofresearchcametoanendin1967,withtheno-gotheoremofColeman
and Mandula [33], which stated that, under reasonable assumptions, the
spacetime symmetries (the Poincar´e group in this case) and the ‘internal’
symmetriessuchasflavororgaugesymmetries,couldonlyappeartogether
combined as a direct product. This was a very strong result that also
seemedto putatheoreticalbarrierforthe unificationofGeneralRelativity
with the Standard Model.
Gol’fand and Likhtman [77] were the first to use a new object to try to
circumventtheColeman-Mandulatheorem. TheyproposedaLiesuperalge-
bra, as opposedto a Lie algebra,as the algebraof infinitesimal symmetries
of a physical theory. Lie superalgebras are vector spaces with a bilinear
bracket that is symmetric or antisymmetric depending on the elements on
which it acts. The Jacobi identity needs only a modification to work also
on Lie superalgebras.
Pioneering works on superalgebras applied to physical theories are
Volkov and Akulov [151], Wess and Zumino [158,159], Ferrara and Zu-
mino [49] and Salam and Strathdee [131,132] to mention just a few.
This prompted a development of physics in a different direction and
ultimatelyledtothediscoveryofsupersymmetrictheoriesandsupergravity.
In supergravity, all of the interactions stem out of a geometric principle so
they are all treated in the same way.
The most obvious restriction that physics imposes on the suitable Lie
superalgebras is that the elements or generators that have a symmetric
bracket (or anticommutator) among them must be in a spinorial repre-
sentation of the Lorentz algebra, so(1,3). These are representations that
do not lift to representations of the Lorentz group SO(1,3) but of its spin
group,SL(2,C)R,whichisitsdoublecoverand,consequently,hasthesame
Lie algebra. A simpler example of this phenomenon is found with the
group SO(3). It has the same Lie algebra as SU(2), its double (universal)
cover,butthe doubletofSU(2)isnotarepresentationofSO(3). Then, the
doublet is a spinorial representation of the orthogonal Lie algebra so(3).
The restriction comes from the spin-statistics connection: ordinary (non
spinorial) representations are associated to generators with antisymmetric
bracketandspinorialrepresentationstogeneratorswithsymmetricbracket
