Foundations of Supersymmetry Claudio Carmeli Lauren Caston Rita Fioresi
- Hardcover: 301 pages
- Language: English
- ISBN : 978-3-03719-097-5
Supersymmetry wa s discovered by physicists in the 1970s. The mathematical treatment
of it began much later and grew out of the works notably of Berezin, Kostant, Leites,
Manin, Bernstein, Freed, Deligne, Morgan, Varadarajan and others. These works are
all in what one may call the differential category and contain many additional references
to the subject.
This monograph has grown out of the desire to present a moderately brief and
focussed account of the mathematical foundations of supersymmetry both in the dif-ferential and algebraic categories. Our view is that supergeometry and super Lie theory
are beautiful areas and deserve much attention.
Our intention was not to write an encyclopedic treatment of supersymmetry but to
supply only the foundational material that will allow the reader to penetrate the more
advanced papers in the wide literature on this subject. For this reason we do not treat the
differential and symplectic supergeometry and we are unable to give a comprehensive
treatment of the representation theory of Lie supergroups and Lie superalgebras, which
can be found in more advanced papers by Kac, Serganova , Penkov, Duflo, Cassinelli
et al. and so on.
Our work is primarily directed to second or third year graduate students who have
taken a one year graduate course in algebra and a beginning course in Lie groups and Lie
algebras. We have provided a discussion without proofs of the classical theory, which
will serve as a departure point for our supergeometric treatment. Our book can very
well be used as a one-semester course or a participating seminar on supersymmetry,
directed to second and third year graduate students.
The language used in this monograph is that of the functor of points. Since this lan-guage is not always familiar even to second-year graduate students we have attempted
to explain it even at the level of classical geometry. Apart from being the most natural
medium for understanding supergeometry, it is also, remarkably enough, the language
closest to the physicists’ method of working with supersymmetry.
We wish to thank professor V. S. Varadarajan for introducing us to this beautiful
part of mathematics. He has truly inspired us through his insight and deep under-standing of the subject. We also wish to thank Dr. L. Balduzzi, Prof. G. Cassinelli,
Prof. A. Cattaneo, Prof. M. Duflo, Prof. F. Gavarini, Prof. A. Kresch, Prof. M. A. Lledo,
Prof. L. Migliorini, Prof. I. M. Musson, Prof. V. Ovsienko, Dr. E. Petracci, Prof. A. Vis-toli and Prof. A. Zubkov for helpful remarks. We also want to thank the UCLA De-partment of Mathematics, the Dipartimento di Matematica, Università di Bologna, and
the Dipartimento di Fisica, Università di Genova, for support and hospitality during
the realization of this work.
of it began much later and grew out of the works notably of Berezin, Kostant, Leites,
Manin, Bernstein, Freed, Deligne, Morgan, Varadarajan and others. These works are
all in what one may call the differential category and contain many additional references
to the subject.
This monograph has grown out of the desire to present a moderately brief and
focussed account of the mathematical foundations of supersymmetry both in the dif-ferential and algebraic categories. Our view is that supergeometry and super Lie theory
are beautiful areas and deserve much attention.
Our intention was not to write an encyclopedic treatment of supersymmetry but to
supply only the foundational material that will allow the reader to penetrate the more
advanced papers in the wide literature on this subject. For this reason we do not treat the
differential and symplectic supergeometry and we are unable to give a comprehensive
treatment of the representation theory of Lie supergroups and Lie superalgebras, which
can be found in more advanced papers by Kac, Serganova , Penkov, Duflo, Cassinelli
et al. and so on.
Our work is primarily directed to second or third year graduate students who have
taken a one year graduate course in algebra and a beginning course in Lie groups and Lie
algebras. We have provided a discussion without proofs of the classical theory, which
will serve as a departure point for our supergeometric treatment. Our book can very
well be used as a one-semester course or a participating seminar on supersymmetry,
directed to second and third year graduate students.
The language used in this monograph is that of the functor of points. Since this lan-guage is not always familiar even to second-year graduate students we have attempted
to explain it even at the level of classical geometry. Apart from being the most natural
medium for understanding supergeometry, it is also, remarkably enough, the language
closest to the physicists’ method of working with supersymmetry.
We wish to thank professor V. S. Varadarajan for introducing us to this beautiful
part of mathematics. He has truly inspired us through his insight and deep under-standing of the subject. We also wish to thank Dr. L. Balduzzi, Prof. G. Cassinelli,
Prof. A. Cattaneo, Prof. M. Duflo, Prof. F. Gavarini, Prof. A. Kresch, Prof. M. A. Lledo,
Prof. L. Migliorini, Prof. I. M. Musson, Prof. V. Ovsienko, Dr. E. Petracci, Prof. A. Vis-toli and Prof. A. Zubkov for helpful remarks. We also want to thank the UCLA De-partment of Mathematics, the Dipartimento di Matematica, Università di Bologna, and
the Dipartimento di Fisica, Università di Genova, for support and hospitality during
the realization of this work.
