Symmetry in Mathematics and Physics

267 Pages·2009·2.237 MB·English

by  Donald Babbitt, Vyjayanthi Chari, Rita Fioresi (ed.)

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American Mathematical Society
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Symmetry in
Mathematics and Physics
V. S. Varadarajan
C
ONTEMPORARY
M
ATHEMATICS
490
Symmetry in
Mathematics and Physics
Conference in Honor of V. S. Varadarajan's Birthday
January 18–20, 2008
University of California, Los Angeles, California
Donald Babbitt
Vyjayanthi Chari
Rita Fioresi
Editors
American Mathematical Society
Providence, Rhode Island
Editorial Board
Dennis DeTurck, managing editor
George Andrews Abel Klein Martin J. Strauss
2000 Mathematics Subject Classification. Primary16Sxx, 17Bxx, 20Gxx, 32Cxx, 51Bxx,
60Fxx, 81Pxx, 81Txx, 81–02.
Photo courtesy of Rahul N. Fernandez
Library of Congress Cataloging-in-Publication Data
Symmetry in mathematics and physics : conference in honor of V.S. Varadarajan’s 70th birth-
day : January 18–20, 2008, University of California, Los Angeles, California / Donald Babbitt,
VyjayanthiChari,RitaFioresi,editors.
p.cm. —(Contemporarymathematics;v.490)
Includesbibliographicalreferences.
ISBN978-0-8218-4731-2(alk.paper)
1.Symmetry(Mathematics)—Congresses. 2. Symmetry(Physics)—Congresses. I.Varadara-
jan,V.S. II.Babbitt,DonaldG. III.Chari,Vyjayanthi. IV.Fioresi,Rita,1966–
QA174.7.S96S96 2009
510—dc22 2009004434
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Contents
Preface vii
Probability
The Role of Weak Convergence in Probability Theory
S. R. S. Varadhan 3
Foundations of Quantum Mechanics
Statistical models on a simplex
Enrico G. Beltrametti 13
Quantum Error Correcting Codes
Quantum Error Correcting Codes and Weyl Commutation Relations
K. R. Parthasarathy 29
Finite and Infinite Dimensional Lie Groups and Algebras
and their Representations
Ideals in parabolic subalgebras of simple Lie algebras
Vyjayanthi Chari, R. J. Dolbin, and T. Ridenour 47
Quantizations of modules of differential operators
Charles H. Conley 61
Geometric realization of PRV components and the Littlewood-Richardson cone
Ivan Dimitrov and Mike Roth 83
Ka¸c algebras, quantum doubles and planar algebras
S. Jijo and V. S. Sunder 97
On a theorem of Ranee Brylinski
Bertram Kostant and Nolan Wallach 105
Conjugacy of maximal toral subalgebras of direct limits of loop algebras
Hadi Salmasian 133
v
vi CONTENTS
A multiplicity formula for tensor products of SL modules and an explicit
2
Sp2n to Sp2n−2×Sp2 branching formula
Nolan Wallach and Oded Yacobi 151
Super Lie Groups and Supergeometry
Super G-spaces
Luigi Balduzzi, Claudio Carmeli, and Gianni Cassinelli 159
Quotients in Supergeometry
Luigi Balduzzi, Claudio Carmeli, and Rita Fioresi 177
Special geometry of D =4,5 supersymmetry
Mar´ıa Antonia Lledo´ and Jos´e Antonio Lo´pez 189
Supersymmetry
Symmetric Spaces in Supergravity
Sergio Ferrara and Alessio Marrani 203
Symmetry and Superstring Phenomenology
Mary K. Gaillard 229
Supersymmetry: A personal view
B. Zumino 241
Preface
ThearticlesinthiscollectionmainlygrewoutofthetalksgivenataConference
held at UCLA in January 2008, which honored V. S. Varadarajan on his 70th
birthday. The main theme of the Conference was symmetry in mathematics and
physics. More precisely, the talks at the conference were dedicated to the interplay
between geometry, group theory, and fundamental physics. In addition to the
speakerstherewereanumberofdoctoralandpostdoctoralfellowsincludingseveral
studentsofVaradarajanwhohadworkedunderhimonthesetopicsthroughouthis
career.
Varadarajan’sworkoverthepast50yearsrepresentsabroadspectrumofmath-
ematics but its main emphasis has been on symmetry in mathematics and math-
ematical physics, broadly interpreted. Some of his significant achievements are:
developmentoftheinfinitesimalmethodinthetheoryofinfinitedimensionalrepre-
sentationsofrealsemisimpleLiealgebras;Fouriertransformtheoryinthecomplex
domainonRiemanniansymmetricspaces;theoryoflocalmoduliforordinarymero-
morphic linear differential equations at an irregular singularity; theory of unitary
representations of super Lie groups and the classification of super particles; and
more recently, studies on the physics associated to non-archimedean space-time.
The relevance of the representation theory of Lie groups and Lie algebras to
the physics of elementary particles and fields has been known for a very long time,
going back to the famous 1939 paper of E. P. Wigner on the representations of the
Poincaregroup. Sincethenthislinkbetweenrepresentationtheoryandphysicshas
deepenedenormously,andincludesquantumfieldtheoryandconformalfieldtheory.
Then something marvelous happened. In the 1970’s the physicists created a new
extensionofgeometrywheretheunderlyingmanifoldsacquiredanti-commutingco-
ordinates in addition to the usual commuting ones, reflecting the Fermionic struc-
ture of matter. This introduced supergeometry and super Lie groups into the mix
and made the connection between geometry and physics much richer. Together
with his students, he has made many important contributions to this area.
It thus seemed appropriate to have a conference at UCLA devoted to some
of these themes. The Conference turned out to be very exciting and stimulating
because of the contributions of the participants who came from the United States
and abroad. Most of the articles in this volume are thus naturally concerned with
the above-mentioned themes: representations of finite and infinite dimensional Lie
groups and Lie algebras, super Lie groups and supergeometry, which are at the
interface of mathematics and fundamental particle physics, and supersymmetry.
The discussions on supergeometry and supersymmetry are especially relevant at
this time since some of the experiments at the Large Hadron Collider at CERN
mayhelpdeterminewhethersupersymmetryisafeatureoftheworldofelementary
vii
viii PREFACE
particles. A few of the articles are in probability and the foundations of quantum
mechanics, areas in which Varadarajan worked early in his career.
The scientific organizing committee consisted of S. R. S. Varadhan (Chair), E.
Beltrametti, T. J. Enright, S. Ferrara, K. R. Parthasarathy, and N. R. Wallach.
The conference was funded by a generous grant from NSF, from a grant from the
Goldman-Sachscorporation,andamatchingcontributionfromaprivatedonor. We
are very grateful to these sources for their generosity, which made the conference
go on in a very smooth manner.
Theeditorswishtoexpresstheirthankstomanypeoplewhoseeffortsmadethis
conference a success, including all the participants. They wish to thank IPAM for
permissiontousetheirfacilitiesfortheconference;ProfessorsChristophThieleand
Robert Steinberg for inaugurating the conference; the department of mathematics
at UCLA for providing help at all stages of the conference and for organizing the
web site, especially Babette Dalton, Robert Amodeo, and Natasja Saint-Satyr;
and Christine Thivierge of the AMS for her editorial assistance in producing this
collection. The editors are also grateful to T. Kibble and Imperial College Press
for permission to reprint B. Zumino’s paper “Supersymmetry: A Personal View”,
which appears by their courtesy in these pages.
Donald Babbitt, UCLA
Vyjayanthi Chari, UC Riverside
Rita Fioresi, Universita` di Bologna, Italy