Quantum Field Theory and Noncommutative Geometry

302 Pages·2005·3.352 MB·English

by  G. Landi (auth.), Ursula Carow-Watamura, Yoshiaki Maeda, Satoshi Watamura (eds.)

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U.Carow-Watamura Y.Maeda S.Watamura (Eds.)
Quantum Field Theory
and Noncommutative
Geometry
123
Editors
UrsulaCarow-Watamura SatoshiWatamura
TohokuUniversity TohokuUniversity
DepartmentofPhysics DepartmentofPhysics
GraduateSchoolofScience GraduateSchoolofScience
Aoba-ku Aoba-ku
Sendai980-8578 Sendai980-8578
Japan Japan
YoshiakiMaeda
KeioUniversity
DepartmentofMathematics
FacultyofScienceandTechnology
HiyoshiCampus
4-1-1Hiyoshi,Kohoku-ku
Yokohama223-8825
Japan
U.Carow-Watamura Y.Maeda S.Watamura(Eds.),QuantumFieldTheoryandNoncom-
mutativeGeometry,Lect.NotesPhys.662(Springer,BerlinHeidelberg2005),
DOI10.1007/b102320
LibraryofCongressControlNumber:2004115524
ISSN0075-8450
ISBN3-540-23900-6SpringerBerlinHeidelbergNewYork
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Preface
This book is based on the workshop “Quantum Field Theory and Noncom-
mutative Geometry” held in November 2002 at Tohoku University, Sendai,
Japan. This workshop was the third in a series, the first one having been
held at the Shonan International Village at Hayama in Kanagawa-ken in
1999, and the second one at Keio University, Yokohama in 2001. The main
aim of these meetings is to enhance the discussion and cooperation between
mathematicians and physicists working on various problems in deformation
quantization, noncommutative geometry and related fields.
The workshop held in Sendai was focused on the topics of noncommuta-
tive geometry and an algebraic approaches to quantum field theory, which
includesthedeformationquantization,symplecticgeometryandapplications
to physics as well as topological field theories.
The idea to treat quantized theories by using an algebraic language can
be traced back to the early days of quantum mechanics, when Heisenberg,
Born and Jordan formulated quantum theory in terms of matrices (matrix
mechanics). Since then, a continuous effort has been made to develop an
algebraic language and tools which would also allow the inclusion gravity.
Among the physicist is point of view, the concept of a minimum length is
discussedmanytimesinvarioustheories,especiallyinthetheoriesofquantum
gravity.Sincethestringisanextendedobject,stringtheorystronglysuggests
the existence of a minimum length, and this brought the discussion on the
quantization of space into this field. However, this discussion raised several
problems, in particular, how such a geometry with minimum length should
be formulated and how a quantization should be performed in a systematic
way.
A hint in this direction came from the theory of quantum groups, which
had been developed in the 1980s and which gave a method to deform an
algebra to become noncommutative, thereby preserving its symmetry as a
q-deformedstructure.NearlyatthesametimeA.Connespublishedhiswork
on noncommutative differential geometry. It was the impact from these two
newfields,thatputforwardtheresearchonquantizedspaces,anddrewmore
and more the physicists’ attention towards this field.
Noncommutativedifferentialgeometry(NCDG)ledtostrikingextensions
of the Atiyah-Singer index theorem and it also shows several common points
VI Preface
withdeformationquantization.Anotherresultisthedevelopmentofnoncom-
mutative gauge theory, which became a very promising candidate as an the
effective theory of the so-called D-brane; a D-brane is a configuration which
evolvedinthecourseofthedevelopmentofstringtheory,leadingtosolutions
of nonperturbative configurations of the string in the D-brane background.
Inspired by the possibilities opened by NCDG; there is now a number of
physicists developing the “matrix theory”, about 80 years after the “matrix
mechanics”.
Deformation quantization is a quantization scheme which has been in-
troduced by Bayen, Flato, Fronsdal, Lichnerowicz and Sternheimer. In this
approach the algebras of quantum observables are defined by a formal de-
formation of the classical observables as formal power series. The expansion
parameteris(cid:1)andtheproductofthesedeformedalgebrasisthestarproduct.
Symplectic geometry and Poisson geometry fit very well to this quantization
scheme since they possess a Poisson structure, and thus deformation quan-
tization is regarded as a quantization from an algebraic point of view. As
we know from the theorem of Gel’fand and Naimark, we can often realize
a classical space from a suitable algebra of the classical observables. From
this point of view, we expect the deformation quantization may give a rea-
sonablequantumspace,whoseinvestigationwillcontributeadevelopmentto
noncommutative geometry.
We collected here the lectures and talks presented in the meeting. When
preparingthisproceedingswemadeefforttomakethisbookinterestingfora
widercommunityofreaders.Therefore,theintroductionstothelecturesand
talksaremoredetailedthanintheworkshop.Alsosomederivationsofresults
are given more explicitly than in the original lecture, such that this volume
becomesaccessibletoresearchersandgraduatestudentswhodidnotjointhe
workshop. A large number of contributions are devoted to presentations of
new results which have not appeared previously in professional journals, or
tocomprehensivereviews(includinganoriginalpart)ofrecentdevelopments
in those topics.
Now we would like to thank all speakers for their continuous effort to
prepare these articles. Also we would like to thank all participants of the
workshop for sticking together until the end of the last talk, thus creating
a good atmosphere and the basis for many fruitful discussions during this
workshop. We also greatly acknowledge the Ministry of Education, Culture,
Sports, Science and Technology, Japan, who supported this workshop by a
Grant-in-Aid for Scientific Research (No. 13135202).
Sendai and Yokohama Ursula Carow-Watamura
January 2005 Yoshiaki Maeda
Satoshi Watamura
Contents
Part I Noncommutative Geometry
Noncommutative Spheres and Instantons
G. Landi....................................................... 3
Some Noncommutative Spheres
T. Natsume .................................................... 57
From Quantum Tori to Quantum Homogeneous Spaces
S. Kamimura................................................... 67
Part II Poisson Geometry
and Deformation Quantization
Local Models for Manifolds with Symplectic Connections
of Ricci Type
M. Cahen...................................................... 77
On Gauge Transformations of Poisson Structures
H. Bursztyn .................................................... 89
Classification of All Quadratic Star Products on a Plane
N. Miyazaki .................................................... 113
Universal Deformation Formulae
for Three-Dimensional Solvable Lie Groups
P. Bieliavsky, P. Bonneau, Y. Maeda.............................. 127
Morita Equivalence, Picard Groupoids
and Noncommutative Field Theories
S. Waldmann................................................... 143
Secondary Characteristic Classes of Lie Algebroids
M. Crainic, R.L. Fernandes ...................................... 157
VIII Contents
Part III Applications in Physics
Gauge Theories on Noncommutative Spacetime Treated
by the Seiberg-Witten Method
J. Wess........................................................ 179
Noncommutative Line Bundles and Gerbes
B. Jurˇco....................................................... 193
Lectures on Two-Dimensional Noncommutative
Gauge Theory
L.D. Paniak, R.J. Szabo ......................................... 205
Part IV Topological Quantum Field Theory
Topological Quantum Field Theories and Operator Algebras
Y. Kawahigashi................................................. 241
Topological Quantum Field Theory and Algebraic Structures
T. Kimura ..................................................... 255
An Infinite Family of Isospectral Pairs
N. Iiyori, T. Itoh, M. Iwami, K. Nakada, T. Masuda ................ 289
List of Contributors
B. Jurˇco L.D. Paniak
Theoretische Physik, Universita¨t Michigan Center for Theoretical
Mu¨nchen, Theresienstr. 37, Physics, University of
80333 Mu¨nchen, Germany Michigan, Ann Arbor, Michigan
[email protected]. 48109-1120, U.S.A.
uni-muenchen.de [email protected]
G. Landi M. Cahen
Dipartimento di Scienze Matem- Universit´e Libre de Bruxelles,
atiche, Universit`a di Trieste, Via Campus Plaine, CP 218,
Valerio 12/b, 34127 Trieste, Italia, 1050 Brussels, Belgium
and INFN, Sezione di Napoli, [email protected]
Napoli, Italia
[email protected] M. Crainic
Depart.ofMath.,UtrechtUniversity
H. Bursztyn 3508 TA Utrecht, The Netherlands
Department of Mathematics, [email protected]
University of Toronto, Toronto,
Ontario M5S 3G3, Canada M. Iwami
[email protected] Graduate School of Pure and
Applied Sciences, University of
J. Wess Tsukuba, Japan
Sektion Physik der Ludwig- [email protected]
Maximillians-Universita¨t,
Theresienstr. 37, 80333 Mu¨nchen, N. Iiyori
Germany, and Max-Planck-Institut Unit of Mathematics and Informa-
fu¨r Physik (Werner-Heisenberg- tion Science, Yamaguchi University,
Institut), Fo¨hringer Ring 6, 80805 Japan
Mu¨nchen, Germany [email protected]
K. Nakada N. Miyazaki
Department of Pure and Applied Department of Mathematics,
Mathematics, General School of Faculty of Economics,
Information Science and Technology, Keio University, 4-1-1, Hiyoshi,
Osaka University, Japan Yokohama, 223-8521, JAPAN
[email protected] [email protected]