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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 Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthe material is concerned, specifically the rights of translation, reprinting, reuse of illustra- tions, recitation, broadcasting, reproduction on microfilm or in any other way, and storageindatabanks.Duplicationofthispublicationorpartsthereofispermittedonly under the provisions of the German Copyright Law of September 9, 1965, in its cur- rentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violations areliabletoprosecutionundertheGermanCopyrightLaw. 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Balloons,ParadigmsofThermodynamics DissipativeSolitons Vol.638:Y.Kosmann-Schwarzbach,B.Grammaticos, Vol.662:U.Carow-Watamura,Y.Maeda,S.Watamura K.M.Tamizhmani(Eds.),IntegrabilityofNonlinear (Eds.),QuantumFieldTheoryandNoncommutative Systems Geometry Vol.639:G.Ripka,DualSuperconductorModelsof ColorConfinement 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 noncommutative 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 deformation 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 quantization 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- Université Libre de Bruxelles, atiche, Università 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]