Perspectives On Noncommutative Geometry

This volume represents the proceedings of the Noncommutative Geometry Workshop that was held as part of the thematic program on operator algebras at the Fields Institute in May 2008. Pioneered by Alain Connes starting in the late 1970s, noncommutative geometry was originally inspired by global analysis, topology, operator algebras, and quantum physics. Its main applications were to settle some long-standing conjectures, such as the Novikov conjecture and the Baum-Connes conjecture. Next came the impact of spectral geometry and the way the spectrum of a geometric operator, like the Laplacian, holds information about the geometry and topology of a manifold, as in the celebrated Weyl law. This has now been vastly generalized through Connes' notion of spectral triples. Finally, recent years have witnessed the impact of number theory, algebraic geometry and the theory of motives, and quantum field theory on noncommutative geometry. Almost all of these aspects are touched upon with new results in the papers of this volume. This book is intended for graduate students and researchers in both mathematics and theoretical physics who are interested in noncommutative geometry and its applications.

This book provides a systematic, comprehensive and up-to-date account of the recent developments in non-commutative geometry, at a pedagogical level. It does not go into the details of rigorous (advanced level) mathematical formulation of non-commutative geometry; rather, it restricts itself to the domain of strings and quantum fields.Since non-commutative geometry has recently aroused renewed interest in open string theory, the author motivates the text from the viewpoint of a string theory. He begins with an introduction to the subject, explaining what one means by non-commutative geometry and why it is relevant to study such geometry, and discussing its possible origin in a string theory.The book comprises nine chapters. Chapter 1 gives a sound mathematical ntroduction to non-commutative spacetime coordinates in classical and quantum physics. In Chapter 2, non-commutativity in a string theory is discussed at a pedagogic level. Chapter 3 deals with an aribitrary D-brane dynamics and Chapter 4 describes the non-commutative gauge theories on a D-brane. In Chapters 5-9, non-commutative quantum field theory (NCQFT) is addressed. In particular, Chapter 5 deals with the real scalar NCQFT, Chapter 6 with that of complex scalar field, Chapter 7 describes spontaneous symmetry breaking in scalar NCQFT, Chapter 8 deals with the U(1) Gauge theory and Chapter 9 with SU(n) Gauge theories.Students will find this book useful as a bridge between string and field theories. In addition, it will prove invaluable for interdisciplinary areas of study.

In recent years, number theory and arithmetic geometry have been enriched by new techniques from noncommutative geometry, operator algebras, dynamical systems, and K-Theory. This volume collects and presents up-to-date research topics in arithmetic and noncommutative geometry and ideas from physics that point to possible new connections between the fields of number theory, algebraic geometry and noncommutative geometry. The articles collected in this volume present new noncommutative geometry perspectives on classical topics of number theory and arithmetic such as modular forms, class field theory, the theory of reductive p-adic groups, Shimura varieties, the local L-factors of arithmetic varieties. They also show how arithmetic appears naturally in noncommutative geometry and in physics, in the residues of Feynman graphs, in the properties of noncommutative tori, and in the quantum Hall effect.

This book is aimed at presenting different methods and perspectives in the theory of Quantum Groups, bridging between the algebraic, representation theoretic, analytic, and differential-geometric approaches. It also covers recent developments in Noncommutative Geometry, which have close relations to quantization and quantum group symmetries. The volume collects surveys by experts which originate from an acitvity at the Max-Planck-Institute for Mathematics in Bonn.

Noncommutative differential geometry is a novel approach to geometry, aimed in part at applications in physics. It was founded in the early eighties by the 1982 Fields Medalist Alain Connes on the basis of his fundamental works in operator algebras. It is now a very active branch of mathematics with actual and potential applications to a variety of domains in physics ranging from solid state to quantization of gravity. The strategy is to formulate usual differential geometry in a somewhat unusual manner, using in particular operator algebras and related concepts, so as to be able to plug in noncommutativity in a natural way. Algebraic tools such as K-theory and cyclic cohomology and homology play an important role in this field. It is an important topic both for mathematics and physics. Contents:K-Theory and D-Branes, Shonan:The Local Index Formula in Noncommutative Geometry Revisited (Alan L Carey, John Phillips, Adam Rennie and Fedor A Sukochev)Semi-Finite Noncommutative Geometry and Some Applications (Alan L Carey, John Phillips and Adam Rennie)Generalized Geometries in String Compactification Scenarios (Tetsuji Kimura)What Happen to Gauge Theories under Noncommutative Deformation? (Akifumi Sako)D-Branes and Bivariant K-Theory (Richard J Szabo)Two-Sided Bar Constructions for Partial Monoids and Applications to K-Homology Theory (Dai Tamaki)Twisting Segal's K-Homology Theory (Dai Tamaki)Spectrum of Non-Commutative Harmonic Oscillators and Residual Modular Forms (Kazufumi Kimoto and Masato Wakayama)Coarse Embeddings and Higher Index Problems for Expanders (Qin Wang)Deformation Quantization and Noncommutative Geometry, RIMS:Enriched Fell Bundles and Spaceoids (Paolo Bertozzini, Roberto Conti and Wicharn Lewkeeratiyutkul)Weyl Character Formula in KK-Theory (Jonathan Block and Nigel Higson)Recent Advances in the Study of the Equivariant Brauer Group (Peter Bouwknegt, Alan Carey and Rishni Ratnam)Entire Cyclic Cohomology of Noncommutative Manifolds (Katsutoshi Kawashima)Geometry of Quantum Projective Spaces (Francesco D'Andrea and Giovanni Landi)On Yang–Mills Theory for Quantum Heisenberg Manifolds (Hyun Ho Lee)Dilatational Equivalence Classes and Novikov–Shubin Type Capacities of Groups, and Random Walks (Shin-ichi Oguni)Deformation Quantization of Gauge Theory in ℝ4 and U(1) Instanton Problems (Yoshiaki Maeda and Akifumi Sako)Dualities in Field Theories and the Role of K-Theory (Jonathan Rosenberg)Dualities in Field Theories and the Role of K-Theory (Jonathan Rosenberg) Readership: Researchers and graduate students in Mathematical Physics and Applied Mathematics. Keywords:Noncommutative Geometry;Deformation Quantizations;D-Brane;K-Theory;T-Duality

This book aims to provide a friendly introduction to non-commutative geometry. It studies index theory from a classical differential geometry perspective up to the point where classical differential geometry methods become insufficient. It then presents non-commutative geometry as a natural continuation of classical differential geometry. It thereby aims to provide a natural link between classical differential geometry and non-commutative geometry. The book shows that the index formula is a topological statement, and ends with non-commutative topology.