Proceedings of the Summer School Geometric and Topological Methods for Quantum Field Theory : Villa de Leyva, Colombia, 9-27 July 2001 / editors, Alexander Cardona, Sylvie Paycha, Hernan Ocampo.

Format:

Book

Conference/Event

Contributor:

Ocampo, Hernan.

Paycha, Sylvie.

Cardona, Alexander.

Conference Name:

Summer School Geometric and Topological Methods for Quantum Field Theory (2001 : Leiva, Boyaca, Colombia)

Language:

English

Subjects (All):

Quantum field theory--Congresses.

Quantum field theory.

Algebraic topology--Congresses.

Algebraic topology.

Geometry, Algebraic--Congresses.

Geometry, Algebraic.

Physical Description:

1 online resource (495 p.)

Other Title:

Geometric and topological methods for quantum field theory

Place of Publication:

Singapore ; River Edge, NJ : World Scientific, c2003.

Language Note:

English

Summary:

This volume offers an introduction to recent developments in several active topics of research at the interface between geometry, topology and quantum field theory. These include Hopf algebras underlying renormalization schemes in quantum field theory, noncommutative geometry with applications to index theory on one hand and the study of aperiodic solids on the other, geometry and topology of low dimensional manifolds with applications to topological field theory, Chern-Simons supergravity and the anti de Sitter/conformal field theory correspondence. It comprises seven lectures organized aroun

Contents:

INTRODUCTION; CONTENTS; Lectures; Noncommutative Geometry; Topological Field Theory; Supergravity and String Theory; Short Communications; Hopf Algebras in Noncommutative Geometry Joseph C. Varilly; Introduction; 1 Noncommutative Geometry and Hopf Algebras; 1.1 The algebraic tools of noncommutative geometry; 1.2 Hopf algebras: introduction; 1.3 Hopf actions of differential operators: an example; 2 The Hopf Algebras of Connes and Kreimer; 2.1 The Gonnes-Kreimer algebra of rooted trees; 2.2 Hopf algebras of Feynman graphs and renormalization; 3 Cyclic Cohomology

3.1 Hochschild and cyclic cohomology of algebras3.2 Cyclic cohomology of Hopf algebras; 4 Noncommutative Homogeneous Spaces; 4.1 Chern characters and noncommutative spheres; 4.2 How Moyal products yield compact quantum groups; 4.3 Isospectral deformations of homogeneous spin geometries; References; The Noncommutative Geometry of Aperiodic Solids Jean Bellissard; Introduction; Acknowledgments:; 1 Mathematical Description of Aperiodic Solids; 1.1 Examples of Aperiodic Solids; 1.2 The Hull; 1.3 Properties of the Hull; 1.4 Atomic Gibbs groundstates; 1.5 Bloch Theory

1.6 The Noncommutative Brillouin zone1.7 Electrons and Phonons; 2 Examples of Hulls; 2.1 Perfect Crystals; 2.2 Disordered Systems; 2.3 Quasicrystals; 2.4 Finite type Tilings; 3 The Gap Labelling Theorems; 3.1 K-theory; 3.1.1 The Group Ko; 3.1.2 Higher K-groups and exact sequences; 3.1.3 The Connes-Thorn isomorphism; 3.1.4 The Pimsner Voiculescu exact sequence; 3.1.5 Morita equivalence; 3.2 Gap Labels; 3.3 Computing Gap Labels; 4 The Quantum Hall Effect; 4.1 Physics; 4.2 The Chern-Kubo formula; 4.3 The Four Traces Way; 4.4 Localization; References

Noncommutative Geometry and Abstract Integration Theory Moulay-Tahar BenameurIntroduction; Chapter I. Review of the classical index theory; 1 Some preliminaries and examples; 1.1 The index theorem for Toeplitz operators; 1.2 Some properties of pseudodigerential operators; 1.3 The index of elliptic complexes; 1.4 Some geometric operators; 2 The Atiyah-Singer index theorem; 2.1 Review of the splitting principle; 2.2 Statement of the index theorem; 2.3 Sketch of the original K-theory proof; 2.4 The heat expansion method; 3 Corollaries and examples; Chapter II. The Murray-Von Neumann index theory

4 Dixmier traces in von Neumann algebras4.1 Dixmier trace and residue of zeta functions; 5 Index theory in von Neumann algebras; 5.1 Definitions and properties; 5.2 An experimental example: Suspended actions; 6 Type II non commutative geometry; 6.1 Von Neumann spectral triples; 6.2 The analytic Chern- Connes character; 6.3 The equivariant case; 6.4 Back to the equivariant periodic homology; References; Introduction to Quantum Invariants of 3-Manifolds, Topological Quantum Field Theories and Modular Categories Christian Blanchet; Introduction; 1 LINKS, 3-MANIFOLDS AND THEIR INVARIANTS

1.1 Knots, links and diagrams

Notes:

Description based upon print version of record.

Includes bibliographical references.

ISBN:

9786611908560

9781281908568

1281908568

9789812705068

9812705066