Geometric and Topological Methods for Quantum Field Theory (Lecture Notes in Physics, 668)

✍ Scribed by Hernan Ocampo (editor), Sylvie Paycha (editor), Andrés Vargas (editor)

• Publisher: Springer
• Year: 2005
• Tongue: English
• Leaves: 242
• Edition: 2005
• Category: Library

✦ Synopsis

This volume offers an introduction, in the form of four extensive lectures, to some recent developments in several active topics at the interface between geometry, topology and quantum field theory. The first lecture is by Christine Lescop on knot invariants and configuration spaces, in which a universal finite-type invariant for knots is constructed as a series of integrals over configuration spaces. This is followed by the contribution of Raimar Wulkenhaar on Euclidean quantum field theory from a statistical point of view. The author also discusses possible renormalization techniques on noncommutative spaces. The third lecture is by Anamaria Font and Stefan Theisen on string compactification with unbroken supersymmetry. The authors show that this requirement leads to internal spaces of special holonomy and describe Calabi-Yau manifolds in detail. The last lecture, by Thierry Fack, is devoted to a K-theory proof of the Atiyah-Singer index theorem and discusses some applications of K-theory to noncommutative geometry. These lectures notes, which are aimed in particular at graduate students in physics and mathematics, start with introductory material before presenting more advanced results. Each chapter is self-contained and can be read independently.

✦ Table of Contents

front-matter
Chapter 1
Knot Invariants and Configuration Space Integrals
C. Lescop
1 First Steps in the Folklore of Knots, Links and Knot Invariants
1.1 Knots and Links
1.2 Link Invariants
1.3 An Easy-to-Compute Non-Trivial Link Invariant: The Jones Polynomial
2 Finite Type Knot Invariants
2.1 Definition and First Examples
2.2 Chord Diagrams
3 Some Properties of Jacobi-Feynman Diagrams
3.1 Jacobi Diagrams
3.2 The Relation IHX in Atn
3.3 A Useful Trick in Diagram Spaces
3.4 Proof of Proposition 3.3
4 The ``Kontsevich, Bott, Taubes, Bar-Natan, Altschuler,Freidel, D. Thurston" Universal Link Invariant Z
4.1 Introduction to Configuration Space Integrals:The Gauss Integrals
4.2 The Chern--Simons Series
5 More on Configuration Spaces
5.1 Compactifications of Configuration Spaces
5.2 Back to Configuration Space Integrals for Links
5.3 The Anomaly
5.4 Universality of Z0CS
5.5 Rationality of Z0CS
6 Diagrams and Lie Algebras. Questions and Problems
6.1 Lie Algebras
6.2 More Spaces of Diagrams
6.3 Linear Forms on Spaces of Diagrams
6.4 Questions
7 Complements
7.1 Complements to Sect. 1
7.2 An Application of the Jones Polynomialto Alternating Knots
7.3 Complements to Sect. 2
7.4 Complements to Subsect. 6.4
References
Chapter 2
Euclidean Quantum Field Theoryon Commutative and Noncommutative Spaces
R. Wulkenhaar
1 From Classical Actions to Lattice Quantum Field Theory
1.1 Introduction
1.2 Classical Action Functionals
1.3 A Reminder of Thermodynamics
1.4 The Partition Function for Discrete Actions
1.5 Field Theory on the Lattice
2 Field Theory in the Continuum
2.1 Generating Functionals
2.2 Perturbative Solution
2.3 Calculation of Simple Feynman Graphs
2.4 Treatment of Subdivergences
3 Renormalisation by Flow Equations
3.1 Introduction
3.2 Derivation of the Polchinski Equation
3.3 The Strategy of Renormalisation
3.4 Perturbative Solution of the Flow Equations
4 Quantum Field Theoryon Noncommutative Geometries
4.1 Motivation
4.2 The Noncommutative RD
4.3 Field Theory on Noncommutative RD
5 Renormalisation Group Approachto Noncommutative Scalar Models
5.1 Introduction
5.2 Matrix Representation
5.3 The Polchinski Equation for Matrix Models
5.4 4-Theory on Noncommutative R2
5.5 4-Theory on Noncommutative R4
References
Chapter 3
Introduction to String Compactification
A. Font and S. Theisen
1 Introduction
2 Kaluza-Klein Fundamentals
2.1 Dimensional Reduction
2.2 Compactification, Supersymmetryand Calabi-Yau Manifolds
2.3 Zero Modes
3 Complex Manifolds, Kähler Manifolds,Calabi-Yau Manifolds
3.1 Complex Manifolds
3.2 Kähler Manifolds
3.3 Holonomy Group of Kähler Manifolds
3.4 Cohomology of Kähler Manifolds
3.5 Calabi-Yau Manifolds
3.6 Calabi-Yau Moduli Space
3.7 Compactification of Type II Supergravitieson a CY Three-Fold
4 Strings on Orbifolds
4.1 Orbifold Geometry
4.2 Orbifold Hilbert Space
4.3 Bosons on TD/ZN
4.4 Type II Strings on Toroidal ZN Symmetric Orbifolds
5 Recent Developments
Appendix A. Conventions and Definitions
A.1 Spinors
A.2 Differential Geometry
Appendix B. First Chern Class of Hypersurfaces of Pn
Appendix C. Partition Function of Type II Strings on T10-d/ZN
References
Chapter 4
Index Theorems and Noncommutative Topology
T. Fack
1 Index of a Fredholm Operator
1.1 Fredholm Operators
1.2 Toeplitz Operators
1.3 The Index of a Fredholm Operator
2 Elliptic Operators on Manifolds
2.1 Pseudodifferential Operators on IRn
2.2 Pseudodifferential Operators on Manifolds
2.3 Analytical Index of an Elliptic Operator
3 Topological K-Theory
3.1 The Group K0 (X)
3.2 Fredholm Operators and Atiyah's Picture of K0 (X)
3.3 Excision in K-Theory
3.4 The Chern Character
3.5 Topological K-Theory for C-Algebras
3.6 Main Properties of the Topological K-Theoryfor C-Algebras
3.7 Kasparov's Picture of K0 (A)
4 The Atiyah-Singer Index Theorem
4.1 Statement of the Theorem
4.2 Construction of the Analytical Index Map
4.3 Construction of the Topological Index Map
4.4 Coincidence of the Analytical and Topological Index Maps
4.5 Cohomological Formula for the Topological Index
4.6 The Hirzebruch Signature Formula
5 The Index Theorem for Foliations
5.1 Index Theorem for Elliptic Families
5.2 The Index Theorem for Foliations
References