拓扑和几何的概念与方法在物理学中的应用有助于加深人们对物理学的许多重要领域(如凝聚态物理、宇宙学、引力和粒子物理)的理解，本书是一本关于几何和拓扑在这些领域中的应用与发展的高级教材，书中分章节相对独立地介绍了规范理论、BRST量子化、手性异常、超对称孤子和非交换几何中的拓扑概念。本书适用于相关专业的研究生、对本领域感兴趣的读者以及讲授相关课程的教师。

本书是《国外物理名著系列》之一。本书几乎涉及了近代物理学中拓扑和几何的所有领域，既有阐述学科基本理论的经典名著，也有反映某一学科专题前沿的专著。基础理论方面强调“经典”，选择了那些经得起时间检验、对物理学的发展产生重要影响、现在还不“过时”的著作；反映物理学某一领域进展的方面强调“前沿”和“热点”，根据国内物理学研究发展的实际情况，选择了能够体现相关学科最新进展，对有关方向的科研人员和研究生有重要参考价值的图书。本书还对部分目录标题和练习题进行了少量的翻译和注释，以方便国内读者的阅读和理解。

Introduction and Overview

E.Bick, F.D.Steffen

1、Topology and Geometry in Physics

2、An Outline of the Book

3、Complementary Literature

Topological Concepts in Gauge Theories

F.Lenz

1、Introduction

2、Nielsen-Olesen Vortex

 2.1 Abelian Higgs Model

 2.2 Topological Excitations

3、Homotopy

 3.1 The Fundamental Group

 3.2 Higher Homotopy Groups

 3.3 Quotient Spaces

 3.4 Degree of Maps

 3.5 Topological Groups

 3.6 Transformation Groups

 3.7 Defects in Ordered Media

4、Yang-Mills Theory

5、't Hooft-Polyakov Monopole

 5.1 Non-Abelian Higgs Model

 5.2 The Higgs Phase

 5.3 Topological Excitations

6、Quantization of Yang-Mills Theory

7、Instantons

 7.1 Vacuum Degeneracy

 7.2 Tunneling

 7.3 Fermions in Topologically Non-trivial Gauge Fields

 7.4 Instanton Gas

 7.5 Topological Charge and Link Invariants

8、Center Symmetry and Confinement

 8.1 Gauge Fields at Finite Temperature and Finite Extension

 8.2 Residual Gauge Symmetries in QED

 8.3 Center Symmetry in SU(2) Yang-Mills Theory

 8.4 Center Vortices

 8.5 The Spectrum of the SU(2) Yang-Mills Theory

9、QCD in Axial Gauge

 9.1 Gauge Fixing

 9.2 Perturbation Theory in the Center-Symmetric Phase

 9.3 Polyakov Loops in the Plasma Phase

 9.4 Monopoles

 9.5 Monopoles and Instantons

 9.6 Elements of Monopole Dynamics

 9.7 Monopoles in Diagonalization Gauges

10、Conclusions

Aspects of BRST Quantization

J.W.van Holten

1、Symmetries and Constraints

 1.1 Dynamical Systems with Constraints

 1.2 Symmetries and Noether's Theorems

 1.3 Canonical Formalism

 1.4 Quantum Dynamics

 1.5 The Relativistic Particle

 1.6 The Electro-magnetic Field

 1.7 Yang-Mills Theory

 1.8 The Relativistic String

2、Canonical BRST Construction

 2.1 Grassmann Variables

 2.2 Classical BRST Transformations

 2.3 Examples

 2.4 Quantum BRST Cohomology

 2.5 BRST-Hodge Decomposition of States

 2.6 BRST Operator Cohomology

 2.7 Lie-Algebra Cohomology

3、Action Formalism

 3.1 BRST Invariance from Hamilton's Principle

 3.2 Examples

 3.3 Lagrangean BRST Formalism

 3.4 The Master Equation

 3.5 Path-Integral Quantization

4、Applications of BRST Methods

 4.1 BRST Field Theory

 4.2 Anomalies and BRST Cohomology

Appendix.Conventions

Chiral Anomalies and Topology

J.Zinn-Justin

1、Symmetries, Regularization, Anomalies

2、Momentum Cut-Off Regularization

 2.1 Matter Fields:Propagator Modification

 2.2 Regulator Fields

 2.3 Abelian Gauge Theory

 2.4 Non-Abelian Gauge Theories

3、Other Regularization Schemes

 3.1 Dimensional Regularization

 3.2 Lattice Regularization

 3.3 Boson Field Theories

 3.4 Fermions and the Doubling Problem

4、The Abelian Anomaly

 4.1 Abelian Axial Current and Abelian Vector Gauge Fields

 4.2 Explicit Calculation

 4.3 Two Dimensions

 4.4 Non-Abelian Vector Gauge Fields and Abelian Axial Current

 4.5 Anomaly and Eigenvalues of the Dirac Operator

5、Instantons, Anomalies, and θ-Vacua

 5.1 The Periodic Cosine Potential

 5.2 Instantons and Anomaly:CP(N-1) Models

 5.3 Instantons and Anomaly:Non-Abelian Gauge Theories

 5.4 Fermions in an Instanton Background

6、Non-Abelian Anomaly

 6.1 General Axial Current

 6.2 Obstruction to Gauge Invariance

 6.3 Wess-Zumino Consistency Conditions

7、Lattice Fermions:Ginsparg-Wilson Relation

 7.1 Chiral Symmetry and Index

 7.2 Explicit Construction:Overlap Fermions

8、Supersymmetric Quantum Mechanics and Domain Wall Fermions

 8.1 Supersymmetric Quantum Mechanics

 8.2 Field Theory in Two Dimensions

 8.3 Domain Wall Fermions

Appendix A. Trace Formula for Periodic Potentials

Appendix B. Resolvent of the Hamiltonian in Supersymmetric QM

Supersymmetric Solitons and Topology

M.Shifman

1、Introduction

2、D= 1+1;N=1

 2.1 Critical(BPS) Kinks

 2.2 The Kink Mass (Classical)

 2.3 Interpretation of the BPS Equations.Morse Theory

 2.4 Quantization.Zero Modes:Bosonic and Fermionic

 2.5 Cancelation of Nonzero Modes

 2.6 Anomaly Ⅰ

 2.7 Anomaly Ⅱ(Shortening Supermultiplet Down to One State)

3、Domain Walls in (3+1)-Dimensional Theories

 3.1 Superspace and Superfields

 3.2 Wess Zumino Models

 3.3 Critical Domain Walls

 3.4 Finding the Solution to the BPS Equation

 3.5 Does the BPS Equation Follow from the Second Order Equation of Motion?

 3.6 Living on a Wall

4、Extended Supersymmetry in Two Dimensions:The Supersymmetric CP(1) Model

 4.1 Twisted Mass

 4.2 BPS Solitons at the Classical Level

 4.3 Quantization of the Bosonic Moduli

 4.4 The Soliton Mass and Holomorphy

 4.5 Switching On Fermions

 4.6 Combining Bosonic and Fermionic Moduli

5、Conclusions

Appendix A. CP(1) Model=O(3) Model (Af=1 Superfields N)

Appendix B. Getting Started (Supersymmetry for Beginners)

 B.1 Promises of Supersymmetry

 B.2 Cosmological Term

 B.3 Hierarchy Problem

Forces from Connes' Geometry

T.Schiicker

1、Introduction

2、Gravity from Riemannian Geometry

 2.1 First Stroke:Kinematics

 2.2 Second Stroke:Dynamics

3、Slot Machines and the Standard Model

 3.1 Input

 3.2 Rules

 3.3 The Winner

 3.4 Wick Rotation

4、Connes' Noncommutative Geometry

 4.1 Motivation:Quantum Mechanics

 4.2 The Calibrating Example:Riemannian Spin Geometry

 4.3 Spin Groups

5、The Spectral Action

 5.1 Repeating Einstein's Derivation in the Commutative Case

 5.2 Almost Commutative Geometry

 5.3 The Minimax Example

 5.4 A Central Extension

6、Connes' Do-It-Yourself Kit

 6.1 Input

 6.2 Output

 6.3 The Standard Model

 6.4 Beyond the Standard Model

7、Outlook and Conclusion

Appendix

 A.1 Groups

 A.2 Group Representations

 A.3 Semi-Direct Product and Poincare Group

 A.4 Algebras

Index