## The Role of Topology in Classical and Quantum PhysicsIn solid-state physics especially topological techniques have turned out to be extremely useful for modelling and explaining physical properties of matter. This book illustrates various applications of algebraic topology in classical field theory (non-linear sigma-models) and in quantizationsin multiply connected spaces (anyons). It treats Chern-Simon Lagrangians, Berry's phase, the polarization of light and the fractional quantum Hall effect. |

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### Contents

AN ELEMENTARY INTRODUCTION TO ALGEBRAIC TOPOLOGY | 1 |

12 Some examples | 2 |

13 Topology of Defects the case of planar spins | 4 |

14 The Fundamental Group | 8 |

15 The Abstract Fundamental Group Conjugacy Classes and freely Homotopic Loops | 18 |

16 The OrderParameter Space as a Coset Space | 28 |

17 Preliminary Theorems Concerning Fundamental Groups | 42 |

18 Higher Homotopy Groups | 46 |

34 Dynamical Implementation of Braid Statistics | 135 |

TOPICS IN CHERNSIMONS PHYSICS | 145 |

42 ChernSimons Lagrangians | 146 |

43 Charged Particles Interacting with a CS Field | 153 |

44 Gauge Fixing and an Explicit Solution for A𝜇 | 159 |

45 Effective Lafrangian for Statistical Particle Interaction | 167 |

A SHORT INTRODUCTION TO CONNECTIONS ON UI BUNDLES AND BERRYS PHASE | 172 |

52 Light Polarization the Hopf Bundle and Pancharatnams Phase | 173 |

19 Relative Homotopy and Relative Homotopy Groups | 57 |

110 The Exact Homotopy Sequence | 58 |

TOPOLOGICAL METHODS IN CLASSICAL FIELD THEORY | 67 |

22 The Pontrjagin Index and the Hopf Invariant | 74 |

23 The SO3 Nonlinear SigmaModels in One and Two Space Dimensions | 85 |

24 The d2 SO3 Nonlinear SigmaModel as a Gauge Theory | 95 |

25 The GinzburgLandau Theory of Superconductivity | 104 |

INEQUIVALENT QUANTIZATIONS IN MULTIPLY CONNECTED SPACES BRAID GROUPS AND ANYONS | 114 |

32 Quantum Mechanics in Nonsimplyconnected Spaces | 117 |

33 The Case of Identical Particles | 124 |

53 The Quantum Adiabatic Phase | 181 |

ELECTRONS IN A MAGNETIC FIELD AND A CURSORY LOOK AT THE QUANTUM HALL EFFECT | 191 |

62 Preliminaries | 194 |

63 The Classical Hall Effect | 201 |

64 Bloch Electrons in a Magnetic Field | 203 |

65 The Integer Quantum Hall Effect | 210 |

66 The Fractional Quantum Hall Effect | 217 |

67 FractionallyCharged Quasiparticles and the Hierarchy of Quantum Hall States | 222 |

REFERENCES | 231 |

### Common terms and phrases

abelian adiabatic algebra angle associated Berry's phase boundary conditions bundle charge classical configuration space conjugacy class connected spaces consider constraint coordinates corresponding cosets defined denote density Differential discussed electrons elementary excitations f and g Fermi field configurations field equations field theories flux tube FQHE fractional statistics freely homotopic fundamental group gauge field gauge transformation given globally Hall conductivity Hamiltonian hence homotopy class homotopy group homotopy sector Hopf fibration Hopf invariant identical particles implies integer interaction IQHE isomorphism Landau level lattice loop f magnetic field manifold nonlinear nonsimply connected normal obtain one-form order parameter order-parameter space parallel transport path Phys Physics planar spins plane Poincare Pontrjagin index proof prove quantization Quantum Hall Effect Quantum Mechanics quasiholes quasiparticles scalar Sect simply connected solution spacetime sphere subgroup superconducting Theorem topological terms vector potential wave function wavefunction WILCZEK winding number