## Geometry, Topology and Physics, Second EditionDifferential geometry and topology have become essential tools for many theoretical physicists. In particular, they are indispensable in theoretical studies of condensed matter physics, gravity, and particle physics. Geometry, Topology and Physics, Second Edition introduces the ideas and techniques of differential geometry and topology at a level suitable for postgraduate students and researchers in these fields. The second edition of this popular and established text incorporates a number of changes designed to meet the needs of the reader and reflect the development of the subject. The book features a considerably expanded first chapter, reviewing aspects of path integral quantization and gauge theories. Chapter 2 introduces the mathematical concepts of maps, vector spaces, and topology. The following chapters focus on more elaborate concepts in geometry and topology and discuss the application of these concepts to liquid crystals, superfluid helium, general relativity, and bosonic string theory. Later chapters unify geometry and topology, exploring fiber bundles, characteristic classes, and index theorems. New to this second edition is the proof of the index theorem in terms of supersymmetric quantum mechanics. The final two chapters are devoted to the most fascinating applications of geometry and topology in contemporary physics, namely the study of anomalies in gauge field theories and the analysis of Polakov's bosonic string theory from the geometrical point of view. Geometry, Topology and Physics, Second Edition is an ideal introduction to differential geometry and topology for postgraduate students and researchers in theoretical and mathematical physics. |

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### Indice

Quantum Physics | 1 |

3 | 81 |

Homology Groups | 93 |

Homotopy Groups | 121 |

Manifolds | 169 |

de Rham Cohomology Groups | 226 |

Riemannian Geometry | 244 |

torsion tensor | 256 |

Fibre Bundles | 348 |

Connections on Fibre Bundles | 374 |

Characteristic Classes | 419 |

Index Theorems | 453 |

Anomalies in Gauge Field Theories | 501 |

528 | |

560 | |

Complex Manifolds | 308 |

### Parole e frasi comuni

Abelian action arcwise basis boundary called chart Chern complex manifold components compute coordinate system corresponding covariant derivative curvature curve defined definition denoted differential dimensional Dirac eigenvalue equation equivalence class equivalence relation Euler characteristic example Exercise exists exterior derivative fibre bundle figure follows fundamental group gauge potential given Grassmann Hamiltonian hence Hermitian holomorphic holonomy homeomorphic homology groups homotopy class horizontal lift Hr(M identified identity index theorem inner product invariant isomorphic Kahler Lagrangian lemma Let G Let us consider Levi-Civita connection Lie group linear loop matrix metric Note obtain one-form operator parallel transport path integral principal bundle Proof pullback quantum r-form r-simplex Riemann Riemannian satisfies scalar Show spin structure subgroup subset symmetry theory topological space torus transition function trivial two-form unit element vanishes vector bundle vector field vector space verify

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