nLab Geometry, Topology and Physics

This page collects material related to the textbook

• Mikio Nakahara,

  Geometry, Topology and Physics,

  First Edition: Institute of Physics Publishing, 1990. ISBN 0-85274-094-8 (hardcover), 0-85274-095-6 (paperback)

  Second Edition: Institute of Physics Publishing, 2003. ISBN: 9780750306065 (paperback), 9781138413368 (hardcover)

  doi:10.1201/9781315275826,

  pdf

on geometry and topology in mathematical physics, such as in fiber bundles in physics.

Quantum Physics

• quantum physics

Mathematical Preliminaries

Homology Groups

• ordinary homology

• homology groups

Homotopy Groups

Abelian groups

• abelian groups

Simplices and simplicial complexes

• simplex

• simplicial complex

• triangulation

Homology groups of simplicial complexes

• ordinary homology

• homology groups

• homotopy groups

General properties of homology groups

• Betti number

Manifolds

• smooth manifolds

de Rham Cohomology Groups

• de Rham cohomology

Riemannian Geometry

• Riemannian geometry

Complex Manifolds

• complex manifolds

Fibre Bundles

• fiber bundles

• principal bundles

Connections on Fiber Bundles

Connections on principal bundles

• connection on a bundle

Holonomy

• holonomy

Curvature

• curvature

The covariant derivative

• covariant derivative

Gauge theories

• gauge theory

  • electromagnetism

  • magnetic monopole

  • Aharonov-Bohm effect

  • Yang-Mills theory

  • instantons

Berry’s phase

• Berry's phase

Characteristic Classes

• Chern-Weil homomorphism

• characteristic classes

• Chern class

• Pontrjagin class

Index Theorems

• index theory

Anomalies in Gauge Field Theories

• quantum anomaly

• anomaly cancellation

Bosonic String Theory

• string theory

category: reference