Geometry, topology and physics
著者
書誌事項
Geometry, topology and physics
（Graduate student series in physics）
Institute of Physics Publishing, c1990
- : hpb
- : pbk
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注記
Bibliography: p. [492]-496
Includes index
内容説明・目次
- 巻冊次
-
: hpb ISBN 9780852740941
内容説明
This book introduces several current mathematical methods to postgraduate students of theoretical physics. This is achieved by presenting applications of the mathematics to physics, high-energy physics, general relativity and condensed matter physics.
目次
- Part 1 Background in physics: path integral and quantum field theories
- gauge theories
- magnetic monopole
- instanton
- orders in condensed matter systems
- general relativity
- Berry's Phase
- string theory. Part 2 Mathematical preliminary: maps
- vector spaces
- topological spaces
- homeomorphism and topological invariants. Part 3 Homology groups: Abelian groups
- simplexes and simplicial complexes
- homology groups of simplicial complexes
- general properties of homology groups. Part 4 Homotopy groups: fundamental groups
- general properties of fundamental groups
- examples of fundamental groups
- fundamental groups of polyhedra
- higher homotopy groups
- general properties of higher homotopy groups
- examples of higher homotopy groups
- defects in nematic liquid crystals
- textures in superfluid 3HE-A. Part 5 Manifolds: calculus on manifolds
- flows and lie derivatives
- differential forms
- integration of differential forms
- lie groups and lie algebras
- action of lie groups on manifolds. Part 6 De Rham cohomology groups: Stokes' theorem
- De Rham cohomology groups
- Poincare's Lemma
- structure of De Rham cohomology groups. Part 7 Riemannain geometry
- Riemannian manifolds and pseudo-Riemannian manifolds
- parallel transport, connection and covariant derivative
- curvature and torsion
- Levi-Civita connections
- holonomy
- isometries and conformal transformations
- killing vector fields and conformal killing vector fields
- non-co-ordinate basis
- differential forms and Hodge theory
- aspects of general relativity and Polyakov string. Part 8 Complex manifolds: calculus on complex manifolds
- complex differential forms
- Hermitian manifolds and Hermitian differential geometry
- Kahler manifolds and Kahler differential geometry
- harmonic forms
- almost complex manifolds. Part 9 Fibre bundles: tangent bundles
- vector bundles
- principal bundles. Part 10 Connections on fibre bundles: connections on principal bundles
- holonomy
- curvature
- covariant derivative on associated vector bundles
- Gauge theories
- Berry's phase. Part 11 Characteristic classes: invariant polynomials and Chern-Weil homomorphism
- Chern classes and characters
- Pontrjagin and Euler classes
- Chern-Simons forms
- Stiefel-Whitney classes. Part 12 Index theorems: elliptic operators and Fredholm operators
- Atiyah-Singer index theorem
- De Rham complex
- Dolbeault complex
- signature complex
- spin complex
- heat kernel and generalized s-functions
- Atiyah-Patodi-Singer index theorem. Part 13 Anomalies in gauge field theories: Abelian anomalies
- non-Abelian anomalies
- Wess-Zumino consistency conditions
- Abelain anomalies versus non-Abelian anomalies
- parity anomaly in odd-dimensions. Part 14 One-loop string amplitude: differential geometry on Riemann surfaces
- quantum theory of Bosonic strings
- one-loop amplitude.
- 巻冊次
-
: pbk ISBN 9780852740958
内容説明
This textbook provides an introduction to the ideas and techniques of differential geometry and topology. It starts with a brief survey of the physics needed to follow the arguments - including quantum field theory, gauge theory and general relativity - to make sure all readers set off from the same starting point. Basic theory of vector spaces and topology is also included to make the book self-contained. Working from the basics to the more elaborate concepts of topology and geometry, all is carefully explained and illustrated with applications. Explicit calculations and diagrams clarify the abstract ideas involved. Many illustrations, exercises and problems are included. This book will be invaluable to advanced undergraduates and researchers in many areas of physics.
目次
Mathematical preliminaries. Homology groups. Homotopy groups/manifolds. De Rham cohomology groups. Riemannian geometry. Complex manifolds. Fibre bundles. Connections on fibre bundles. Characteristic classes. Index theorems. Anomalies in gauge field theories. Bosonic string theory.
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