Differential 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.
"a very impressive book.." -- Australian and New Zealand Physicists" The clarity of the presentation is enhanced by explicit calculations and diagrams; the proof of a theorem is given only when it is instructive and not very technical. There is also a large number of exercises and problems, and last but not least, an index superb layout" -- Zentralblatt fur Mathematick un ihre Grenzgebiete "I believe that the book will not only boost modernization of the traditional courses of theoretical physics but will prompt the specialist in topology and differential geometry to have a closer look at the applications. So I welcome this second edition." --Christopher Gilmour
Preface to the First Edition | p. xvii |
Preface to the Second Edition | p. xix |
How to Read this Book | p. xxi |
Notation and Conventions | p. xxii |
Quantum Physics | p. 1 |
Analytical mechanics | p. 1 |
Newtonian mechanics | p. 1 |
Lagrangian formalism | p. 2 |
Hamiltonian formalism | p. 5 |
Canonical quantization | p. 9 |
Hilbert space, bras and kets | p. 9 |
Axioms of canonical quantization | p. 10 |
Heisenberg equation, Heisenberg picture and Schrodinger picture | p. 13 |
Wavefunction | p. 13 |
Harmonic oscillator | p. 17 |
Path integral quantization of a Bose particle | p. 19 |
Path integral quantization | p. 19 |
Imaginary time and partition function | p. 26 |
Time-ordered product and generating functional | p. 28 |
Harmonic oscillator | p. 31 |
Transition amplitude | p. 31 |
Partition function | p. 35 |
Path integral quantization of a Fermi particle | p. 38 |
Fermionic harmonic oscillator | p. 39 |
Calculus of Grassmann numbers | p. 40 |
Differentiation | p. 41 |
Integration | p. 42 |
Delta-function | p. 43 |
Gaussian integral | p. 44 |
Functional derivative | p. 45 |
Complex conjugation | p. 45 |
Coherent states and completeness relation | p. 46 |
Partition function of a fermionic oscillator | p. 47 |
Quantization of a scalar field | p. 51 |
Free scalar field | p. 51 |
Interacting scalar field | p. 54 |
Quantization of a Dirac field | p. 55 |
Gauge theories | p. 56 |
Abelian gauge theories | p. 56 |
Non-Abelian gauge theories | p. 58 |
Higgs fields | p. 60 |
Magnetic monopoles | p. 60 |
Dirac monopole | p. 61 |
The Wu-Yang monopole | p. 62 |
Charge quantization | p. 62 |
Instantons | p. 63 |
Introduction | p. 63 |
The (anti-)self-dual solution | p. 64 |
Problems | p. 66 |
Mathematical Preliminaries | p. 67 |
Maps | p. 67 |
Definitions | p. 67 |
Equivalence relation and equivalence class | p. 70 |
Vector spaces | p. 75 |
Vectors and vector spaces | p. 75 |
Linear maps, images and kernels | p. 76 |
Dual vector space | p. 77 |
Inner product and adjoint | p. 78 |
Tensors | p. 80 |
Topological spaces | p. 81 |
Definitions | p. 81 |
Continuous maps | p. 82 |
Neighbourhoods and Hausdorff spaces | p. 82 |
Closed set | p. 83 |
Compactness | p. 83 |
Connectedness | p. 85 |
Homeomorphisms and topological invariants | p. 85 |
Homeomorphisms | p. 85 |
Topological invariants | p. 86 |
Homotopy type | p. 88 |
Euler characteristic: an example | p. 88 |
Problems | p. 91 |
Homology Groups | p. 93 |
Abelian groups | p. 93 |
Elementary group theory | p. 93 |
Finitely generated Abelian groups and free Abelian groups | p. 96 |
Cyclic groups | p. 96 |
Simplexes and simplicial complexes | p. 98 |
Simplexes | p. 98 |
Simplicial complexes and polyhedra | p. 99 |
Homology groups of simplicial complexes | p. 100 |
Oriented simplexes | p. 100 |
Chain group, cycle group and boundary group | p. 102 |
Homology groups | p. 106 |
Computation of H[subscript 0](K) | p. 110 |
More homology computations | p. 111 |
General properties of homology groups | p. 117 |
Connectedness and homology groups | p. 117 |
Structure of homology groups | p. 118 |
Betti numbers and the Euler-Poincare theorem | p. 118 |
Problems | p. 120 |
Homotopy Groups | p. 121 |
Fundamental groups | p. 121 |
Basic ideas | p. 121 |
Paths and loops | p. 122 |
Homotopy | p. 123 |
Fundamental groups | p. 125 |
General properties of fundamental groups | p. 127 |
Arcwise connectedness and fundamental groups | p. 127 |
Homotopic invariance of fundamental groups | p. 128 |
Examples of fundamental groups | p. 131 |
Fundamental group of torus | p. 133 |
Fundamental groups of polyhedra | p. 134 |
Free groups and relations | p. 134 |
Calculating fundamental groups of polyhedra | p. 136 |
Relations between H[subscript 1](K) and [pi subscript 1]([vertical bar]K[vertical bar]) | p. 144 |
Higher homotopy groups | p. 145 |
Definitions | p. 146 |
General properties of higher homotopy groups | p. 148 |
Abelian nature of higher homotopy groups | p. 148 |
Arcwise connectedness and higher homotopy groups | p. 148 |
Homotopy invariance of higher homotopy groups | p. 148 |
Higher homotopy groups of a product space | p. 148 |
Universal covering spaces and higher homotopy groups | p. 148 |
Examples of higher homotopy groups | p. 150 |
Orders in condensed matter systems | p. 153 |
Order parameter | p. 153 |
Superfluid [superscript 4]He and superconductors | p. 154 |
General consideration | p. 157 |
Defects in nematic liquid crystals | p. 159 |
Order parameter of nematic liquid crystals | p. 159 |
Line defects in nematic liquid crystals | p. 160 |
Point defects in nematic liquid crystals | p. 161 |
Higher dimensional texture | p. 162 |
Textures in superfluid [superscript 3]He-A | p. 163 |
Superfluid [superscript 3]He-A | p. 163 |
Line defects and non-singular vortices in [superscript 3]He-A | p. 165 |
Shankar monopole in [superscript 3]He-A | p. 166 |
Problems | p. 167 |
Manifolds | p. 169 |
Manifolds | p. 169 |
Heuristic introduction | p. 169 |
Definitions | p. 171 |
Examples | p. 173 |
The calculus on manifolds | p. 178 |
Differentiable maps | p. 179 |
Vectors | p. 181 |
One-forms | p. 184 |
Tensors | p. 185 |
Tensor fields | p. 185 |
Induced maps | p. 186 |
Submanifolds | p. 188 |
Flows and Lie derivatives | p. 188 |
One-parameter group of transformations | p. 190 |
Lie derivatives | p. 191 |
Differential forms | p. 196 |
Definitions | p. 196 |
Exterior derivatives | p. 198 |
Interior product and Lie derivative of forms | p. 201 |
Integration of differential forms | p. 204 |
Orientation | p. 204 |
Integration of forms | p. 205 |
Lie groups and Lie algebras | p. 207 |
Lie groups | p. 207 |
Lie algebras | p. 209 |
The one-parameter subgroup | p. 212 |
Frames and structure equation | p. 215 |
The action of Lie groups on manifolds | p. 216 |
Definitions | p. 216 |
Orbits and isotropy groups | p. 219 |
Induced vector fields | p. 223 |
The adjoint representation | p. 224 |
Problems | p. 224 |
de Rham Cohomology Groups | p. 226 |
Stokes' theorem | p. 226 |
Preliminary consideration | p. 226 |
Stokes' theorem | p. 228 |
de Rham cohomology groups | p. 230 |
Definitions | p. 230 |
Duality of H[subscript r](M) and H[superscript r](M); de Rham's theorem | p. 233 |
Poincare's lemma | p. 235 |
Structure of de Rham cohomology groups | p. 237 |
Poincare duality | p. 237 |
Cohomology rings | p. 238 |
The Kunneth formula | p. 238 |
Pullback of de Rham cohomology groups | p. 240 |
Homotopy and H[superscript 1](M) | p. 240 |
Riemannian Geometry | p. 244 |
Riemannian manifolds and pseudo-Riemannian manifolds | p. 244 |
Metric tensors | p. 244 |
Induced metric | p. 246 |
Parallel transport, connection and covariant derivative | p. 247 |
Heuristic introduction | p. 247 |
Affine connections | p. 249 |
Parallel transport and geodesics | p. 250 |
The covariant derivative of tensor fields | p. 251 |
The transformation properties of connection coefficients | p. 252 |
The metric connection | p. 253 |
Curvature and torsion | p. 254 |
Definitions | p. 254 |
Geometrical meaning of the Riemann tensor and the torsion tensor | p. 256 |
The Ricci tensor and the scalar curvature | p. 260 |
Levi-Civita connections | p. 261 |
The fundamental theorem | p. 261 |
The Levi-Civita connection in the classical geometry of surfaces | p. 262 |
Geodesics | p. 263 |
The normal coordinate system | p. 266 |
Riemann curvature tensor with Levi-Civita connection | p. 268 |
Holonomy | p. 271 |
Isometries and conformal transformations | p. 273 |
Isometries | p. 273 |
Conformal transformations | p. 274 |
Killing vector fields and conformal Killing vector fields | p. 279 |
Killing vector fields | p. 279 |
Conformal Killing vector fields | p. 282 |
Non-coordinate bases | p. 283 |
Definitions | p. 283 |
Cartan's structure equations | p. 284 |
The local frame | p. 285 |
The Levi-Civita connection in a non-coordinate basis | p. 287 |
Differential forms and Hodge theory | p. 289 |
Invariant volume elements | p. 289 |
Duality transformations (Hodge star) | p. 290 |
Inner products of r-forms | p. 291 |
Adjoints of exterior derivatives | p. 293 |
The Laplacian, harmonic forms and the Hodge decomposition theorem | p. 294 |
Harmonic forms and de Rham cohomology groups | p. 296 |
Aspects of general relativity | p. 297 |
Introduction to general relativity | p. 297 |
Einstein-Hilbert action | p. 298 |
Spinors in curved spacetime | p. 300 |
Bosonic string theory | p. 302 |
The string action | p. 303 |
Symmetries of the Polyakov strings | p. 305 |
Problems | p. 307 |
Complex Manifolds | p. 308 |
Complex manifolds | p. 308 |
Definitions | p. 308 |
Examples | p. 309 |
Calculus on complex manifolds | p. 315 |
Holomorphic maps | p. 315 |
Complexifications | p. 316 |
Almost complex structure | p. 317 |
Complex differential forms | p. 320 |
Complexification of real differential forms | p. 320 |
Differential forms on complex manifolds | p. 321 |
Dolbeault operators | p. 322 |
Hermitian manifolds and Hermitian differential geometry | p. 324 |
The Hermitian metric | p. 325 |
Kahler form | p. 326 |
Covariant derivatives | p. 327 |
Torsion and curvature | p. 329 |
Kahler manifolds and Kahler differential geometry | p. 330 |
Definitions | p. 330 |
Kahler geometry | p. 334 |
The holonomy group of Kahler manifolds | p. 335 |
Harmonic forms and [characters not reproducible]-cohomology groups | p. 336 |
The adjoint operators [characters not reproducible] and [characters not reproducible] | p. 337 |
Laplacians and the Hodge theorem | p. 338 |
Laplacians on a Kahler manifold | p. 339 |
The Hodge numbers of Kahler manifolds | p. 339 |
Almost complex manifolds | p. 341 |
Definitions | p. 342 |
Orbifolds | p. 344 |
One-dimensional examples | p. 344 |
Three-dimensional examples | p. 346 |
Fibre Bundles | p. 348 |
Tangent bundles | p. 348 |
Fibre bundles | p. 350 |
Definitions | p. 350 |
Reconstruction of fibre bundles | p. 353 |
Bundle maps | p. 354 |
Equivalent bundles | p. 355 |
Pullback bundles | p. 355 |
Homotopy axiom | p. 357 |
Vector bundles | p. 357 |
Definitions and examples | p. 357 |
Frames | p. 359 |
Cotangent bundles and dual bundles | p. 360 |
Sections of vector bundles | p. 361 |
The product bundle and Whitney sum bundle | p. 361 |
Tensor product bundles | p. 363 |
Principal bundles | p. 363 |
Definitions | p. 363 |
Associated bundles | p. 370 |
Triviality of bundles | p. 372 |
Problems | p. 372 |
Connections on Fibre Bundles | p. 374 |
Connections on principal bundles | p. 374 |
Definitions | p. 375 |
The connection one-form | p. 376 |
The local connection form and gauge potential | p. 377 |
Horizontal lift and parallel transport | p. 381 |
Holonomy | p. 384 |
Definitions | p. 384 |
Curvature | p. 385 |
Covariant derivatives in principal bundles | p. 385 |
Curvature | p. 386 |
Geometrical meaning of the curvature and the Ambrose-Singer theorem | p. 388 |
Local form of the curvature | p. 389 |
The Bianchi identity | p. 390 |
The covariant derivative on associated vector bundles | p. 391 |
The covariant derivative on associated bundles | p. 391 |
A local expression for the covariant derivative | p. 393 |
Curvature rederived | p. 396 |
A connection which preserves the inner product | p. 396 |
Holomorphic vector bundles and Hermitian inner products | p. 397 |
Gauge theories | p. 399 |
U(1) gauge theory | p. 399 |
The Dirac magnetic monopole | p. 400 |
The Aharonov-Bohm effect | p. 402 |
Yang-Mills theory | p. 404 |
Instantons | p. 405 |
Berry's phase | p. 409 |
Derivation of Berry's phase | p. 410 |
Berry's phase, Berry's connection and Berry's curvature | p. 411 |
Problems | p. 418 |
Characteristic Classes | p. 419 |
Invariant polynomials and the Chern-Weil homomorphism | p. 419 |
Invariant polynomials | p. 420 |
Chern classes | p. 426 |
Definitions | p. 426 |
Properties of Chern classes | p. 428 |
Splitting principle | p. 429 |
Universal bundles and classifying spaces | p. 430 |
Chern characters | p. 431 |
Definitions | p. 431 |
Properties of the Chern characters | p. 434 |
Todd classes | p. 435 |
Pontrjagin and Euler classes | p. 436 |
Pontrjagin classes | p. 436 |
Euler classes | p. 439 |
Hirzebruch L-polynomial and A-genus | p. 442 |
Chern-Simons forms | p. 443 |
Definition | p. 443 |
The Chern-Simons form of the Chern character | p. 444 |
Cartan's homotopy operator and applications | p. 445 |
Stiefel-Whitney classes | p. 448 |
Spin bundles | p. 449 |
Cech cohomology groups | p. 449 |
Stiefel-Whitney classes | p. 450 |
Index Theorems | p. 453 |
Elliptic operators and Fredholm operators | p. 453 |
Elliptic operators | p. 454 |
Fredholm operators | p. 456 |
Elliptic complexes | p. 457 |
The Atiyah-Singer index theorem | p. 459 |
Statement of the theorem | p. 459 |
The de Rham complex | p. 460 |
The Dolbeault complex | p. 462 |
The twisted Dolbeault complex and the Hirzebruch-Riemann-Roch theorem | p. 463 |
The signature complex | p. 464 |
The Hirzebruch signature | p. 464 |
The signature complex and the Hirzebruch signature theorem | p. 465 |
Spin complexes | p. 467 |
Dirac operator | p. 468 |
Twisted spin complexes | p. 471 |
The heat kernel and generalized [zeta]-functions | p. 472 |
The heat kernel and index theorem | p. 472 |
Spectral [zeta]-functions | p. 475 |
The Atiyah-Patodi-Singer index theorem | p. 477 |
[eta]-invariant and spectral flow | p. 477 |
The Atiyah-Patodi-Singer (APS) index theorem | p. 478 |
Supersymmetric quantum mechanics | p. 481 |
Clifford algebra and fermions | p. 481 |
Supersymmetric quantum mechanics in flat space | p. 482 |
Supersymmetric quantum mechanics in a general manifold | p. 485 |
Supersymmetric proof of index theorem | p. 487 |
The index | p. 487 |
Path integral and index theorem | p. 490 |
Problems | p. 500 |
Anomalies in Gauge Field Theories | p. 501 |
Introduction | p. 501 |
Abelian anomalies | p. 503 |
Fujikawa's method | p. 503 |
Non-Abelian anomalies | p. 508 |
The Wess-Zumino consistency conditions | p. 512 |
The Becchi-Rouet-Stora operator and the Faddeev-Popov ghost | p. 512 |
The BRS operator, FP ghost and moduli space | p. 513 |
The Wess-Zumino conditions | p. 515 |
Descent equations and solutions of WZ conditions | p. 515 |
Abelian anomalies versus non-Abelian anomalies | p. 518 |
m dimensions versus m + 2 dimensions | p. 520 |
The parity anomaly in odd-dimensional spaces | p. 523 |
The parity anomaly | p. 524 |
The dimensional ladder: 4-3-2 | p. 525 |
Bosonic String Theory | p. 528 |
Differential geometry on Riemann surfaces | p. 528 |
Metric and complex structure | p. 528 |
Vectors, forms and tensors | p. 529 |
Covariant derivatives | p. 531 |
The Riemann-Roch theorem | p. 533 |
Quantum theory of bosonic strings | p. 535 |
Vacuum amplitude of Polyakov strings | p. 535 |
Measures of integration | p. 538 |
Complex tensor calculus and string measure | p. 550 |
Moduli spaces of Riemann surfaces | p. 554 |
One-loop amplitudes | p. 555 |
Moduli spaces, CKV, Beltrami and quadratic differentials | p. 555 |
The evaluation of determinants | p. 557 |
References | p. 560 |
Index | p. 565 |
Table of Contents provided by Rittenhouse. All Rights Reserved. |
ISBN: 9780750306065
ISBN-10: 0750306068
Series: Graduate Student Series in Physics
Audience:
Professional
Format:
Paperback
Language:
English
Number Of Pages: 596
Published: 4th June 2003
Publisher: Taylor & Francis Ltd
Country of Publication: GB
Dimensions (cm): 23.4 x 15.9
x 3.2
Weight (kg): 1.03
Edition Number: 2
Edition Type: New edition