Geometry, Topology and Physics, Second Edition

by Nakahara; Mikio

Edition: 2nd

ISBN13: 9780750306065

ISBN10: 0750306068

Format: Nonspecific Binding

Pub. Date: 2003-06-04

Publisher(s): CRC Press

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Summary

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.

Preface to the First Edition

xvii

Preface to the Second Edition

xix

How to Read this Book

xxi

Notation and Conventions

xxii

Quantum Physics

(66)

Analytical mechanics

(8)

Newtonian mechanics

(1)

Lagrangian formalism

(3)

Hamiltonian formalism

(4)

Canonical quantization

(10)

Hilbert space, bras and kets

(1)

Axioms of canonical quantization

(3)

Heisenberg equation, Heisenberg picture and Schrodinger picture

(1)

Wavefunction

(4)

Harmonic oscillator

(2)

Path integral quantization of a Bose particle

(12)

Path integral quantization

(7)

Imaginary time and partition function

(2)

Time-ordered product and generating functional

(3)

Harmonic oscillator

(7)

Transition amplitude

(4)

Partition function

(3)

Path integral quantization of a Fermi particle

(13)

Fermionic harmonic oscillator

(1)

Calculus of Grassmann numbers

(1)

Differentiation

(1)

Integration

(1)

Delta-function

(1)

Gaussian integral

(1)

Functional derivative

(1)

Complex conjugation

(1)

Coherent states and completeness relation

(1)

Partition function of a fermionic oscillator

(4)

Quantization of a scalar field

(4)

Free scalar field

(3)

Interacting scalar field

(1)

Quantization of a Dirac field

(1)

Gauge theories

(4)

Abelian gauge theories

(2)

Non-Abelian gauge theories

(2)

Higgs fields

(1)

Magnetic monopoles

(3)

Dirac monopole

(1)

The Wu-Yang monopole

(1)

Charge quantization

(1)

Instantons

(4)

Introduction

(1)

The (anti-) self-dual solution

(2)

Problems

(1)

Mathematical Preliminaries

(26)

Maps

(8)

Definitions

(3)

Equivalence relation and equivalence class

(5)

Vector spaces

(6)

Vectors and vector spaces

(1)

Linear maps, images and kernels

(1)

Dual vector space

(1)

Inner product and adjoint

(2)

Tensors

(1)

Topological spaces

(4)

Definitions

(1)

Continuous maps

(1)

Neighbourhoods and Hausdorff spaces

(1)

Closed set

(1)

Compactness

(2)

Connectedness

(1)

Homeomorphisms and topological invariants

(8)

Homeomorphisms

(1)

Topological invariants

(2)

Homotopy type

(1)

Euler characteristic: an example

(3)

Problems

(2)

Homology Groups

(28)

Abelian groups

(5)

Elementary group theory

(3)

Finitely generated Abelian groups and free Abelian groups

(1)

Cyclic groups

(2)

Simplexes and simplicial complexes

(2)

Simplexes

(1)

Simplicial complexes and polyhedra

(1)

Homology groups of simplicial complexes

100

(17)

Oriented simplexes

100

(2)

Chain group, cycle group and boundary group

102

(4)

Homology groups

106

(4)

Computation of H0(K)

110

(1)

More homology computations

111

(6)

General properties of homology groups

117

(4)

Connectedness and homology groups

117

(1)

Structure of homology groups

118

(1)

Betti numbers and the Euler--Poincare theorem

118

(2)

Problems

120

(1)

Homotopy Groups

121

(48)

Fundamental groups

121

(6)

Basic ideas

121

(1)

Paths and loops

122

(1)

Homotopy

123

(2)

Fundamental groups

125

(2)

General properties of fundamental groups

127

(4)

Arcwise connectedness and fundamental groups

127

(1)

Homotopic invariance of fundamental groups

128

(3)

Examples of fundamental groups

131

(3)

Fundamental group of torus

133

(1)

Fundamental groups of polyhedra

134

(11)

Free groups and relations

134

(2)

Calculating fundamental groups of polyhedra

136

(8)

Relations between H1(K) and π1(|K|)

144

(1)

Higher homotopy groups

145

(3)

Definitions

146

(2)

General properties of higher homotopy groups

148

(2)

Abelian nature of higher homotopy groups

148

(1)

Arcwise connectedness and higher homotopy groups

148

(1)

Homotopy invariance of higher homotopy groups

148

(1)

Higher homotopy groups of a product space

148

(1)

Universal covering spaces and higher homotopy groups

148

(2)

Examples of higher homotopy groups

150

(3)

Orders in condensed matter systems

153

(6)

Order parameter

153

(1)

Superfluid 4He and superconductors

154

(3)

General consideration

157

(2)

Defects in nematic liquid crystals

159

(4)

Order parameter of nematic liquid crystals

159

(1)

Line defects in nematic liquid crystals

160

(1)

Point defects in nematic liquid crystals

161

(1)

Higher dimensional texture

162

(1)

Textures in superfluid 3He-A

163

(6)

Superfluid 3He-A

163

(2)

Line defects and non-singular vortices in 3He-A

165

(1)

Shankar monopole in 3He-A

166

(1)

Problems

167

(2)

Manifolds

169

(57)

Manifolds

169

(9)

Heuristic introduction

169

(2)

Definitions

171

(2)

Examples

173

(5)

The calculus on manifolds

178

(10)

Differentiable maps

179

(2)

Vectors

181

(3)

One-forms

184

(1)

Tensors

185

(1)

Tensor fields

185

(1)

Induced maps

186

(2)

Submanifolds

188

(1)

Flows and Lie derivatives

188

(8)

One-parameter group of transformations

190

(1)

Lie derivatives

191

(5)

Differential forms

196

(8)

Definitions

196

(2)

Exterior derivatives

198

(3)

Interior product and Lie derivative of forms

201

(3)

Integration of differential forms

204

(3)

Orientation

204

(1)

Integration of forms

205

(2)

Lie groups and Lie algebras

207

(9)

Lie groups

207

(2)

Lie algebras

209

(3)

The one-parameter subgroup

212

(3)

Frames and structure equation

215

(1)

The action of Lie groups on manifolds

216

(10)

Definitions

216

(3)

Orbits and isotropy groups

219

(4)

Induced vector fields

223

(1)

The adjoint representation

224

(1)

Problems

224

(2)

de Rham Cohomology Groups

226

(18)

Stokes' theorem

226

(4)

Preliminary consideration

226

(2)

Stokes' theorem

228

(2)

de Rham cohomology groups

230

(5)

Definitions

230

(3)

Duality of Hr(M) and Hr(M); de Rham's theorem

233

(2)

Poincare's lemma

235

(2)

Structure of de Rham cohomology groups

237

(7)

Poincare duality

237

(1)

Cohomology rings

238

(1)

The Kunneth formula

238

(2)

Pullback of de Rham cohomology groups

240

(1)

Homotopy and H1(M)

240

(4)

Riemannian Geometry

244

(64)

Riemannian manifolds and pseudo-Riemannian manifolds

244

(3)

Metric tensors

244

(2)

Induced metric

246

(1)

Parallel transport, connection and covariant derivative

247

(7)

Heuristic introduction

247

(2)

Affine connections

249

(1)

Parallel transport and geodesics

250

(1)

The covariant derivative of tensor fields

251

(1)

The transformation properties of connection coefficients

252

(1)

The metric connection

253

(1)

Curvature and torsion

254

(7)

Definitions

254

(2)

Geometrical meaning of the Riemann tensor and the torsion tensor

256

(4)

The Ricci tensor and the scalar curvature

260

(1)

Levi-Civita connections

261

(10)

The fundamental theorem

261

(1)

The Levi-Civita connection in the classical geometry of surfaces

262

(1)

Geodesics

263

(3)

The normal coordinate system

266

(2)

Riemann curvature tensor with Levi-Civita connection

268

(3)

Holonomy

271

(2)

Isometries and conformal transformations

273

(6)

Isometries

273

(1)

Conformal transformations

274

(5)

Killing vector fields and conformal Killing vector fields

279

(4)

Killing vector fields

279

(3)

Conformal Killing vector fields

282

(1)

Non-coordinate bases

283

(6)

Definitions

283

(1)

Cartan's structure equations

284

(1)

The local frame

285

(2)

The Levi-Civita connection in a non-coordinate basis

287

(2)

Differential forms and Hodge theory

289

(8)

Invariant volume elements

289

(1)

Duality transformations (Hodge star)

290

(1)

Inner products of r-forms

291

(2)

Adjoints of exterior derivatives

293

(1)

The Laplacian, harmonic forms and the Hodge decomposition theorem

294

(2)

Harmonic forms and de Rham cohomology groups

296

(1)

Aspects of general relativity

297

(5)

Introduction to general relativity

297

(1)

Einstein--Hilbert action

298

(2)

Spinors in curved spacetime

300

(2)

Bosonic string theory

302

(6)

The string action

303

(2)

Symmetries of the Polyakov strings

305

(2)

Problems

307

(1)

Complex Manifolds

308

(40)

Complex manifolds

308

(7)

Definitions

308

(1)

Examples

309

(6)

Calculus on complex manifolds

315

(5)

Holomorphic maps

315

(1)

Complexifications

316

(1)

Almost complex structure

317

(3)

Complex differential forms

320

(4)

Complexification of real differential forms

320

(1)

Differential forms on complex manifolds

321

(1)

Dolbeault operators

322

(2)

Hermitian manifolds and Hermitian differential geometry

324

(6)

The Hermitian metric

325

(1)

Kahler form

326

(1)

Covariant derivatives

327

(2)

Torsion and curvature

329

(1)

Kahler manifolds and Kahler differential geometry

330

(6)

Definitions

330

(4)

Kahler geometry

334

(1)

The holonomy group of Kahler manifolds

335

(1)

Harmonic forms and ∂-cohomology groups

336

(5)

The adjoint operators ∂† and ∂†

337

(1)

Laplacians and the Hodge theorem

338

(1)

Laplacians on a Kahler manifold

339

(1)

The Hodge numbers of Kahler manifolds

339

(2)

Almost complex manifolds

341

(3)

Definitions

342

(2)

Orbifolds

344

(4)

One-dimensional examples

344

(2)

Three-dimensional examples

346

(2)

Fibre Bundles

348

(26)

Tangent bundles

348

(2)

Fibre bundles

350

(7)

Definitions

350

(3)

Reconstruction of fibre bundles

353

(1)

Bundle maps

354

(1)

Equivalent bundles

355

(1)

Pullback bundles

355

(2)

Homotopy axiom

357

(1)

Vector bundles

357

(6)

Definitions and examples

357

(2)

Frames

359

(1)

Cotangent bundles and dual bundles

360

(1)

Sections of vector bundles

361

(1)

The product bundle and Whitney sum bundle

361

(2)

Tensor product bundles

363

(1)

Principal bundles

363

(11)

Definitions

363

(7)

Associated bundles

370

(2)

Triviality of bundles

372

(1)

Problems

372

(2)

Connections on Fibre Bundles

374

(45)

Connections on principal bundles

374

(10)

Definitions

375

(1)

The connection one-form

376

(1)

The local connection form and gauge potential

377

(4)

Horizontal lift and parallel transport

381

(3)

Holonomy

384

(1)

Definitions

384

(1)

Curvature

385

(6)

Covariant derivatives in principal bundles

385

(1)

Curvature

386

(2)

Geometrical meaning of the curvature and the Ambrose--Singer theorem

388

(1)

Local form of the curvature

389

(1)

The Bianchi identity

390

(1)

The covariant derivative on associated vector bundles

391

(8)

The covariant derivative on associated bundles

391

(2)

A local expression for the covariant derivative

393

(3)

Curvature rederived

396

(1)

A connection which preserves the inner product

396

(1)

Holomorphic vector bundles and Hermitian inner products

397

(2)

Gauge theories

399

(10)

U(1) gauge theory

399

(1)

The Dirac magnetic monopole

400

(2)

The Aharonov--Bohm effect

402

(2)

Yang--Mills theory

404

(1)

Instantons

405

(4)

Berry's phase

409

(10)

Derivation of Berry's phase

410

(1)

Berry's phase, Berry's connection and Berry's curvature

411

(7)

Problems

418

(1)

Characteristic Classes

419

(34)

Invariant polynomials and the Chern--Weil homomorphism

419

(7)

Invariant polynomials

420

(6)

Chern classes

426

(5)

Definitions

426

(2)

Properties of Chern classes

428

(1)

Splitting principle

429

(1)

Universal bundles and classifying spaces

430

(1)

Chern characters

431

(5)

Definitions

431

(3)

Properties of the Chern characters

434

(1)

Todd classes

435

(1)

Pontrjagin and Euler classes

436

(7)

Pontrjagin classes

436

(3)

Euler classes

439

(3)

Hirzebruch L-polynomial and A-genus

442

(1)

Chern-Simons forms

443

(5)

Definition

443

(1)

The Chern-Simons form of the Chern character

444

(1)

Cartan's homotopy operator and applications

445

(3)

Stiefel--Whitney classes

448

(5)

Spin bundles

449

(1)

Cech cohomology groups

449

(1)

Stiefel--Whitney classes

450

(3)

Index Theorems

453

(48)

Elliptic operators and Fredholm operators

453

(6)

Elliptic operators

454

(2)

Fredholm operators

456

(1)

Elliptic complexes

457

(2)

The Atiyah--Singer index theorem

459

(1)

Statement of the theorem

459

(1)

The de Rham complex

460

(2)

The Dolbeault complex

462

(2)

The twisted Dolbeault complex and the Hirzebruch--Riemann--Roch theorem

463

(1)

The signature complex

464

(3)

The Hirzebruch signature

464

(1)

The signature complex and the Hirzebruch signature theorem

465

(2)

Spin complexes

467

(5)

Dirac operator

468

(3)

Twisted spin complexes

471

(1)

The heat kernel and generalized ζ-functions

472

(5)

The heat kernel and index theorem

472

(3)

Spectral ζ -functions

475

(2)

The Atiyah--Patodi--Singer index theorem

477

(4)

η-invariant and spectral flow

477

(1)

The Atiyah--Patodi--Singer (APS) index theorem

478

(3)

Supersymmetric quantum mechanics

481

(6)

Clifford algebra and fermions

481

(1)

Supersymmetric quantum mechanics in flat space

482

(3)

Supersymmetric quantum mechanics in a general manifold

485

(2)

Supersymmetric proof of index theorem

487

(14)

The index

487

(3)

Path integral and index theorem

490

(10)

Problems

500

(1)

Anomalies in Gauge Field Theories

501

(27)

Introduction

501

(2)

Abelian anomalies

503

(5)

Fujikawa's method

503

(5)

Non-Abelian anomalies

508

(4)

The Wess--Zumino consistency conditions

512

(6)

The Becchi--Rouet--Stora operator and the Faddeev--Popov ghost

512

(1)

The BRS operator, FP ghost and moduli space

513

(2)

The Wess--Zumino conditions

515

(1)

Descent equations and solutions of WZ conditions

515

(3)

Abelian anomalies versus non-Abelian anomalies

518

(5)

m dimensions versus m + 2 dimensions

520

(3)

The parity anomaly in odd-dimensional spaces

523

(5)

The parity anomaly

524

(1)

The dimensional ladder: 4--3--2

525

(3)

Bosonic String Theory

528

(32)

Differential geometry on Riemann surfaces

528

(7)

Metric and complex structure

528

(1)

Vectors, forms and tensors

529

(2)

Covariant derivatives

531

(2)

The Riemann-Roch theorem

533

(2)

Quantum theory of bosonic strings

535

(20)

Vacuum amplitude of Polyakov strings

535

(3)

Measures of integration

538

(12)

Complex tensor calculus and string measure

550

(4)

Moduli spaces of Riemann surfaces

554

(1)

One-loop amplitudes

555

(5)

Moduli spaces, CKV, Beltrami and quadratic differentials

555

(2)

The evaluation of determinants

557

(3)

References

560

(5)

Index

565

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Geometry, Topology and Physics, Second Edition

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