D-Branes

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ISBN: 9780521030052 | 0521030056
Cover: Paperback
Copyright: 11/2/2006

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D-branes represent a key theoretical tool in the understanding of strongly coupled superstring theory and M-theory. They have led to many striking discoveries, including the precise microphysics underlying the thermodynamic behaviour of certain black holes, and remarkable holographic dualities between large-N gauge theories and gravity. This book provides a self-contained introduction to the technology of D-branes, presenting the recent developments and ideas in a pedagogical manner. It is suitable for use as a textbook in graduate courses on modern string theory and theoretical particle physics, and will also be an indispensable reference for seasoned practitioners. The introductory material is developed by first starting with the main features of string theory needed to get rapidly to grips with D-branes, uncovering further aspects while actually working with D-branes. Many advanced applications are covered, with discussions of open problems which could form the basis for new avenues of research.

List of inserts

xviii

Preface

1 Overview and overture

(23)

1.1 The classical dynamics of geometry

(6)

1.2 Gravitons and photons

(4)

1.3 Beyond classical gravity: perturbative strings

(4)

1.4 Beyond perturbative strings: branes

(4)

1.5 The quantum dynamics of geometry

(1)

1.6 Things to do in the meantime

(2)

1.7 On with the show

(2)

2 Relativistic strings

(46)

2.1 Motion of classical point particles

(3)

2.1.1 Two actions

(2)

2.1.2 Symmetries

(1)

2.2 Classical bosonic strings

(13)

2.2.1 Two actions

(2)

2.2.2 Symmetries

(1)

2.2.3 String equations of motion

(1)

2.2.4 Further aspects of the two dimensional perspective

(4)

2.2.5 The stress tensor

(1)

2.2.6 Gauge fixing

(2)

2.2.7 The mode decomposition

(1)

2.2.8 Conformal invariance as a residual symmetry

(1)

2.2.9 Some Hamiltonian dynamics

(2)

2.3 Quantised bosonic strings

(7)

2.3.1 The constraints and physical states

(1)

2.3.2 The intercept and critical dimensions

(3)

2.3.3 A glance at more sophisticated techniques

(2)

2.4 The sphere, the plane and the vertex operator

(4)

2.4.1 States and operators

(3)

2.5 Chan-Paton factors

(1)

2.6 Unoriented strings

(4)

2.6.1 Unoriented open strings

(2)

2.6.2 Unoriented closed strings

(1)

2.6.3 World-sheet diagrams

(1)

2.7 Strings in curved backgrounds

(5)

2.8 A quick look at geometry

(9)

2.8.1 Working with the local tangent frames

(2)

2.8.2 Differential forms

(2)

2.8.3 Coordinate vs. orthonormal bases

(2)

2.8.4 The Lorentz group as a gauge group

(1)

2.8.5 Fermions in curved spacetime

(1)

2.8.6 Comparison to differential geometry

(2)

3 A closer look at the world-sheet

(24)

3.1 Conformal invariance

(10)

3.1.1 Diverse dimensions

(3)

3.1.2 The special case of two dimensions

(1)

3.1.3 States and operators

(1)

3.1.4 The operator product expansion

(1)

3.1.5 The stress tensor and the Virasoro algebra

(4)

3.2 Revisiting the relativistic string

(5)

3.3 Fixing the conformal gauge

(2)

3.3.1 Conformal ghosts

(1)

3.3.2 The critical dimension

(1)

3.4 The closed string partition function

(7)

4 Strings on circles and T-duality

(35)

4.1 Fields and strings on a circle

(5)

4.1.1 The Kaluza --Klein reduction

(1)

4.1.2 Closed strings on a circle

(3)

4.2 T-duality for closed strings

(1)

4.3 A special radius: enhanced gauge symmetry

100

(3)

4.4 The circle partition function

103

(1)

4.5 Toriodal cmnpactifications

104

(4)

4.6 More on enhanced gauge symmetry

108

(5)

4.6.1 Lie algebras and groups

108

(3)

4.6.2 The classical Lie algebras

111

(2)

4.6.3 Physical realisations with vertex operators

113

(1)

4.7 Another special radius: bosonisation

113

(4)

4.8 String theory on an orbifold

117

(2)

4.9 T-duality for open strings: D-branes

119

(4)

4.9.1 Chan—Paton factors and Wilson lines

121

(2)

4.10 D-brane collective coordinates

123

(2)

4.11 T-duality for unoriented strings: orientifolds

125

(4)

5 Background fields and world-volume actions

129

(12)

5.1 T-duality in background fields

129

(2)

5.2 A first look at the D-brane world-volume action

131

(4)

5.2.1 World-volume actions from tilted D-branes

133

(2)

5.3 The Dirac—Born—Infeld action

135

(1)

5.4 The action of T-duality

136

(1)

5.5 Non-Abelian extensions

136

(2)

5.6 D-braves and gauge theory

138

(1)

5.7 BPS lumps on the world-volume

138

(3)

6 D-brane tension and boundary states

141

(14)

6.1 The D-brane tension

142

(6)

6.1.1 An open string partition function

142

(3)

6.1.2 A background field computation

145

(3)

6.2 The orientifold tension

148

(2)

6.2.1 Another open string partition function

148

(2)

6.3 The boundary state formalism

150

(5)

7 Supersymmetric strings

155

(37)

7.1 The three basic superstring theories

155

(14)

7.1.1 Open superstrings: type I

155

(5)

7.1.2 Closed superstrings: type II

160

(5)

7.1.3 Type I from type IIB, the prototype orientifold

165

(1)

7.1.4 The Green—Schwarz mechanism

166

(3)

7.2 The two basic heterotic string theories

169

(5)

7.2.1 SO(32) and E8 x E8 from self-dual lattices

171

(1)

7.2.2 The massless spectrum

172

(2)

7.3 The ten dimensional supergravities

174

(2)

7.4 Heterotic toroidal compactifications

176

(2)

7.5 Superstring toroidal compactification

178

(1)

7.6 A superstring orbifold: discovering the K3 manifold

179

(13)

7.6.1 The orbifold spectrum

180

(3)

7.6.2 Another miraculous anomaly cancellation

183

(1)

7.6.3 The K3 manifold

184

(1)

7.6.4 Blowing up the orbifold

185

(4)

7.6.5 Some other K3 orbifolds

189

(2)

7.6.6 Anticipating D-manifolds

191

(1)

8 Supersymmetric strings and T-duality

192

(13)

8.1 T-duality of supersymmetric strings

192

(3)

8.1.1 T-duality of type II superstrings

192

(1)

8.1.2 T-duality of type I superstrings

193

(1)

8.1.3 T-duality for the heterotic strings

194

(1)

8.2 D-branes as BPS solitons

195

(2)

8.3 The D-brane charge and tension

197

(3)

8.4 The orientifold charge and tension

200

(1)

8.5 Type I from type IIB, revisited

201

(1)

8.6 Dirac charge quantisation

201

(1)

8.7 D-branes in type I

202

(3)

9 World-volume curvature couplings

205

(19)

9.1 Tilted D-branes and branes within branes

205

(1)

9.2 Anomalous gauge couplings

206

(4)

9.3 Characteristic classes and invariant polynomials

210

(6)

9.4 Anomalous curvature couplings

216

(2)

9.5 A relation to anomalies

218

(2)

9.6 D-branes and K-theory

220

(1)

9.7 Further non-Abelian extensions

221

(1)

9.8 Further curvature couplings

222

(2)

10 The geometry of D-branes

224

(25)

10.1 A look at black holes in four dimensions

224

(14)

10.1.1 A brief study of the Einstein-Maxwell system

224

(1)

10.1.2 Basic properties of Schwarzschild

225

(3)

10.1.3 Basic properties of R,eissner-Nordstrom

228

(1)

10.1.4 Extremality, supersymmetry, and the BPS condition

228

(4)

10.1.5 Multiple black holes and multicentre solutions

232

(1)

10.1.6 Near horizon geometry and an infinite throat

233

(1)

10.1.7 Cosmological constant; de Sitter and anti-dc Sitter

233

(1)

10.1.8 de-Sitter spacetime and the sphere

234

(1)

10.1.9 Anti-de Sitter in various coordinate systems

235

(1)

10.1.10 Anti-de Sitter as a hyperbolic slice

236

(1)

10.1.11 Revisiting the extremal solution

237

(1)

10.2 Tle geometry of D-branes

238

(5)

10.2.1 A family of 'p-brane' solutions

238

(1)

10.2.2 The boost form of solution

239

(1)

10.2.3 The extremal limit and coincident D-branes

240

(3)

10.3 Probing p-brane geometry with Dp-branes

243

(3)

10.3.1 Thought experiment: building p with Dp

243

(1)

10.3.2 Effective Lagrangian from the world-volume action

244

(1)

10.3.3 A metric on moduli space

245

(1)

10.4 T-duality and supergravity solutions

246

(3)

10.4.1 D(p + 1) from Dp

246

(2)

10.4.2 D(p — 1) from Dp

248

(1)

11 Multiple D-branes and bound states

249

(12)

11.1 Dp and Dp' from boundary conditions

249

(3)

11.2 The BPS bound for the Dp—Dp' system

252

(2)

11.3 Bound states of fundamental strings and D-strings

254

(1)

11.4 The three-string junction

255

(3)

11.5 Aspects of D-brane bound states

258

(3)

11.5.1 0-0 bound states

258

(1)

11.5.2 0-2 bound states

258

(1)

11.5.3 0-4 bound states

259

(1)

11.5.4 0-6 bound states

260

(1)

11.5.5 0-8 bound states

260

(1)

12 Strong coupling and string duality

261

(21)

12.1 Type IIB/type IIB duality

261

(3)

12.1.1 D1-brane collective coordinates

261

(2)

12.1.2 S-duality and SL(2,Z)

263

(1)

12.2 SO(32) Type I/heterotic duality

264

(1)

12.2.1 D1-brane collective coordinates

264

(1)

12.3 Dual branes from 10D string—string duality

265

(6)

12.3.1 The heterotic NS-fivebrane

267

(1)

12.3.2 The type IIA and type IIB NS5-brave

268

(3)

12.4 Type IIA/M-theory duality

271

(2)

12.4.1 A closer look at D0-branes

271

(1)

12.4.2 Eleven dimensional supergravity

271

(2)

12.5 E8 x E8 heterotic string/M-theory duality

273

(3)

12.6 M2-branes and M5-branes

276

(2)

12.6.1 Supergravity solutions

276

(1)

12.6.2 From D-branes and NS5-branes to M-branes and back

277

(1)

12.7 U-duality

278

(4)

12.7.1 Type II strings on T5 and E6(6)

278

(1)

12.7.2 U-duality and bound states

279

(3)

13 D-branes and geometry I

282

(40)

13.1 D-branes as probes of ALE spaces

282

(9)

13.1.1 Basic setup and a quiver gauge theory

282

(3)

13.1.2 The moduli space of vacua

285

(1)

13.1.3 ALE space as metric on moduli space

286

(3)

13.1.4 D-branes and the hyper-Kahler quotient

289

(2)

13.2 Fractional D-branes and wrapped D-branes

291

(3)

13.2.1 Fractional branes

291

(1)

13.2.2 Wrapped branes

292

(2)

13.3 Wrapped, fractional and stretched branes

294

(6)

13.3.1 NS5-branes from ALE spaces

295

(1)

13.3.2 Dual realisations of quivers

296

(4)

13.4 D-branes as instantons

300

(6)

13.4.1 Seeing the instanton with a probe

301

(4)

13.4.2 Small instantons

305

(1)

13.5 D-branes as monopoles

306

(8)

13.5.1 Adjoint Higgs and monopoles

309

(2)

13.5.2 BPS monopole solution from Nahm data

311

(3)

13.6 The D-brane dielectric effect

314

(8)

13.6.1 Non-Abelian world-volume interactions

314

(2)

13.6.2 Stable fuzzy spherical D-branes

316

(2)

13.6.3 Stable smooth spherical D-branes

318

(4)

14 K3 orientifolds and compactification

322

(23)

14.1 ZN orientifolds and Chan–Paton factors

322

(2)

14.2 Loops and tadpoles for ALE ZAJ singularities

324

(9)

14.2.1 One-loop diagrams and tadpoles

324

(1)

14.2.2 Computing the one-loop diagrams

325

(5)

14.2.3 Extracting the tadpoles

330

(3)

14.3 Solving the tadpole equations

333

(3)

14.3.1 T-duality relations

333

(1)

14.3.2 Explicit solutions

334

(2)

14.4 Closed string spectra

336

(3)

14.5 Open string spectra

339

(2)

14.6 Anomalies for N = 1 in six dimensions

341

(4)

15 D-branes and geometry II

345

(22)

15.1 Probing p with D(p — 4)

345

(1)

15.2 Probing six-branes: Kaluza—Klein monopoles and M-theory

346

(2)

15.3 The moduli space of 3D supersymmetric gauge theory

348

(4)

15.4 Wrapped branes and the enhancon mechanism

352

(8)

15.1.1 Wrapping D6-branes

353

(1)

15.1.2 The repulson geometry

354

(2)

15.1.3 Probing with a wrapped D6-brane

356

(4)

15.5 The consistency of excision in supergravity

360

(2)

15.6 The moduli space of pure glue in 3D

362

(5)

15.6.1 Multi-monopole moduli space

363

(4)

16 Towards M- and F-theory

367

(42)

16.1 The type IIB string and F-theory

367

(27)

16.1.1 SL(2,Z) duality

368

(1)

16.1.2 The (p,q) strings

369

(2)

16.1.3 String networks

371

(2)

16.1.4 The self-duality of D3-branes

373

(2)

16.1.5 (p,q) Fivebranes

375

(1)

16.1.6 SL(2 Z) and D7-branes

376

(3)

16.1.7 Some algebraic geometry

379

(4)

16.1.8 F-theory, and a dual heterotic description

383

(1)

16.1.9 p,q) Sevenbranes

384

(2)

16.1.10 Enhanced gauge symmetry and singularities of K3

386

(1)

16.1.11 F-theory at constant coupling

387

(5)

16.1.12 The moduli space of N = 2 SU (N) with Nf = 4

392

(2)

16.2 M-theory origins of F-theory

394

(6)

16.2.1 M-branes and odd D-branes

396

(3)

16.2.2 M-theory on K3 and heterotic on T³

399

(1)

16.2.3 Type IIA on K3 and heterotic on T4

400

(1)

16.3 Matrix theory

400

(9)

16.3.1 Another look at D0-branes

401

(1)

16.3.2 The infinite momentum frame

402

(2)

16.3.3 Matrix string theory

404

(5)

17 D-branes and black holes

409

(31)

17.1 Black hole thermodynamics

409

(5)

17.1.1 The path integral and the Euclidean calculus

409

(2)

17.1.2 The semiclassical approximation

411

(1)

17.1.3 The temperature of black holes

412

(2)

17.2 The Euclidean action calculus

414

(4)

17.2.1 The action for Schwarzschild

414

(2)

17.2.2 The action for Reissner-Nordström

416

(1)

17.2.3 The laws of thermodynamics

417

(1)

17.3 D = 5 Reissner-Nordström black holes

418

(11)

17.3.1 Making the black hole

420

(5)

17.3.2 Microscopic entropy and a 2D field theory

425

(2)

17.3.3 Non-extremality and a 2D dilute gas limit

427

(2)

17.4 Near horizon geometry

429

(3)

17.5 Replacing T4 with K3

432

(9)

17.5.1 The geometry

432

(1)

17.5.2 The microscopic entropy

433

(1)

17.5.3 Probing the black hole with branes

434

(3)

17.5.4 The enhançon and the second law

437

(3)

18 D-branes, gravity and gauge theory

440

(27)

18.1 The AdS/CFT correspondence

441

(11)

18.1.1 Branes and the decoupling limit

441

(2)

18.1.2 Sphere reduction and gauged supergravity

443

(3)

18.1.3 Extracts from the dictionary

446

(3)

18.1.4 The action, counterterms, and the stress tensor

449

(3)

18.2 The correspondence at finite temperature

452

(3)

18.2.1 Limits of the non-extremal D3-brane

452

(1)

18.2.2 The AdS-Schwarzschild black hole in global coordinates

453

(2)

18.3 The correspondence with a chemical potential

455

(9)

18.3.1 Spinning D3-branes and charged AdS black holes

455

(4)

18.3.2 The AdS Reissner Nordstrom black hole

459

(1)

18.3.3 Thermodynamic phase structure

459

(5)

18.4 The holographic principle

464

(3)

19 The holographic renormalisation group

467

(37)

19.1 Benormalisation group flows from gravity

467

(5)

19.1.1 A BPS domain wall and supersymmetry

469

(3)

19.2 Flowing on the Coulomb branch

472

(8)

19.2.1 A five dimensional solution

472

(3)

19.2.2 A ten dimensional solution

475

(1)

19.2.3 Probing the geometry

475

(3)

19.2.4 Brane distributions

478

(2)

19.3 An N = 1 gauge dual RG flow

480

(14)

19.3.1 The five dimensional solution

482

(4)

19.3.2 The ten dimensional solution

486

(1)

19.3.3 Probing with a D3-brane

487

(1)

19.3.4 The Coulomb branch

488

(1)

19.3.5 Kähler structure of the Coulomb branch

489

(5)

19.4 An N=2 gauge dual RG flow and the enhançon

494

(8)

19.4.1 The five dimensional solution

494

(4)

19.4.2 The ten dimensional solution

498

(1)

19.4.3 Probing with a D3-brane

499

(1)

19.4.4 The moduli space

500

(2)

19.5 Beyond gravity duals

502

(2)

20 Taking stock

504

(6)

References

510

(19)

Index

529

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D-Branes

Summary

Table of Contents

Supplemental Materials