D-Branes

by Clifford V. Johnson

• ISBN13: 9780521030052
• ISBN10: 0521030056
• Format: Paperback
• Copyright: 2006-11-02
• Publisher: Cambridge University Press

Summary

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.

Table of Contents

List of inserts	xviii
Preface	xx
1 Overview and overture	1	(23)
1.1 The classical dynamics of geometry	1	(6)
1.2 Gravitons and photons	7	(4)
1.3 Beyond classical gravity: perturbative strings	11	(4)
1.4 Beyond perturbative strings: branes	15	(4)
1.5 The quantum dynamics of geometry	19	(1)
1.6 Things to do in the meantime	20	(2)
1.7 On with the show	22	(2)
2 Relativistic strings	24	(46)
2.1 Motion of classical point particles	24	(3)
2.1.1 Two actions	24	(2)
2.1.2 Symmetries	26	(1)
2.2 Classical bosonic strings	27	(13)
2.2.1 Two actions	27	(2)
2.2.2 Symmetries	29	(1)
2.2.3 String equations of motion	30	(1)
2.2.4 Further aspects of the two dimensional perspective	31	(4)
2.2.5 The stress tensor	35	(1)
2.2.6 Gauge fixing	35	(2)
2.2.7 The mode decomposition	37	(1)
2.2.8 Conformal invariance as a residual symmetry	37	(1)
2.2.9 Some Hamiltonian dynamics	38	(2)
2.3 Quantised bosonic strings	40	(7)
2.3.1 The constraints and physical states	41	(1)
2.3.2 The intercept and critical dimensions	42	(3)
2.3.3 A glance at more sophisticated techniques	45	(2)
2.4 The sphere, the plane and the vertex operator	47	(4)
2.4.1 States and operators	48	(3)
2.5 Chan-Paton factors	51	(1)
2.6 Unoriented strings	52	(4)
2.6.1 Unoriented open strings	52	(2)
2.6.2 Unoriented closed strings	54	(1)
2.6.3 World-sheet diagrams	55	(1)
2.7 Strings in curved backgrounds	56	(5)
2.8 A quick look at geometry	61	(9)
2.8.1 Working with the local tangent frames	61	(2)
2.8.2 Differential forms	63	(2)
2.8.3 Coordinate vs. orthonormal bases	65	(2)
2.8.4 The Lorentz group as a gauge group	67	(1)
2.8.5 Fermions in curved spacetime	68	(1)
2.8.6 Comparison to differential geometry	68	(2)
3 A closer look at the world-sheet	70	(24)
3.1 Conformal invariance	70	(10)
3.1.1 Diverse dimensions	70	(3)
3.1.2 The special case of two dimensions	73	(1)
3.1.3 States and operators	74	(1)
3.1.4 The operator product expansion	75	(1)
3.1.5 The stress tensor and the Virasoro algebra	76	(4)
3.2 Revisiting the relativistic string	80	(5)
3.3 Fixing the conformal gauge	85	(2)
3.3.1 Conformal ghosts	85	(1)
3.3.2 The critical dimension	86	(1)
3.4 The closed string partition function	87	(7)
4 Strings on circles and T-duality	94	(35)
4.1 Fields and strings on a circle	94	(5)
4.1.1 The Kaluza --Klein reduction	95	(1)
4.1.2 Closed strings on a circle	96	(3)
4.2 T-duality for closed strings	99	(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