Non-Perturbative Field Theory

Half-title......Page 3
Series-title......Page 4
Title......Page 7
Copyright......Page 8
Dedication......Page 9
Contents......Page 11
Preface......Page 17
Acknowledgements......Page 20
PART I Non-perturbative methods in two-dimensional field theory......Page 21
1.1 Complex geometry......Page 23
1.2 Free massless scalar field......Page 24
1.3 Symmetries of the classical action......Page 25
1.4 Mode expansion......Page 26
1.6 Canonical quantization......Page 27
1.7 Radial quantization......Page 29
1.8 Operator product expansion......Page 31
1.9 Path integral quantization......Page 32
1.10 Affine current algebra......Page 33
1.11 Virasoro algebra......Page 34
2.1 Conformal symmetry in two dimensions......Page 37
2.2 Primary fields......Page 38
2.3 Conformal properties of the energy-momentum tensor......Page 40
2.4 Virasoro algebra for CFT......Page 41
2.5 Descendant operators......Page 42
2.6 Hilbert space of states......Page 43
2.7 Unitary CFT and Kac determinant......Page 45
2.8 Characters......Page 48
2.9 Correlators and the conformal Ward identity......Page 49
2.10 Crossing symmetry, duality and bootstrap......Page 51
2.11 Verlinde's formula......Page 53
2.12 Free Majorana fermions – an example of a CFT......Page 54
2.13 The Ising model – the m = 3 unitary minimal model......Page 57
3.1 Simple finite-dimensional Lie algebras......Page 59
3.1.2 Cartan matrix and Dynkin diagrams......Page 61
3.1.3 Highest weight states......Page 63
3.2 Affine current algebra......Page 64
3.2.1 Cartan matrix and Dynkin diagrams......Page 66
3.2.2 The Weyl group......Page 67
3.2.3 Highest weight representations......Page 68
3.3 Current OPEs and the Sugawara construction......Page 69
3.4 Primary fields......Page 71
3.5 ALA characters......Page 72
3.6 Correlators, null vectors and the Knizhnik-Zamolodchikov equation......Page 73
3.7.1 Free Majorana fermions and…......Page 75
3.7.2 Primary fields......Page 77
3.8 Free Dirac fermions and the…......Page 78
4.1 From free massless scalar theory to the WZW model......Page 81
4.2 Perturbative conformal invariance......Page 85
4.3 ALA, Sugawara construction and the Virasoro algebra......Page 86
4.4 Correlation functions of primary fields......Page 87
4.5 WZW models with boundaries – D branes......Page 91
4.6 G/H coset models......Page 93
4.7 G/G coset models......Page 95
5.2 From the theory of a massive free scalar field to integrable models......Page 99
5.3 Classical solitons......Page 101
5.4 Breathers or "doublets"......Page 106
5.5 Quantum solitons......Page 108
5.5.1 Quantization of the breather......Page 110
5.6 Integrability and factorized S-matrix......Page 112
5.7 Yang-Baxter equations......Page 114
5.8 The general solution of the S-matrix......Page 115
5.8.1 The S-matrix of the sine-Gordon model......Page 117
5.9 From conformal field theories to integrable models......Page 119
5.10 Conserved charges and classical integrability......Page 121
5.10.1 The Lax pair method......Page 122
5.10.2 The generating function method......Page 123
5.11.1 Multilocal charges from integral equation......Page 124
5.11.2 Charges by inductive procedure......Page 126
5.12 Quantum integrable charges in the O(N) model......Page 127
5.13 Non-local charges and quantum groups......Page 128
5.14.1 The XXX1/2 model......Page 131
5.14.2 Bethe ansatz equations......Page 134
5.14.3 The thermodynamic Bethe ansatz......Page 138
5.14.4 Spin chain model in discrete time......Page 142
5.14.5 The discretized version of the sine-Gordon model......Page 143
5.15 The continuum thermodynamic Bethe ansatz......Page 145
6 Bosonization......Page 151
6.1.1 Bosonization of a free massless Dirac fermion......Page 152
6.2 Duality between the Thirring model and the sine-Gordon model......Page 156
6.3.1 Bosonization of Majorana fermions......Page 159
6.3.2 Bosonization of Dirac fermions......Page 161
6.3.3 The bosonization of a mass bilinear of Dirac fermions......Page 162
6.3.4 Bosonization of Dirac fermions with color and flavor......Page 163
6.3.5 Bosonization of mass bilinears in the product scheme......Page 166
6.3.6 Bosonization of the U(NF NC) WZW action......Page 167
6.4 Chiral bosons......Page 168
6.4.1 Chiral boson via coupling to fictitious "light-cone gravity"......Page 169
6.4.2 Non manifestly Lorentz invariant classical action......Page 172
6.4.3 Coupling to abelian gauge fields......Page 177
6.4.4 Chiral WZW and coupling to non-abelian gauge fields......Page 178
6.5 Bosonization of systems of operators of high conformal dimension......Page 179
6.5.1 The bosonization of the "b,c" free CFT......Page 180
6.5.2 The bosonization of the beta, gamma system......Page 182
6.5.3 The Wakimoto bosonization......Page 183
7.1 Introduction......Page 185
7.2 The Gross-Neveu model......Page 186
7.3 The CPN-1 model......Page 191
PART II Two-dimensional non-perturbative gauge dynamics......Page 195
8.1 Pure Maxwell theory......Page 197
8.2 QED2 – Schwinger's model......Page 198
8.3 Yang-Mills theory......Page 199
8.4 Quantum chromodynamics......Page 200
9.1 QED2 – The massive Schwinger model......Page 203
9.2 Abelian bosonization of flavored QCD2......Page 205
9.3.1 Gauging the WZW action......Page 207
9.3.2 Multiflavor QCD2 using the U(NFxNC) scheme......Page 209
10 The 't Hooft solution of 2d QCD......Page 211
10.1 Scattering of mesons......Page 218
10.2 Higher 1/N corrections......Page 221
11.2 Universality of conformal field theories coupled to YM2......Page 223
11.3.1 The basic setup......Page 226
11.3.2 't Hooft-like equation for the two-current wave function......Page 229
11.3.3 The two-current mesonic spectrum......Page 232
11.3.4 Special cases: Nf = 1, Nf = Nc and Nf >> Nc......Page 233
11.4 The adjoint vacuum and its one-current state......Page 236
11.4.1 The action of M2 on the one-current states......Page 238
12.1 Discretized light-cone quantization......Page 243
12.2 Application of DLCQ to QCD2 with fundamental fermions......Page 244
12.3 The spectrum of QCD2 with adjoint fermions......Page 248
13.1 The strong coupling limit......Page 257
13.2 Classical soliton solutions......Page 259
13.3 Semi-classical quantization and the baryons......Page 260
13.5 Quark flavor content of the baryons......Page 267
13.6 Multibaryons......Page 269
13.7 States, wave functions and binding energies......Page 270
13.8 Meson-baryon scattering......Page 272
13.8.1 Abelian case......Page 274
13.8.2 The non-abelian case......Page 276
13.8.3 Extension to arbitrary coupling......Page 279
14.1 The string tension of the massive Schwinger model......Page 285
14.3 Beyond the small mass abelian string tension......Page 288
14.4 Correction to the leading long distance abelian potential......Page 289
14.5 Finite temperature......Page 291
14.6 Two-dimensional QCD......Page 292
14.7 Symmetric and antisymmetric representations......Page 296
15.2 The action......Page 299
15.3 Two-dimensional Yang-Mills theory......Page 302
15.4 Schwinger model revisited......Page 303
15.5 Back to the YM theory......Page 305
15.6 An alternative formulation......Page 307
15.7 The resolution of the puzzle......Page 308
15.8 On bosonized QCD2......Page 309
15.9 Summary and discussion......Page 310
16.1 Introduction......Page 311
16.2 The partition function of the YM2 theory......Page 312
16.3 The partition function of gYM2 theories......Page 316
16.4 Loop averages in the generalized case......Page 317
16.5 Stringy YM2 theory......Page 319
16.6 Toward the stringy generalized YM2......Page 321
16.7 Examples......Page 322
16.8 Summary......Page 324
PART III From two to four dimensions......Page 327
17 Conformal invariance in four-dimensional field theories and in QCD......Page 329
17.1 Conformal symmetry algebra in four dimensions......Page 330
17.2 Conformal invariance of fields, Noether currents and conservation laws......Page 332
17.3 Collinear and transverse conformal transformations of fields......Page 334
17.4 Collinear primary fields and descendants......Page 336
17.5 Conformal operator product expansion......Page 338
17.6 Conformal Ward identities......Page 339
17.7 Conformal invariance and QCD4......Page 342
18 Integrability in four-dimensional gauge dynamics......Page 349
18.1 Integrability of large N four-dimensional N = 4 SYM......Page 350
18.2 High energy scattering and integrability......Page 353
19.1 Large N QCD in four dimensions......Page 357
19.1.1 Counting rules for correlation functions......Page 362
19.2 Meson phenomenology......Page 363
19.2.1 Axial U(1) and the mass of the eta'......Page 365
19.3 Baryons in the large N expansion......Page 366
19.3.1 The Hartree approximation......Page 367
19.3.2 Baryons made out of heavy quarks......Page 368
19.3.3 Baryons made out of light quarks......Page 369
19.3.4 Baryonic excited states......Page 371
19.4 Scattering processes......Page 372
20.2 The Skyrme action......Page 375
20.2.1 The Sigma term......Page 376
20.2.2 The WZ term......Page 377
20.2.3 The Skyrme term......Page 378
20.2.4 A mass term......Page 379
20.2.5 Gauging the Skyrme action......Page 380
20.3.1 The classical Skyrmion......Page 381
20.3.2 Semiclassical quantization of the soliton......Page 384
20.3.3 The Skyrme model and large Nc QCD......Page 386
20.4 The Skyrme model for Nf = 3......Page 387
21.1 Introduction......Page 391
21.2 The Yang-Mills Higgs theory - basics......Page 392
21.3 Topological solitons and magnetic monopoles......Page 393
21.4 The 't Hooft-Polyakov magnetic monopole solution......Page 396
21.5 Charge quantization......Page 397
21.6 Zero modes, time-dependent solutions and dyons......Page 398
21.7 BPS monopoles and dyons......Page 401
21.8 Montonen Olive duality......Page 402
21.9 Nahm construction of multimonopole solutions......Page 403
21.9.1 SU(2) two-monopole solutions......Page 405
21.10 Moduli space of monopoles......Page 406
22.1 The basic properties of the instanton......Page 409
22.2 The ADHM construction of instantons......Page 414
22.3 On the moduli space of instantons......Page 416
22.4 Instantons and tunneling between the vacua of the YM theory......Page 420
22.5 Instantons, theta vacua and the UA(1) anomaly......Page 423
23.1 General......Page 427
23.2 Conformal invariance......Page 428
23.3 Integrability......Page 430
23.4 Bosonization......Page 431
23.5 Topological field configurations......Page 432
23.6 Confinement versus screening......Page 434
23.7.1 Mesons......Page 436
23.7.2 Baryons......Page 438
23.8.1 Further progress in the application of the methods discussed in the book......Page 440
23.8.3 Developments in gauge dynamics due to other methods......Page 441
References......Page 443
Index......Page 453