Non-Perturbative Field Theory (Cambridge Monographs on Mathematical Physics)

Cover
Half-title page
Series Page
Title page
Copyright page
Dedication page
Contents
Preface
Acknowledgements
PART I NON-PERTURBATIVE METHODS IN TWO-DIMENSIONAL FIELD THEORY
1 From massless free scalar field to conformal fieldtheories
	1.1 Complex geometry
	1.2 Free massless scalar field
	1.3 Symmetries of the classical action
	1.4 Mode expansion
	1.5 Noether currents and charges
	1.6 Canonical quantization
	1.7 Radial quantization
	1.8 Operator product expansion
	1.9 Path integral quantization
	1.10 Affine current algebra
	1.11 Virasoro algebra
2 Conformal field theory
	2.1 Conformal symmetry in two dimensions
	2.2 Primary fields
	2.3 Conformal properties of the energy-momentum tensor
	2.4 Virasoro algebra for CFT
	2.5 Descendant operators
	2.6 Hilbert space of states
	2.7 Unitary CFT and Kac determinant
	2.8 Characters
	2.9 Correlators and the conformal Ward identity
	2.10 Crossing symmetry, duality and bootstrap
	2.11 Verlinde’s formula
	2.12 Free Majorana fermions – an example of a CFT
	2.13 The Ising model – the m = 3 unitary minimal model
3 Theories invariant under affine current algebras
	3.1 Simple finite-dimensional Lie algebras
	3.2 Affine current algebra
	3.3 Current OPEs and the Sugawara construction
	3.4 Primary fields
	3.5 ALA characters
	3.6 Correlators, null vectors and the Knizhnik–Zamolodchikov equation
	3.7 Free fermion realization
	3.8 Free Dirac fermions and the U(N)
4 Wess–Zumino–Witten model and coset models
	4.1 From free massless scalar theory to the WZW model
	4.2 Perturbative conformal invariance
	4.3 ALA, Sugawara construction and the Virasoro algebra
	4.4 Correlation functions of primary fields
	4.5 WZW models with boundaries – D branes
	4.6 G/H coset models
	4.7 G/G coset models
5 Solitons and two-dimensional integrable models
	5.1 Introduction
	5.2 From the theory of a massive free scalar field to integrable models
	5.3 Classical solitons
	5.4 Breathers or “doublets”
	5.5 Quantum solitons
	5.6 Integrability and factorized S-matrix
	5.7 Yang–Baxter equations
	5.8 The general solution of the S-matrix
	5.9 From conformal field theories to integrable models
	5.10 Conserved charges and classical integrability
	5.11 Multilocal conserved charges
	5.12 Quantum integrable charges in the O(N) model
	5.13 Non-local charges and quantum groups
	5.14 Integrable spin chain models and the algebraic Bethe ansatz
	5.15 The continuum thermodynamic Bethe ansatz
6 Bosonization
	6.1 Abelian bosonization
	6.2 Duality between the Thirring model and the sine-Gordon model
	6.3 Witten’s non-abelian bosonization
	6.4 Chiral bosons
	6.5 Bosonization of systems of operators of high conformal dimension
7 The large N limit of two-dimensional models
	7.1 Introduction
	7.2 The Gross–Neveu model
	7.3 The CPN−1 model
PART II TWO-DIMENSIONAL NON-PERTURBATIVE GAUGE DYNAMICS
8 Gauge theories in two dimensions – basics
	8.1 Pure Maxwell theory
	8.2 QED2 – Schwinger’s model
	8.3 Yang–Mills theory
	8.4 Quantum chromodynamics
9 Bosonized gauge theories
	9.1 QED2 – The massive Schwinger model
	9.2 Abelian bosonization of flavored QCD2
	9.3 Non-abelian bosonization of QCD2
10 The ’t Hooft solution of 2d QCD
	10.1 Scattering of mesons
	10.2 Higher 1/N corrections
11 Mesonic spectrum from current algebra
	11.1 Introduction
	11.2 Universality of conformal field theories coupled to YM2
	11.3 Mesonic spectra of two-current states
	11.4 The adjoint vacuum and its one-current state
12 DLCQ and the spectra of QCD with fundamental and adjoint fermions
	12.1 Discretized light-cone quantization
	12.2 Application of DLCQ to QCD2 with fundamental fermions
	12.3 The spectrum of QCD2 with adjoint fermions
13 The baryonic spectrum of multiflavor QCD2 in the strong coupling limit
	13.1 The strong coupling limit
	13.2 Classical soliton solutions
	13.3 Semi-classical quantization and the baryons
	13.4 The baryonic spectrum
	13.5 Quark flavor content of the baryons
	13.6 Multibaryons
	13.7 States, wave functions and binding energies
	13.8 Meson-baryon scattering
14 Confinement versus screening
	14.1 The string tension of the massive Schwinger model
	14.2 The Schwinger model in bosonic form
	14.3 Beyond the small mass abelian string tension
	14.4 Correction to the leading long distance abelian potential
	14.5 Finite temperature
	14.6 Two-dimensional QCD
	14.7 Symmetric and antisymmetric representations
15 QCD2 , coset models and BRST quantization
	15.1 Introduction
	15.2 The action
	15.3 Two-dimensional Yang–Mills theory
	15.4 Schwinger model revisited
	15.5 Back to the YM theory
	15.6 An alternative formulation
	15.7 The resolution of the puzzle
	15.8 On bosonized QCD2
	15.9 Summary and discussion
16 Generalized Yang–Mills theory on a Riemann surface
	16.1 Introduction
	16.2 The partition function of the YM2 theory
	16.3 The partition function of gYM2 theories
	16.4 Loop averages in the generalized case
	16.5 Stringy YM2 theory
	16.6 Toward the stringy generalized YM2
	16.7 Examples
	16.8 Summary
PART III FROM TWO TO FOUR DIMENSIONS
17 Conformal invariance in four-dimensional field theories and in QCD
	17.1 Conformal symmetry algebra in four dimensions
	17.2 Conformal invariance of fields, Noether currents and conservation laws
	17.3 Collinear and transverse conformal transformations of fields
	17.4 Collinear primary fields and descendants
	17.5 Conformal operator product expansion
	17.6 Conformal Ward identities
	17.7 Conformal invariance and QCD4
18 Integrability in four-dimensional gauge dynamics
	18.1 Integrability of large N four-dimensional N = 4 SYM
	18.2 High energy scattering and integrability
19 Large N methods in QCD4
	19.1 Large N QCD in four dimensions
	19.2 Meson phenomenology
	19.3 Baryons in the large N expansion
	19.4 Scattering processes
20 From 2d bosonized baryons to 4d Skyrmions
	20.1 Introduction
	20.2 The Skyrme action
	20.3 The baryon as a Skyrmion
	20.4 The Skyrme model for Nf = 3
21 From two-dimensional solitons to four-dimensional magnetic monopoles
	21.1 Introduction
	21.2 The Yang–Mills Higgs theory – basics
	21.3 Topological solitons and magnetic monopoles
	21.4 The ’t Hooft–Polyakov magnetic monopole solution
	21.5 Charge quantization
	21.6 Zero modes, time-dependent solutions and dyons
	21.7 BPS monopoles and dyons
	21.8 Montonen Olive duality
	21.9 Nahm construction of multimonopole solutions
	21.10 Moduli space of monopoles
22 Instantons of QCD
	22.1 The basic properties of the instanton
	22.2 The ADHM construction of instantons
	22.3 On the moduli space of instantons
	22.4 Instantons and tunneling between the vacua of the YM theory
	22.5 Instantons, theta vacua and the UA(1) anomaly
23 Summary, conclusions and outlook
	23.1 General
	23.2 Conformal invariance
	23.3 Integrability
	23.4 Bosonization
	23.5 Topological field configurations
	23.6 Confinement versus screening
	23.7 Hadronic phenomenology of two dimensions versus four dimensions
	23.8 Outlook
References
Index