Contents -- Preface -- Chapter 1 Theoretical Introduction -- 1.1. Gauge Fields -- 1.1.1. Gauge "symmetry" -- 1.1.2. Gauge fields on the lattice -- 1.1.3. Fixing the gauge -- 1.1.4. Hamiltonian quantization -- 1.1.5. Rotator - a toy model for periodic coordinates -- 1.2. Road to QCD -- 1.2.1. Quarks and the "colors" -- 1.2.2. Renormalizability and asymptotic freedom -- 1.3. Path Integrals and Euclidean Time -- 1.3.1. Green functions and Feynman path integrals -- 1.3.2. Perturbation theory and Euclidean path integrals -- 1.3.3. Numerical evaluation of Euclidean path integrals -- 1.4. Gauge Fields on the Lattice -- 1.4.1. Renormalization group and asymptotic freedom -- 1.4.2. Continuum limit of lattice gauge theory -- 1.4.3. Path integrals for fermions -- 1.5. Light Quarks and Symmetries of QCD -- 1.5.1. Exact and approximate symmetries of QCD -- 1.5.2. Chiral anomalies, the UV approach -- 1.5.3. Chiral anomalies, the IR approach -- 1.5.4. Other applications of chiral anomalies -- 1.5.5. Scale anomaly -- 1.6. Heavy Quarks, New Symmetry and Effective Theory -- 1.7. Changing the Number of Colors Nc -- 1.7.1. Large number of colors -- 1.7.2. QCD with the smallest (Nc = 2) number of colors -- Chapter 2 Phenomenology of the QCD Vacuum -- 2.1. Phenomenology of the Hadronic World -- 2.1.1. Brief history -- 2.1.2. The "usual" hadrons -- 2.1.3. The "unusual" mesons -- 2.1.4. The exotic hadrons -- 2.1.5. Remarks about highly excited states -- 2.2. Models of Hadronic Structure -- 2.2.1. Generalities -- 2.2.2. MIT bag -- 2.2.3. Skyrmions -- 2.2.4. Chiral bags -- 2.2.5. Evolving views on the nature of the spin forces -- 2.3. Models of the QCD Vacuum: An Overview -- 2.3.1. Condensates and scales -- 2.3.2. Condensate factorization and stochastic vacuum model -- 2.3.3. An example of a highly inhomogeneous model: the instanton vacuum.

2.4. Chiral Symmetry Breaking and Effective Low Energy Theory -- 2.4.1. Spontaneous breaking of the chiral symmetry -- 2.4.2. The Goldstone modes: oscillations of the quark condensate -- 2.4.3. Quark condensate and Dirac eigenvalue spectrum -- 2.4.4. Elements of chiral perturbation theory -- 2.4.5. Effective chiral Lagrangian -- 2.4.6. Nambu-Jona-Lasinio model -- 2.5. Color Confinement -- 2.5.1. Static potential -- 2.5.2. Dual superconductivity -- 2.5.3. Structure of flux tubes -- 2.5.4. Interaction of flux tubes -- Chapter 3 Euclidean Theory of Tunneling: From Quantum Mechanics to Gauge Theories -- 3.1. Tunneling in Quantum Mechanics -- 3.1.1. Brief history of tunneling -- 3.1.2. Double-well problem and instantons -- 3.1.3. Pre-exponent and zero modes -- 3.1.4. Instanton gas -- 3.1.5. Two-loop quantum corrections -- 3.2. A Digression: Tunneling Versus Perturbative Series -- 3.2.1. Convergence of perturbative series -- 3.2.1.1. Dyson instability -- 3.2.1.2. Perturbative series in high orders -- 3.2.1.3. Semiclassical evaluation of Dyson's instability -- 3.2.1.4. High orders of perturbative series in field theories -- 3.2.2. Instanton-anti-instanton interaction and one more correction to the ground state energy -- 3.3. Fermions Coupled to the Double-Well Potential -- 3.4. Instantons in Gauge Theories -- 3.4.1. Topologically nontrivial objects -- 3.4.2. Topologically distinct pure gauge configurations -- 3.4.3. Digression: spherically symmetric Yang-Mills fields -- 3.4.4. Static magnetic configurations and their minimal energy -- 3.5. Tunneling and BPST Instanton -- 3.5.1. Instanton solution -- 3.5.2. Theta vacua -- 3.5.3. Tunneling amplitude -- Chapter 4 Instanton Ensemble in QCD -- 4.1. Brief History of Instantons -- 4.1.1. Discovery and early applications -- 4.1.2. Phenomenology leads to a qualitative picture.

4.1.3. Technical development during 1980s -- 4.1.4. Recent progress -- 4.1.5. Instantons at finite temperatures and chiral restoration -- 4.1.6. Instantons and color superconductivity at high densities -- 4.2. Tunneling and Light Quarks -- 4.2.1. Relating gauge field topology to the axial charge -- 4.2.2. Fermionic zero modes -- 4.2.3. The 't Hooft effective interaction -- 4.2.4. Baryon number violation in the standard model -- 4.3. Instanton Ensemble -- 4.3.1. Qualitative discussion of the instanton ensembles -- 4.3.2. Mean field approximation: pure glue -- 4.3.3. Quark condensate in the mean field approximation -- 4.3.4. The single instanton approximation -- 4.4. The Interacting Instanton Liquid Model -- 4.4.1. Screening of the topological charge -- 4.5. Instantons for Larger Number of Colors -- 4.5.1. Naive counting and expectations -- 4.5.2. Mean field arguments and the chiral condensate -- 4.5.3. Fluctuations in the interacting instanton liquid -- 4.5.4. Do instantons cluster at large Nc? -- Chapter 5 Lattice QCD -- 5.1. Generalities -- 5.1.1. Brief history -- 5.1.2. Lattice limitations -- 5.1.3. Mesoscopic regime and the random matrix theory -- 5.1.4. Art of numerical simulation of multi-dimensional integrals -- 5.2. Fermions on the lattice -- 5.2.1. Fermionic doublers -- 5.2.2. Wilson fermions -- 5.2.3. Ginsparg-Wilson relation and lattice chiral symmetry -- 5.2.4. Known solutions to GW relation -- 5.2.5. Domain wall fermions -- 5.3. Hadronic spectroscopy on the lattice -- 5.3.1. Glueballs in gluodynamics -- 5.3.2. Light quark spectroscopy in quenched approximation -- 5.3.3. Spectroscopy with dynamical quarks -- 5.4. Topology on the lattice -- 5.4.1. Quantum-mechanical topology and perfect actions -- 5.4.2. Naive and geometric methods for gauge fields -- 5.4.3. Are the lowest Dirac eigenstates locally chiral?.

5.4.4. Testing the large Nc limit on the lattice -- Chapter 6 QCD Correlation Functions -- 6.1. Generalities -- 6.1.1. Why the correlation functions? -- 6.1.2. Different representations of the correlation functions -- 6.1.3. Quantum numbers and inequalities -- 6.1.4. Correlators with chirality flips -- 6.2. Phenomenology of Mesonic Correlation Functions -- 6.2.1. Vector and axial correlators -- 6.2.2. Comparing axial and vector channels -- 6.2.3. Pseudoscalar SU(3) octet (π, K, η) channels -- 6.2.4. SU(3) singlet pseudoscalars -- 6.2.5. Hadron-parton duality -- 6.3. Operator Product Expansion and QCD Sum Rules -- 6.3.1. Brief history and overview -- 6.3.2. Separation of scales -- 6.3.3. OPE in a background field -- 6.3.4. Sum rules for heavy-light mesons -- 6.3.5. OPE for light quark baryons -- 6.3.6. OPE for mesons made of light quarks -- 6.4. Instantons and the Correlators: Analytic Results -- 6.4.1. Propagator in the field of a single instanton -- 6.4.2. First order in the 't Hooft effective vertex -- 6.4.3. Propagator in the instanton ensemble -- 6.4.4. Propagator in the mean field approximation -- 6.4.5. Correlators in the random phase approximation -- 6.5. Correlators in the Instanton Liquid -- 6.5.1. Quark propagator in the instanton liquid -- 6.5.2. Mesonic correlators -- 6.5.3. Baryonic correlation functions -- 6.5.4. Comparison to correlators on the lattice -- 6.5.5. Gluonic correlation functions -- 6.6. Hadronic Structure and n-Point Correlators -- 6.6.1. Wave functions -- 6.6.2. Form factors -- Chapter 7 High Energy Hadronic Collisions -- 7.1. Introduction -- 7.1.1. Reggions and the Pomeron -- 7.1.2. High energy collisions in pQCD and its "phases" -- 7.1.3. Evolving descriptions of soft Pomeron dynamics -- 7.2. Instanton-Induced Processes at High Energies -- 7.2.1. Toward the "holy grail".

7.2.2. Exciting a quantum system from under the barrier -- 7.2.3. Semiclassical production of sphaleron-like clusters -- 7.2.4. Explosion of the turning states -- 7.2.5. Semiclassical evaluation of the cross section -- 7.2.6. Semiclassical Wilson lines -- 7.2.7. Pomeron from instantons -- 7.3. Pomeron Structure and Interactions -- 7.3.1. Clustering in inclusive pp collisions -- 7.3.2. Inclusive production of clusters in double-Pomeron processes -- 7.3.3. Exclusive production of hadrons in double-Pomeron processes -- Chapter 8 QCD at Finite Temperatures -- 8.1. Introduction -- 8.1.1. Brief history and the basic scales -- 8.1.2. From field theory to thermodynamics -- 8.1.3. A quantum particle at finite T -- 8.1.4. Gauge and fermion fields at finite T -- 8.2. QCD at High Temperatures -- 8.2.1. Screening versus anti-screening -- 8.2.2. Thermodynamical potential in the lowest order -- 8.2.3. Ring diagram re-summation -- 8.2.4. IR divergences in general -- 8.2.5. Are perturbative series useful in practice? -- 8.2.6. HTL re-summations and the quasiparticle gas -- 8.2.7. Viscosity of the QGP -- 8.3. Hadronic Matter -- 8.3.1. Pion gas at low T -- 8.3.2. Resonance gas -- 8.3.3. Pion liquid -- 8.4. QCD Phase Transitions at Finite T -- 8.4.1. Deconfinement -- 8.4.2. Chiral symmetry restoration -- 8.4.3. Static quark potential at high T -- 8.4.4. Equation of state in the transition region -- 8.5. Instantons at Finite T -- 8.5.1. Finite temperature field theory and the caloron solution -- 8.5.2. Instanton density at high temperature -- 8.5.3. Instantons at low temperature -- 8.5.4. Chiral symmetry restoration and instantons -- 8.5.5. Instanton ensemble in the phase transition region -- 8.5.6. Critical behavior in the instanton liquid -- 8.6. Hadronic Correlation Functions at Finite Temperature -- 8.6.1. Screening masses -- 8.6.2. Temporal correlation functions.

8.6.3. U(1)A breaking at high T.