"This is a very interesting book on an important topic both for physics and for mathematics." - Bulletin of the London Mathematic Society

• Definitions and Notations.- 1 The Simplest Lagrangians.- 2 Quadratic Lagrangians.- 3 Internal Symmetries.- 4 Gauge Fields.- 5 Particles Corresponding to Nonquadratic Lagrangians.- 6 Lagrangians of Strong, Weak and Electromagnetic Interactions.- 7 Grand Unifications.- 8 Topologically Stable Defects.- 9 Topological Integrals of Motion.- 10 A Two-Dimensional Model. Abrikosov Vortices.- 11 ’t Hooft—Polyakov Monopoles.- 12 Topological Integrals of Motion in Gauge Theory.- 13 Particles in Gauge Theories.- 14 The Magnetic Charge.- 15 Electromagnetic Field Strength and Magnetic Charge in Gauge Theories.- 16 Extrema of Symmetric Functionals.- 17 Symmetric Gauge Fields.- 18 Estimates of the Energy of a Magnetic Monopole.- 19 Topologically Non-Trivial Strings.- 20 Particles in the Presence of Strings.- 21 Nonlinear Fields.- 22 Multivalued Action Integrals.- 23 Functional Integrals.- 24 Applications of Functional Integrals to Quantum Theory.- 25 Quantization of Gauge Theories.- 26 Elliptic Operators.- 27 The Index and Other Properties of Elliptic Operators.- 28 Determinants of Elliptic Operators.- 29 Quantum Anomalies.- 30 Instantons.- 31 The Number of Instanton Parameters.- 32 Computation of the Instanton Contribution.- 33 Functional Integrals for a Theory Containing Fermion Fields.- 34 Instantons in Quantum Chromodynamics.- 35 Topological Spaces.- 36 Groups.- 37 Gluings.- 38 Equivalence Relations and Quotient Spaces.- 39 Group Representations.- 40 Group Actions.- 41 The Adjoint Representation of a Lie Group.- 42 Elements of Homotopy Theory.- 43 Applications of Topology to Physics.- Bibliographical Remarks.- References.