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In recent years topology has firmly established itself as an important part of the physicist's mathematical arsenal. It has many applications, first of all in quantum field theory, but increasingly also in other areas of physics. The main focus of this book is on the results of quantum field theory that are obtained by topological methods. Some aspects of the theory of condensed matter are also discussed. Part I is an introduction to quantum field theory: it discusses the basic Lagrangians used in the theory of elementary particles. Part II is devoted to the applications of topology to quantum field theory. Part III covers the necessary mathematical background in summary form. The book is aimed at physicists interested in applications of topology to physics and at mathematicians wishing to familiarize themselves with quantum field theory and the mathematical methods used in this field. It is accessible to graduate students in physics and mathematics.

Quantum Field Theory and Topology

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.