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Quantum Field Theory and Topology [1993]

1
Introduction
6
Definitions and Notations
13
The Simplest Lagrangians
17
Quadratic Lagrangians
19
Internal Symmetries
24
Gauge Fields
28
Particles Corresponding to Nonquadratic Lagrangians
30
Lagrangians of Strong, Weak and Electromagnetic Interactions
37
Grand Unifications
43
Topologically Stable Defects
56
Topological Integrals of Motion
62
A Two-Dimensional Model. Abrikosov Vortices
68
’t Hooft—Polyakov Monopoles
74
Topological Integrals of Motion in Gauge Theory
80
Particles in Gauge Theories
83
The Magnetic Charge
89
Electromagnetic Field Strength and Magnetic Charge in Gauge Theories
94
Extrema of Symmetric Functionals
97
Symmetric Gauge Fields
104
Estimates of the Energy of a Magnetic Monopole
109
Topologically Non-Trivial Strings
115
Particles in the Presence of Strings
122
Nonlinear Fields
128
Multivalued Action Integrals
132
Functional Integrals
138
Applications of Functional Integrals to Quantum Theory
146
Quantization of Gauge Theories
158
Elliptic Operators
163
The Index and Other Properties of Elliptic Operators
169
Determinants of Elliptic Operators
173
Quantum Anomalies
183
Instantons
194
The Number of Instanton Parameters
199
Computation of the Instanton Contribution
207
Functional Integrals for a Theory Containing Fermion Fields
216
Instantons in Quantum Chromodynamics
223
Topological Spaces
225
Groups
229
Gluings
233
Equivalence Relations and Quotient Spaces
235
Group Representations
241
Group Actions
245
The Adjoint Representation of a Lie Group
247
Elements of Homotopy Theory
257
Applications of Topology to Physics
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