In particle physics, symmetry principles are some of the most important concepts in model building. Symmetries play crucial roles for the theory to be renormalizable and unitary. The Lagrangian must be chosen so that it fulfils the observed symmetry. Note, however, that the symmetry of the Lagrangian is classical. There is no warranty that symmetry of the Lagrangian may be elevated to a quantum symmetry, i.e., the symmetry of the effective action. If the classical symmetry of the Lagrangian cannot be maintained in the process of quantization, the theory is said to have an anomaly. There are many types of anomaly: the chiral anomaly, gauge anomaly, gravitational anomaly, supersymmetry anomaly and so on. Each adjective refers to the symmetry under consideration. In the present chapter we look at the geometrical and topological structures of the anomalies appearing in gauge theories.
We follow closely Alvarez-Gaumé (1986), Alvarez-Gaumé and Ginsparg (1985) and Sumitani (1985). See Rennie (1990) and Bartlmann (1996) for a complete analysis of the subject. Mickelsson (1989) and Nash (1991) have a section on anomalies from a more mathematical point of view.
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