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This chapter studies the role of anomaly in the topology of gauge theories. It shows that the anomaly also has a ‘natural’ explanation; it occurs as an obstruction in certain nontrivial bundles and it is determined completely by a topological quantity — the index. Section 11.1 discusses the relation of the single anomaly to the Atiyah–Singer index theorem. Section 11.2 describes the geometric-topological character of the non-Abelian index anomaly in the context of index theorems. Section 11.3 shows the connection between the index of the Weyl operator and the heat kernel of the Laplacian, shedding light on Fujikawa's regularization procedure. Section 11.4 presents the Atiyah–Singer index theorem for the case of YM fields. Section 11.15 introduces a special Dirac operator which is equivalent to the Wely operator, and calculates the non-Abelian anomaly, Bardeen's result, by the path integral method. It also explains the procedure of Alvarez–Gaum é — how to determine the non-Abelian anomaly by a generalized index theorem.

Chapter 4 presented a thorough investigation of the anomaly within perturbation theory. This chapter presents the nonperturbative view. Section 5.1 performs a chiral transformation of the path integral and finds the anomalous Ward identity. Section 5.2 regularizes the transformation Jacobian à la Fujikawa, and in this way derives the singlet anomaly; the two-dimensional case is added in Section 5.3. Section 5.4 shows the regularization independence of the anomaly and Section 5.5 discusses the conflict between gauge- and chiral symmetry in the light of an uncertainty principle. Section 5.6 demonstrates the generalization of the path integral method to non-Abelian fields leading to non-Abelian anomaly. Finally, Section 5.7 carries out the regularization of the Jacobian by means of the heat kernel method and by the zeta function procedure.

The Chern–Simons form and the homotopy operator plays an important role in connection with anomalies. In fact, the anomaly can be calculated on pure algebraic grounds from a variation of the Chern–Simons form using a homotopy operator. This chapter begins with a discussion of a symmetric invariant polynomial of fields, which is the starting point for deriving the Chern–Simons form and ‘transgression formula’. It then proves the important Poincaré lemma and introduces in this connection a homotopy operator. A generalization of the ‘transgression’ — the Cartan homotopy formula — follows. The homotopy formula is applied to a Chern–Simons form with gauge transformed fields, and the non-Abelian anomaly is derived in this way. Finally, the chapter presents a general formula for the variation of the Chern–Simons form, which expresses the anomaly.

This chapter shows how the singlet anomaly in 2n dimensions determines the non-Abelian anomaly in (2n - 2) dimensions via a set of equations. These are part of a whole chain of equations, which descend in their form degree, thus called the Stora–Zumino chain of descent equations. This chain is derived in pure mathematical grounds — algebra and differential geometry — and its meaning is physics is discussed. The chapter offers the topological aspect of a chain, described by an index theorem.