Index and anomaly
Index and anomaly
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.
Keywords: gauge theory, Atiyah–Singer index theorem, non-Abelian index anomaly
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