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Anomalies in Quantum Field Theory$
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Reinhold A. Bertlmann

Print publication date: 2000

Print ISBN-13: 9780198507628

Published to Oxford Scholarship Online: February 2010

DOI: 10.1093/acprof:oso/9780198507628.001.0001

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Chern–Simons form, homotopy operator and anomaly

Chern–Simons form, homotopy operator and anomaly

(p.321) 7 Chern–Simons form, homotopy operator and anomaly
Anomalies in Quantum Field Theory

Reinhold A. Bertlmann

Oxford University Press

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.

Keywords:   Chern–Simons form, anomalies, non-Abelian anomaly, Cartan homotopy formula, transgression formula

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