Lectures on Anomalies
Abstract
These lectures on anomalies are relatively selfcontained and intended for graduate students who are familiar with the basics of quantum field theory. We begin with several derivations of the abelian anomaly: anomalous transformation of the measure, explicit computation of the triangle Feynman diagram, relation to the index of the Euclidean Dirac operator. The chiral (nonabelian) gauge anomaly is derived by evaluating the anomalous triangle diagram with three nonabelian gauge fields coupled to a chiral fermion. We discuss in detail the relation between anomaly, current nonconservation and noninvariance of the effective action, with special emphasis on the derivation of the anomalous SlavnovTaylor/Ward identities. We show why anomalies always are finite and local. A general characterization is given of gauge groups and fermion representations which may lead to anomalies in four dimensions, and the issue of anomaly cancellation is discussed, in particular the classical example of the standard model. Then, we move to more formal developments and arbitrary even dimensions. After introducing a few basic notions of differential geometry, in particular characteristic classes, we derive the descent equations. We prove the WessZumino consistency condition and show that relevant anomalies correspond to BRST cohomologies at ghost number one. We discuss why and how anomalies are related to characteristic classes in two more dimensions and outline their computation in terms of the index of an appropriate Dirac operator. Finally we derive the gauge and gravitational anomalies in arbitrary even dimensions from the appropriate index and explain the anomaly cancellations in tendimensional IIB supergravity and in type I and heterotic superstrings.
 Publication:

arXiv eprints
 Pub Date:
 February 2008
 arXiv:
 arXiv:0802.0634
 Bibcode:
 2008arXiv0802.0634B
 Keywords:

 High Energy Physics  Theory
 EPrint:
 108 pages, 10 figures