  These lectures on anomalies are relatively self-contained 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 (non-abelian) gauge
anomaly is derived by evaluating the anomalous triangle diagram with three
non-abelian gauge fields coupled to a chiral fermion. We discuss in detail the
relation between anomaly, current non-conservation and non-invariance of the
effective action, with special emphasis on the derivation of the anomalous
Slavnov-Taylor/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
Wess-Zumino 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 ten-dimensional
IIB supergravity and in type I and heterotic superstrings.
