The chiral/axial and conformal/Weyl/trace anomalies

-symmetry is a fundamental concept in theoretical physics...
-a symmetry is exhibited classically by the invariance of an action principle under some transformations...
-quantum mechanically, a symmetry is exhibited by the invariance of the path integral under the above transformations ..
-if the path integral remains symmetric while the vacuum is not we speak of spontaneous symmetry breaking...
-sometimes, after properly quantizing a theory, the path integral fails to remain symmetric...in this case we speak of an anomaly..
-A classic example is the axial or chiral anomaly which is the non-invariance of the path integral measure under global chiral transformations...everything is encoded in the famous one-loop triangle diagram...this anomaly can be explained as a tunnelling process from one vacuum to another called instanton..this anomaly gives the correct rate for the decay of pion into photons, viz
\pi^0....>\gamma+\gamma
-another important example of anomaly is the conformal or Weyl anomaly..the conformal group SO(D,2) where D=4 contains, beside Lorentz transformations and translations, conformal transformations...rotations and translations can never be broken....scale transformations can be broken by anomaly....most theories which are scale invariant are also conformally invariant..the scale transformation is given by the scaling of the metric
g_{ab}.......>\lambda*g_{ab}
This is a symmetry of the action principle in the absence of mass..We will assume that we have fermions in the fundamental representation of a unitary group....The quantum Weyl anomaly is then signalled by the non-vanishing of the beta function...This is precisely equivalent to the introduction of a dynamically generated confinement mass scale which in turn determines the masses of the hadrons..this conclusion remains unaltered if we have started out with massive fermions...since the masses of the quarks are very small compared to the confinement scale ...
-If we add enough supersymmetry, then conformal symmetry become exact both at the quantum and classical levels..The classical example is N=4 supersymmetry in D=4 dimensions...
-In the case of string theory conformal symmetry can not be broken by anomaly since there they are local symmetries...This is what gives critical strings at D=10 and D=26....local symmetries, such as gauge symmetries, can not be broken by anomaly because of unitarity and causality....
-The conformal or Weyl anomaly is also called the trace anomaly..this is because in the context of general relativity the trace of the stress-energy-momentum tensor is precisely what determines the variation of the action with respect to the conformal rescaling..thus the conformal or Weyl or trace anomaly is given by the vacuum expectation value of the stress-energy-momentum tensor...