Axial anomaly in quantum electro- and chromodynamics and the structure of the vacuum in quantum chromodynamics

In this report, I discuss the current state of the problem of the axial anomaly in quantum electrodynamics (QED) and quantum chromodynamics (QCD) and how the axial anomaly is related to the structure of the vacuum in QCD. In QCD, the vacuum average of the axial anomaly is proportional to a new quantum number n, the winding number. The axial anomaly condition implies that there are zero modes of the Dirac equation for a massless quark and that there is spontaneous breaking of chiral symmetry in QCD, which leads to the formation of a quark condensate. The axial anomaly can be represented in the form of a sum rule the structure function in the dispersion representation of the axial -- vector -- vector (AVV) vertex. On the basis of this sum rule, it is calculated the width of the \pi^0\to 2\gamma decay with an accuracy of 1.5%. It is demonstrated, that 't Hooft conjecture -- the singularities of the amplitudes calculated in perturbative QCD on quark-gluon basis should reproduce themselves in calculations on the hadrons basis -- is not fulfilled generally.