Axial $U_A(1)$ Anomaly: a New Mechanism to Generate Massless Bosons

Prior to the establishment of $QCD$ as the correct theory describing hadronic physics, it was realized that the essential ingredients of the hadronic world at low energies are chiral symmetry and its spontaneous breaking. Spontaneous symmetry breaking is a non-perturbative phenomenon and thanks to massive $QCD$ simulations on the lattice we have at present a good understanding on the vacuum realization of the non-abelian chiral symmetry as a function of the physical temperature. As far as the $U_A(1)$ anomaly is concerned, and especially in the high temperature phase, the current situation is however far from satisfactory. The first part of this article is devoted to review the present status of lattice calculations, in the high temperature phase of $QCD$, of quantities directly related to the $U_A(1)$ axial anomaly. In the second part I will analyze some interesting physical implications of the $U_A(1)$ anomaly, recently suggested, in systems where the non-abelian axial symmetry is fulfilled in the vacuum. More precisely I will argue that, if the $U_A(1)$ symmetry remains effectively broken, the topological properties of the theory can be the basis of a mechanism, other than Goldstone's theorem, to generate a rich spectrum of massless bosons at the chiral limit.