| Summary: | 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 of the vacuum realization of the non-abelian chiral symmetry as a function of the physical temperature. As far as the <inline-formula><math display="inline"><semantics><mrow><msub><mi>U</mi><mi>A</mi></msub><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula> 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 reviewing the present status of lattice calculations, in the high temperature phase of QCD, of quantities directly related to the <inline-formula><math display="inline"><semantics><mrow><msub><mi>U</mi><mi>A</mi></msub><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula> axial anomaly. In the second part, some recently suggested interesting physical implications of the <inline-formula><math display="inline"><semantics><mrow><msub><mi>U</mi><mi>A</mi></msub><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula> anomaly in systems where the non-abelian axial symmetry is fulfilled in the vacuum are analyzed. More precisely it is argued that, if the <inline-formula><math display="inline"><semantics><mrow><msub><mi>U</mi><mi>A</mi></msub><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula> 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.
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