Aspects of Chiral Symmetry

Based on the lectures given at the Advanced School in Quantum Chromodynamics (Benasque, July 2000). To be published in the Festschrift in honor of B.L. Ioffe (ed. M. Shifman); 42 pages LaTeX, 6 eps figuresWe give a pedagogical review of implications of chiral symmetry in QCD. First, we briefly discuss classical textbook subjects such as the axial anomaly, spontaneous breaking of the flavor-nonsinglet chiral symmetry, formation of light pseudo-Goldstone particles, and their effective interactions. Then we proceed to other issues. We explain in some detail a recent discovery how to circumvent the Nielsen--Ninomiya's theorem and implement chirally symmetric fermions on the lattice. We touch upon such classical issues as the Vafa-Witten's theorem and 't Hooft's anomaly matching conditions. We derive a set of exact theorems concerning the dynamics of the theory in a finite Euclidean volume and the behavior of the Dirac spectral density. Finally, we discuss an imaginary world with a nonzero value of the vacuum angle theta