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These lectures concern the properties of topological charge in gauge theories and the physical effects which have been attributed to its existence. No introduction to this subject would be adequate without a discussion of the original work of Belavin, Polyakov, Schwarz, and Tyupkin [1], of the beautiful calculation by ´t Hooft [2,3], and of the occurrence of 8-vacua [4-6]. Other important topics include recent progress on solutions of the Yang-Mills equation of motion [7,8], and the problem of parity and time-reversal invariance in strong interactions [9] (axions [10,11], etc. ). In a few places, I have strayed from the conventional line and in one important case, disagreed with it. The im portant remark concerns the connection between chirality and topological charge first pointed out by ´t Hooft [2]: in the literature, the rule is repeatedly quoted with the wrong sign! If QS is the generator for Abelian chiral transformations of massless quarks with N flavours, the correct form of the rule is ßQs = - 2N {topological charge} (1. 1) where ßQS means the out eigenvalue of QS minus the in eigenvalue. The sign can be checked by consulting the standard WKB calculation [2,3], rotating to Minkowski space, and observing that the sum of right-handed chiralities of operators in a Green´s function equals -ßQS. The wrong sign is an automatie consequence of a standard but incorrect derivation in which the axial charge is misidentified.

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Opening Address.- Elementary Introduction to Gauge Theories.- Effects of Topological Charge in Gauge Theories.- Topological Methods for Gauge Theories.- Lattice Gauge Fields.- Gauge Fields on a Lattice (Selected Topics).- Chromodynamic Theory of Hadrons.- Jets and QCD.- Supersymmetry and Gauge Theories of Weak and Electromagnetic Interactions.- Gauge Theories of Gravitation.- High-Energy Neutrino Experiments.- Hadron Induced Lepton Production.- Charm and Heavy Lepton Production by e+-e? Interactions.- Analysis of T-violation without Assumed CPT.- KSO Decays and Lorentz Invariance.- A Unified Gauge Theory without ?-e Universality.- Classical Limit for Arbitrary Commutation Relations.- Nonlocal Properties of Fermionic Matter in Strong Gravitational Fields.- Low-Energy ?N- and NN-Scattering in Chiral Dynamics with the Use of Superpropagators.- Remarks on Vacuum Structure of Yang-Mills Theory.- Multi-Dimensional Unified Theory.- Remarks on Symanzik´s Approach to Nonrenormalizable Theories.- A Quasi-Metric Associated with SU(2) Yang-Mills Fields.- Observable Consequences of Spontaneously Broken Non-Abelian Symmetries.- Summary.

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