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Some Global Aspects of Gauge Anomalies of Semisimple Gauge Groups and Fermion Generations in GUT and Superstring Theories

We study more extensively and completely for global gauge anomalies with some semisimple gauge groups as initiated in ref.1. A detailed and complete proof or derivation is provided for the Z_2 global gauge anomaly given in ref.1 for a gauge theory with the semisimple gauge group SU(2)\times SU(2)\times SU(2) in D=4 dimensions and Weyl fermions in the irreducible representation (IR) ω=(2,2,2) with 2 denoting the corresponding dimensions. This Z_2 anomaly was used in the discussions related to generic SO(10) and supersymmetric SO(10) unification theories^1 for the total generation numbers of fermions and mirror fermions. Our result^1 that the global anomaly coefficient formula is given by A(ω)=exp[iπQ_2(\Box)]=-1 in this case with Q_2(\Box) being the Dynkin index for SU(8) in the fundamental IR (\Box)=(8) is also discussed, and as shown by our results^1 that the semisimple gauge transformations collectively may have physical consequences which do not correspond to successive simple gauge transformations. The similar result given in ref.1 for the Z_2 global gauge anomaly of gauge group SU(2)\times SU(2) is also discussed. We also give a complete proof for some relevent topological results. Gauge anomalies for the relevant semisimple gauge groups are also briefly discussed in higher dimensions, especially for self-contragredient representations, with discussions involving trace identities relating to ref.14. We also remark the connection of our results and discussions to the total generation numbers in relevant theories.