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

H. Zhang
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Abstract

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.

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H. Zhang. 2005-05-11. Some Global Aspects of Gauge Anomalies of Semisimple Gauge Groups and Fermion Generations in GUT and Superstring Theories. https://doi.org/10.1142/s0217751x06025341

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