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Taking the trace of an operator over all the states/particles effectively means taking a sum over all the eigenvalues of the operator (the charges) over these states/particles. So what matters is how many states you have and with what charges. The numbers that you are referring to are just the appropriate multiplicities of the states. For example the first ...


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The only possible way Bosons admit anomalies in flat space (analogous to the case for fermions you have mentioned) is when they do not admit a covariant lagrangian. These are called by a special name, "Chiral Bosons". The usual Bose lagrangian can always be regulated to give an action free of anomalies. See section 8 of this paper. ...



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