Counting Yang-Mills dyons with index theorems

We count the supersymmetric bound states of many distinct BPS monopoles in $N=4\mathrm{}$ Yang-Mills theories and in pure $N=2\mathrm{}$ Yang-Mills theories. The novelty here is that we work in generic Coulombic vacua where more than one adjoint Higgs fields are turned on. The number of purely magnetic bound states is again found to be consistent with the electromagnetic duality of the $N=4\mathrm{}\mathrm{SU}\left(n\right)\mathrm{}$ theory, as expected. We also count dyons of generic electric charges, which correspond to 1/4 BPS dyons in $N=4\mathrm{}$ theories and 1/2 BPS dyons in $N=2\mathrm{}$ theories. Surprisingly, the degeneracy of dyons is typically much larger than would be accounted for by a single supermultiplet of appropriate angular momentum, implying many supermultiplets of the same charge and the same mass.