On the BPS Spectrum of the Rank-1 Minahan-Nemeschansky Theories

The F-theory realization of the rank-1 Minahan Nemeschansky (MN) $E_6$, $E_7$ and $E_8$ theories leads to a description of the BPS states on the Coulomb branch in terms of Type IIB $(p,q)$-string networks. Subject to a simple ansatz for the types of networks which can occur, we study the representations of the flavor symmetry group which occur in the BPS spectrum. The results we find for the $E_6$ and $E_7$ theories are in perfect agreement with previous calculations by other methods (in particular, we find that arbitrarily large representations occur), but our scheme is easier to implement and more computationally efficient. The string network picture also gives a possible explanation of the experimental observation that in rank-1 MN theories, BPS states whose charge is $n$ times a primitive charge occur with BPS index divisible by $(-1{)}^{n+1}n$.