BPS states in the Minahan-Nemeschansky E6 theory

We use the method of spectral networks to compute BPS state degeneracies in the Minahan-Nemeschansky E6E6 theory, on its Coulomb branch, without turning on a mass deformation. The BPS multiplicities come out in representations of the E6E6 flavor symmetry. For example, along the simplest ray in electromagnetic charge space, we give the first 14 numerical degeneracies, and the first 7 degeneracies as representations of E6E6 . We find a complicated spectrum, exhibiting exponential growth of multiplicities as a function of the electromagnetic charge. There is one unexpected outcome: the spectrum is consistent (in a nontrivial way) with the hypothesis of spin purity, that if a BPS state in this theory has electromagnetic charge equal to n times a primitive charge, then it appears in a spin- n2n2 multiplet.