BPS states in the Minahan-Nemeschansky E6 theory

TL;DR

This paper computes BPS state degeneracies in the Minahan-Nemeschansky E6 theory using spectral networks, revealing a complex spectrum with exponential growth and supporting the spin purity hypothesis.

Contribution

It introduces a spectral network method to compute BPS spectra in the E6 theory without mass deformation, providing new detailed degeneracy data.

Findings

BPS multiplicities form E6 representations.

Spectrum exhibits exponential growth with charge.

Supports the spin purity hypothesis.

Abstract

We use the method of spectral networks to compute BPS state degeneracies in the Minahan-Nemeschansky $E_6$ theory, on its Coulomb branch, without turning on a mass deformation. The BPS multiplicities come out in representations of the $E_6$ 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 $E_6$. 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-$\frac{n}{2}$ multiplet.