Abstract
We use the method of spectral networks to calculate BPS degeneracies in the Minahan-Nemeschansky E7 theory, as representations of the E7 flavor symmetry. Our results provide another example of a pattern noticed earlier in the Minahan-Nemeschansky E6 theory: when the electromagnetic charge is n times a primitive charge, the BPS index is a positive integer multiple of (−1)n+1n. We also calculate BPS degeneracies in the Minahan-Nemeschansky E6 theory for larger charges than were previously computed.
Highlights
We use the method of spectral networks to calculate BPS degeneracies in the Minahan-Nemeschansky E7 theory, as representations of the E7 flavor symmetry
Our results provide another example of a pattern noticed earlier in the Minahan-Nemeschansky E6 theory: when the electromagnetic charge is n times a primitive charge, the BPS index is a positive integer multiple of (−1)n+1n
Since the flavor symmetry is predicted to be enhanced to E7, these characters should a posteriori assemble into characters of representations of E7
Summary
The IR U(1) gauge theory of the theory on its Coulomb branch is described by the SeibergWitten curve, which takes the form det(λ − Φ(z)) = 0,. Ρ determines the Jordan block structure of Φ−1, which in turn determines the form of the meromorphic differentials φd(z); φd(z) has a pole of order at most pd(ρ) at a puncture with partition ρ. The electromagnetic charge lattice of the IR U(1) theory on the Coulomb branch is. Notice that the curve Σ given in (2.4) has Z4 symmetry, generated by the transformation λ → iλ. This generator permutes the primitive electric and magnetic charges: it maps γ1 → −γ2 and γ2 → γ1
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