BPS states in the Minahan-Nemeschansky $E_7$ theory

A plethora of K3 metrics

“…In §4, we carry out this procedure, working at leading order in the blow-up parameters. At this order, we are only able to find weak constraints on the BPS spectra of the 4d SCFTs mentioned above (which are consistent with the results of [16][17][18][19][20]). Furthermore, as in [2] we will find that there are BPS states in the compactified little string theory which do not contribute at this order, and so one will need to proceed (at least) to the next order in order to study the full spectrum of the little string theory.…”

“…Furthermore, since the symplectic forms are only sensitive to the magnitudes of the real FI parameters at the present order, there is some phase freedom in our modifications in (4.15). However, (4.15) does at least lead to a change of variables from ηs to ξs with the property that all of the data from [18,20] agrees with our Poissonresummed Higgs branch results, and it also simplifies some formulae. Rather than obsess (further) over this, we defer its resolution to [39,44], since higher order corrections to the symplectic forms should be sensitive to the phases of the FI parameters, and will therefore hopefully clarify this point, and the perturbation theory about ALE metrics that describes K3 metrics near the fixed points may also be helpful.…”

“…Nevertheless, it is quite gratifying to find characters of E n representations and integral BPS state counts hiding within K3 metrics, in addition to the Spin(8) characters and BPS state counts we found in [2]. This is particularly so because of the tremendous differences between the SU (2) N f = 4 and MN spectra: infinitely many E n [17] and (conjecturally [18][19][20]) supersymmetry representations and charges with arbitrarily large values of gcd(p, q) are present. And, indeed, our constraints, weak though they may be (at the present order), are infinite in number and involve the BPS index at arbitrarily large values of gcd(p, q); this is in contrast with the results of [18,20], which completely determine the BPS index for small values of gcd(p, q).…”

“…The only remaining problem is, when there are singularities with isomorphic corresponding factors in G 3d within a singular fiber, to figure out which singularity, with its associated FI parameters as in (3.24), should be paired with which simple factor of G 3d and its associated ηs. To do so, we will combine the results of our Poisson resummation with a general result of [16] that constrains the flavor representations that can appear in the BPS spectrum with given values of the gauge charges, as well as minimal information from [20] when q = 4.…”

“…This is particularly so because of the tremendous differences between the SU (2) N f = 4 and MN spectra: infinitely many E n [17] and (conjecturally [18][19][20]) supersymmetry representations and charges with arbitrarily large values of gcd(p, q) are present. And, indeed, our constraints, weak though they may be (at the present order), are infinite in number and involve the BPS index at arbitrarily large values of gcd(p, q); this is in contrast with the results of [18,20], which completely determine the BPS index for small values of gcd(p, q). We are also led to conjecture novel strong relationships between the BPS spectra of the MN theories and of the SU (2) N f = 4 SCFT.…”