This may not hold along conifold-like degenerations, where the mass of the magnetic monopole states can be as big as the scale $m_{*}$ defined in (19) [6].

This relation can be satisfied exactly for certain choices of moduli fields, or to arbitrary accuracy for generic ones.

For extremal BPS states, $\gamma$ corresponds to the extremality factor. Given the results of [32, 51], one does not expect RFT-charged particles to be extremal in general, so we henceforth refer to $\gamma$ as charge-to-mass ratio. It would nevertheless be interesting to compare our findings with the divergences for $\gamma$ recently found in [52].

As pointed out already, this may also apply for the field associated to the asymptotic trajectory, but in practice one can check that this direction does not contribute to the curvature divergence [4, 5, 6].

Examples of these are the Seiberg-Witten points discussed in the text, or the so-called coni-LCS singularities [53, 54, 55], which can also be shown to obey both (1) and (27) [6].

The identity (29) can be deduced from the no-force condition ${\mathcal{Q}}_{\mathrm{rigid}}^2=G^{\mu \overline{\nu }}{\partial }_{\mu }{Z}_{\mathrm{rig}}{\overline{\partial }}_{\overline{\nu }}{\overline{Z}}_{\mathrm{rig}}$, which holds for BPS states in 4D $\mathcal{N}=2$ rigid field theories.

When ker $\mathbf{K}$ is trivial, one can also have a set of diverging $\gamma$’s and an RFT-decoupling limit. However, in this case the RFT gauge couplings tend to zero quite fast, and as a result the asymptotic behavior of the metric eigenvalues is such that no curvature divergence is generated. At most, one may have a term of the form (25) that contributes as a constant to the asymptotic curvature, rendering it in some cases positive [4].

With respect to the decoupling criteria discussed in [6], in this case the RFT does not decouple at the level of Pauli interactions. The same applies for RFT sectors made of non-effective divisors.