Localized Towers on the Composite Magnetic Monopole and the Weak Gravity Conjecture$=$Distance Conjecture Connection in Effective Field Theories

We study the energy spectrum of the composite magnetic monopole (CMM) in the model originally constructed in arXiv:1608.06951 to examine how the applicability of the weak gravity conjecture (WGC) propagates to lower energy scales in the effective field theory (EFT) framework. This is motivated by the WGC$=$distance conjecture (DC) connection suggested in the literature. Assuming that the limits of the model parameters correspond to the boundaries of the moduli space in a UV theory, we showed that the parameter dependence of the energy spectrum of the CMM is broadly in accord with the prediction of the WGC$=$DC connection. However, unlike the original DC, the low-energy excitations of the CMM are localized on the CMM. This leads us to propose an extension of the DC to include the localized tower of excited states. We discuss the implications of this extension of the DC to the swampland constraints on EFTs.