Massive modes on magnetized blow-up manifold of $T^2/\mathbb{Z}_N$

We study massive modes on a magnetized blow-up manifold of $T^2/\mathbb{Z}_N$. The blow-up manifold can be constructed by appropriately replacing orbifold singular points with a part of $S^2$. To ensure a smooth connection between the massive modes on magnetized $T^2/\mathbb{Z}_N$ orbifold and those on magnetized $S^2$, it is required that not only the total magnetic flux as well as the total curvature but also the effective magnetic flux on the connected line remain invariant under the blow-up procedure. Furthermore, we find that the number of the localized modes at each orbifold singular point increases by one for each unit increment of the mass level.