Modular Curves and the Refined Distance Conjecture

We test the refined distance conjecture in the vector multiplet moduli space of 4D $\mathcal{N}=2$ compactifications of the type IIA string that admit a dual heterotic description. In the weakly coupled regime of the heterotic string, the moduli space geometry is governed by the perturbative heterotic dualities, which allows for exact computations. This is reflected in the type IIA frame through the existence of a K3 fibration. We identify the degree $d=2N$ of the K3 fiber as a parameter that could potentially lead to large distances, which is substantiated by studying several explicit models. The moduli space geometry degenerates into the modular curve for the congruence subgroup $\Gamma_0(N)^+$. In order to probe the large $N$ regime, we initiate the study of Calabi-Yau threefolds fibered by general degree $d>8$ K3 surfaces by suggesting a construction as complete intersections in Grassmann bundles.