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We focus on 4D $\mathcal{N}=2$ string vacua described both by perturbativeHeterotic theory and by Type IIA theory; a Calabi--Yau three-fold $X_{\rm IIA}$in the Type IIA language is further assumed to have a regular K3-fibration. Itis well-known that one can assign a modular form $\Phi$ to such a vacuum bycounting perturbative BPS states in Heterotic theory or collectingNoether--Lefschetz numbers associated with the K3-fibration of$X_{\mathrm{IIA}}$. In this article, we expand the observations and ideas(using gauge threshold correction) in the literature and formulate a modularform $\Psi$ with full generality for the class of vacua above, which can beused along with $\Phi$ for the purpose of classification of those vacua.Topological invariants of $X_{\mathrm{IIA}}$ can be extracted from $\Phi$ and$\Psi$, and even a pair of diffeomorphic Calabi--Yau's with differentK\"{a}hler cones may be distinguished by introducing the notion of "the set of$\Psi$'s for Higgs cascades/for curve classes". We illustrated these ideas bysimple examples.

Modular Forms as Classification Invariants of 4D N=2 Heterotic--IIA Dual Vacua