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

We focus on 4D $\mathcal{N}=2$ string vacua described both by perturbative Heterotic 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. It is well-known that one can assign a modular form $\Phi$ to such a vacuum by counting perturbative BPS states in Heterotic theory or collecting Noether--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 modular form $\Psi$ with full generality for the class of vacua above, which can be used 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 different K\"{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 by simple examples.