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

TL;DR

This paper introduces modular forms as invariants to classify 4D N=2 heterotic--IIA string vacua, enabling distinction of Calabi-Yau manifolds with similar topological features through new invariants.

Contribution

It formulates a new modular form invariant for classifying 4D N=2 vacua, extending previous ideas and allowing differentiation of Calabi-Yau manifolds with different Kähler cones.

Findings

Modular forms can classify and distinguish string vacua.

New invariants derived from modular forms reveal topological differences.

Examples demonstrate the effectiveness of the classification method.

Abstract

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.