New Examples of Abelian D4D2D0 Indices

TL;DR

This work develops explicit Abelian D4D2D0 index generating functions for several quotient Calabi–Yau threefolds by assuming the modularity of MSW invariants. Building on the GV–PT–DT framework and the BCOV holomorphic anomaly equations, it leverages explicit polar data, torsion in H^2, and a refined AGMP Ansatz to fix the holomorphic ambiguities and produce vector-valued modular forms that encode MSW indices. The paper provides detailed one-parameter examples (Z5 quintic, Z3 bicubic, Z7 Pfaffian, Z5 Hosono–Takagi) and five multiparameter cases, comparing with GV invariants to test modularity and extend predictions to higher genus. Although some models remain speculative due to limited GV data, the results demonstrate how modularity can determine MSW indices from a few polar terms, offering new checks and pointing to directions linking MSW modularity with elliptic fibrations and wall-crossing phenomena.

Abstract

We apply the methods of \cite{Alexandrov:2023zjb} to compute generating series of D4D2D0 indices with a single unit of D4 charge for several compact Calabi-Yau threefolds, assuming modularity of these indices. Our examples include a $\mathbb{Z}_{7}$ quotient of Rødland's pfaffian threefold, a $\mathbb{Z}_{5}$ quotient of Hosono-Takagi's double quintic symmetroid threefold, the $\mathbb{Z}_{3}$ quotient of the bicubic intersection in $\mathbb{P}^{5}$, and the $\mathbb{Z}_{5}$ quotient of the quintic hypersurface in $\mathbb{P}^{4}$. For these examples we compute GV invariants to the highest genus that available boundary conditions make possible, and for the case of the quintic quotient alone this is sufficiently many GV invariants for us to make one nontrivial test of the modularity of these indices. As discovered in \cite {Alexandrov:2023zjb}, the assumption of modularity allows us to compute terms in the topological string genus expansion beyond those obtainable with previously understood boundary data. We also consider five multiparameter examples with $h^{1,1}>1$, for which only a single index needs to be computed for modularity to fix the rest. We propose a modification of the formula in \cite{Alexandrov:2022pgd} that incorporates torsion to solve these models. Our new examples are only tractable because they have sufficiently small triple intersection and second Chern numbers, which happens because all of our examples are suitable quotient manifolds. In an appendix we discuss some aspects of quotient threefolds and their Wall data.