New Examples of Abelian D4D2D0 Indices

TL;DR

This paper computes D4D2D0 indices for various Calabi-Yau threefolds using modularity assumptions, extends the calculation of GV invariants, and tests the modularity hypothesis with new examples including quotients and multiparameter models.

Contribution

It provides new explicit examples of D4D2D0 indices for quotient Calabi-Yau threefolds and extends the computational framework using modularity assumptions and torsion modifications.

Findings

Computed GV invariants to high genus for several quotient Calabi-Yau threefolds.

Confirmed the modularity of D4D2D0 indices through nontrivial tests.

Extended the formula for indices to include torsion effects.

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 R{\o}dland'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.