We apply the methods of S. Alexandrov et al. , Commun. Number Theory Phys. 18, 49 (2024) 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 Z₇ ℤ 7 quotient of Rødland’s pfaffian threefold, a Z₅ ℤ 5 quotient of Hosono-Takagi’s double quintic symmetroid threefold, the Z₃ ℤ 3 quotient of the bicubic intersection in P⁵ ℙ 5, and the Z₅ ℤ 5 quotient of the quintic hypersurface in P⁴ 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 S. Alexandrov et al. , Commun. Number Theory Phys. 18, 49 (2024), 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, 11 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 S. Alexandrov et al. , Adv. Theor. Math. Phys. 27, 683 (2023) 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.
Joseph McGovern (Tue,) studied this question.