Modular bootstrap for D4-D2-D0 indices on compact Calabi–Yau threefolds

We investigate the modularity constraints on the generating series $h_r(\tau)$ of BPS indices counting D4-D2-D0 bound states with fixed D4-brane charge $r$ in type IIA string theory compactified on complete intersection Calabi–Yau threefolds with $b_2 = 1$. For unit D4-brane, $h_1$ transforms as a (vector-valued) modular form under the action of $SL(2,\mathbb{Z})$ and thus is completely determined by its polar terms. We propose an Ansatz for these terms in terms of rank‑1 Donaldson–Thomas invariants, which incorporates contributions from a single $\mathrm{D}6 \textrm{-} \overline{\mathrm{D}6}$ pair. Using an explicit overcomplete basis of the relevant space of weakly holomorphic modular forms (valid for any $r$), we find that for $10$ of the $13$ allowed threefolds, the Ansatz leads to a solution for $h_1$ with integer Fourier coefficients, thereby predicting an infinite series of DT invariants. For $r \gt 1$, $h_r$ is mock modular and determined by its polar part together with its shadow. Restricting to $r = 2$, we use the generating series of Hurwitz class numbers to construct a series $h^{(\mathrm{an})}_2$ with exactly the same modular anomaly as $h_2$, so that the difference $h_2 - h^{(\mathrm{an})}_2$ is an ordinary modular form fixed by its polar terms. For lack of a satisfactory Ansatz, we leave the determination of these polar terms as an open problem.

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Published 6 June 2024