%0 Unpublished work %T Modular bootstrap for D4-D2-D0 indices on compact Calabi-Yau threefolds %+ Laboratoire Charles Coulomb (L2C) %+ Universiteit Utrecht / Utrecht University [Utrecht] %+ Trinity College Dublin %+ Laboratoire de Physique Théorique et Hautes Energies (LPTHE) %A Alexandrov, Sergey %A Gaddam, Nava %A Manschot, Jan %A Pioline, Boris %Z L2C:22-025 %8 2022-04-05 %D 2022 %Z 2204.02207 %Z Physics [physics]/High Energy Physics - Theory [hep-th] %Z Mathematics [math]/Algebraic Geometry [math.AG]Preprints, Working Papers, ... %X 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 $SL(2,Z)$ 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 D6-anti-D6 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 $r > 1$, $h_r$ 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 $h^{\rm an}_2$ with exactly the same modular anomaly as $h_2$, so that the difference $h_{2}-h^{\rm an}_2$ 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. %G English %L hal-03635855 %U https://hal.science/hal-03635855 %~ CNRS %~ LPTHE %~ L2C %~ UNIV-MONTPELLIER %~ SORBONNE-UNIVERSITE %~ SORBONNE-UNIV %~ SU-SCIENCES %~ SU-TI %~ ALLIANCE-SU %~ UM-2015-2021 %~ UM-EPE