Modular bootstrap for D4-D2-D0 indices on compact Calabi-Yau threefolds
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866916466407768064 |
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| author | Alexandrov, Sergei Gaddam, Nava Manschot, Jan Pioline, Boris |
| author_facet | Alexandrov, Sergei Gaddam, Nava Manschot, Jan Pioline, Boris |
| contents | We investigate the modularity constraints on the generating series $h_r(τ)$ 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 $h^{an}_2$ with exactly the same modular anomaly as $h_2$, so that the difference $h_{2}-h^{an}_2$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_02207 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Modular bootstrap for D4-D2-D0 indices on compact Calabi-Yau threefolds Alexandrov, Sergei Gaddam, Nava Manschot, Jan Pioline, Boris High Energy Physics - Theory Algebraic Geometry We investigate the modularity constraints on the generating series $h_r(τ)$ 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 $h^{an}_2$ with exactly the same modular anomaly as $h_2$, so that the difference $h_{2}-h^{an}_2$ 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. |
| title | Modular bootstrap for D4-D2-D0 indices on compact Calabi-Yau threefolds |
| topic | High Energy Physics - Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2204.02207 |