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Résumé |
In type IIB string theory compactified on a Calabi-Yau three-fold, S-duality requires that the sum over D3-D1-
D(-1) instantons (with fixed D3-brane charge) must be modular invariant. Correspondingly, in type IIA string
theory, holography requires that the D4-D2-D0 black hole partition function (with fixed D4-brane charge)
should be modular. In either case, the sum over D1 (resp. D2) charges is given by an indefinite theta series of
signature $r(1,b_2-1)$, where $r$ is the D3 (resp. D4) charge. Using physical insight we have found a general
way of determining the modular completion of a large class of holomorphic indefinite theta series,
generalizing Zweger's method to arbitrary signature. Based on a series of works with Alexandrov, Banerjee
and Manschot. |
