On the proof of S-duality modularity conjecture on quintic threefolds
Also appears in collection : Moduli spaces in geometry / Espaces de modules en géométrie
I will talk about joint work during the recent years with Amin Gholampour, Richard Thomas and Yukinobu Toda, on proving the modularity property of the generating series of certain DT invariants of torsion sheaves with two dimensional support in ambient threefolds. More specifically, I will talk about algebraic-geometric proof of S-duality conjecture in superstring theory made formerly by physicists: Gaiotto, Strominger, Yin, regarding the modularity of DT invariants of sheaves supported on hyperplane sections of the quintic Calabi-Yau threefold. Our strategy is to first use degeneration and localization techniques to reduce the threefold theory to a certain intersection theory over relative Hilbert scheme of points on surfaces and then prove modularity; More precisely, together with Gholampour we have proven that the generating series, associated to the top intersection numbers of the Hibert scheme of points, relative to an effective divisor, on a smooth quasi-projective surface is a modular form. This is a generalization of the result of Okounkov-Carlsson for absolute Hilbert schemes. These intersection numbers, together with the generating series of Noether-Lefschetz numbers, will provide the ingrediants to prove modularity of the above DT invariants over the quintic threefold.
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