Title: Counting curves on surfaces in Calabi-Yau threefolds and proof of S-duality modularity conjecture

2015.12.01 |

Date | Tue 12 Jan |

Time | 14:15 — 15:15 |

Location | 1532-322 (G3.3) |

Abstract: I will talk about recent joint works with Amin Gholampour, Richard Thomas and Yukinobu Toda, on an algebraic-geometric proof of the 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 the 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 Hilbert 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, where they used representation theory and the machinery of vertex operators to prove this statement for absolute Hilbert schemes. These intersection numbers eventually, together with the generating series of Noether-Lefschetz numbers as I will explain, will provide the ingredients to achieve an algebraic-geometric proof of S-duality modularity conjecture.