On some modularity properties of Donaldson-Thomas invariants of compact Calabi-Yau threefolds predicted by supersymmetric string theory

I will talk about one of the special cases 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. In order to approach this problem mathematically, one needs to reduce the threefold theory to a certain intersection theory over the relative Hilbert scheme of points on surfaces and then prove the claimed modularity. More precisely, I will talk about our proof 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.