Donaldson-Thomas invariants of 2-dimensional sheaves inside threefolds and modular forms (Q1694300)

In theoretical physics, \(S\)-duality (also known as strong-weak duality) is an equivalence of two physical theories, which may be either quantum field theories or string theories. \textit{C. Vafa} and \textit{E. Witten} [Nucl. Phys., B 431, No. 1--2, 3--77 (1994; Zbl 0964.81522)] gave a test of S-duality on \(N=4\) SUSY Yang-Mills theory, where they used Weitzenböck formula to identify moduli spaces of solutions of VW equations with moduli spaces of ASD connections on certain 4-manifolds. On algebraic surfaces, the latter spaces can be described using algebraic geometry and the generating series of Euler characteristics of the moduli spaces are easier to compute and can be verified to satisfy modular property. Vafa-Witten equations on 4-manifolds may be regarded as a dimensional reduction of G2 instanton equations on Riemannian 7-manifolds with exceptional holonomy G2. So one may want to give a S-duality test on G2 gauge theory. As mentioned above, algebraic geometric setting is much easier to do calculations and reduction to Calabi-Yau 3-folds seems reasonable. On CY 3-folds, Donaldson-Thomas invariants [\textit{R. P. Thomas}, J. Differ. Geom. 54, No. 2, 367--438 (2000; Zbl 1034.14015)] are counting invariants for \(\mathrm{SU}(3)\)-instantons (or more generally coherent sheaves), which are closely related to BPS numbers in type IIA/B string theories. In this very interesting paper, the authors study \(S\)-duality on \(K3\)-fibered CY 3-folds, where two dimensional stable sheaves supported on fibers are considered. They first showed in the smooth fibration case, DT invariants can be explicitly computed using Euler characteristics of Hilbert schemes of points on \(K3\) surfaces and Noether-Lefschetz numbers. Then they used degeneration formula of \textit{J. Li} and \textit{B. Wu} [Commun. Anal. Geom. 23, No. 4, 841--921 (2015; Zbl 1349.14014)] to handle the case of nodal fibered \(K3\)-surfaces and then explicitly check modularity of the generating series.