S-duality for surfaces with $A_n$-type singularities

Abstract

We show that the generating series of Euler characteristics of Hilbert schemes of points on any algebraic surface with at worst $A_n$-type singularities is described by the theta series determined by integer valued positive definite quadratic forms and the Dedekind eta function. In particular it is a Fourier development of a meromorphic modular form with possibly half integer weight. The key ingredient is to apply the flop transformation formula of Donaldson-Thomas type invariants counting two dimensional torsion sheaves on 3-folds proved in the author's previous paper.

17 pages. arXiv admin note: text overlap with arXiv:1311.7476

References in corpus (3)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• The Hilbert scheme of a plane curve singularity and the HOMFLY polynomial of its link
• Zero dimensional Donaldson-Thomas invariants of threefolds