Localized donaldson-thomas theory of surfaces

Abstract

Let S be a projective simply connected complex surface and L be a line bundle on S. We study the moduli space of stable compactly supported 2-dimensional sheaves on the total spaces of L. The moduli space admits a C∗-action induced by scaling the fibers of L. We identify certain components of the fixed locus of the moduli space with the moduli space of torsion free sheaves and the nested Hilbert schemes on S. We define the localized (reduced) Donaldson-Thomas invariants of L by virtual localization in the case that L twisted by the anti-canonical bundle of S admits a nonzero global section. When pg (S) > 0, in combination with Mochizuki’s formulas, we are able to express these invariants in terms of the invariants from the nested Hilbert schemes defined by the authors, the Seiberg-Witten invariants of S, and the integrals over the products of Hilbert schemes of points on S. When L is the canonical bundle of S, the Vafa-Witten invariants defined recently by Tanaka-Thomas, can be extracted from these localized DT invariants. VW invariants are expected to have modular properties as predicted by S-duality.