Localized Donaldson-Thomas theory of surfaces

Let $S$ be a projective simply connected complex surface and $\cal{L}$ be a line bundle on $S$. We study the moduli space of stable compactly supported2-dimensional sheaves on the total spaces of $\cal{L}$. The moduli space admits a ${\Bbb C}^*$-action induced by scaling the fibers of $\cal{L}$. 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 $\cal{L}$ by virtual localization in the case that $\cal{L}$ twisted by the anti-canonical bundle of $S$ admits a nonzero global section. When $p_g(S)>0$, 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 $\cal{L}$ 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.