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## Localized Donaldson-Thomas theory of surfaces

Research output: Working paper

Amin Gholampour, University of Maryland, United States- Artan Sheshmani
Shing-Tung Yau, Harvard University, United States

Let S be a projective simply connected complex surface and be a line bundle on S. We study the moduli space of stable compactly supported 2-dimensional sheaves on the total spaces of . The moduli space admits a ℂ∗-action induced by scaling the fibers of . 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 Donaldson-Thomas invariants of by virtual localization in the case that 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 the localized DT invariants in terms of the invariants of the nested Hilbert schemes defined by the authors in [GSY17a], the Seiberg-Witten invariants of S, and the integrals over the products of Hilbert schemes of points on S. When 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.

Original language | English |
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Number of pages | 28 |
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State | Published - 2017 |
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ID: 119326972