Donaldson-Thomas theory

introduction

The Donaldson-Thomas invariant of a Calabi-Yau 3-fold Y (complex projective manifold of dimension 3 with nowhere vanishing holomorphic 3-form) can be thought of as a generalization of the Donaldson invariant.
It was defined by virtual integrals on the moduli space of stable sheaves on Y and expected to count algebraic curves in Y.
The categorification conjecture due to Kontsevich-Soibelman, Joyce-Song, Behrend-Bryan-Szendroi and others claims that there should be a cohomology theory on the moduli space of stable sheaves whose Euler number coincides with the Donaldson-Thomas invariant.
I will talk about recent progress about the categorification conjecture by using perverse sheaves. Locally the moduli space is the critical locus of a holomorphic function on a complex manifold called a Chern-Simons chart and we have the perverse sheaf of vanishing cycles on the critical locus. By constructing suitable Chern-Simons charts and homotopies using gauge theory, it is possible to glue the perverse sheaves of vanishing cycles to obtain a globally defined perverse sheaf whose hypercohomology is the desired categorified Donaldson-Thomas invariant. As an application, we can provide a mathematical theory of the Gopakumar-Vafa (BPS) invariant. I will also discuss wall crossing formulas for these invariants.

In 1980s, Donaldson discovered his famous invariant of 4-manifolds which was subsequently proved to be an integral on the moduli space of semistable sheaves when the 4-manifold is an algebraic surface.
In 1994, the Seiberg-Witten invariant was discovered and conjectured to be equivalent to the Donaldson invariant (still open).
In late 1990s, Taubes proved that the Seiberg-Witten invariant also counts pseudo-holomorphic curves.

Nagao, Kentaro. 2010. “Donaldson-Thomas theory and cluster algebras.” 1002.4884 (February 26). http://arxiv.org/abs/1002.4884
Sergio Cecotti, Andrew Neitzke, Cumrun Vafa, 2010, R-Twisting and 4d/2d Correspondences
M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations