Donaldson-Thomas invariants -- Kai Behrend

Donaldson-Thomas invariants were introduced by Simon Donaldson and Richard Thomas as a holomorphic analogue of the Casson invariant of threefolds. The invariants are by definition virtual counts of holomorphic vector bundles on a three-dimensional complex oriented (Calabi-Yau) manifold. Recently, Donaldson-Thomas invariants were shown to have deep connections with the Gromov-Witten theory and Quantum Cohomology of Calabi-Yau threefolds. We will explain what Donaldson-Thomas invariants are. We will discuss their construction, and explain why they have attracted so much interest in recent years. In particular, we will explain how Donaldson-Thomas theory gives a new point of view of the Hilbert scheme of points on a Calabi-Yau threefold. We will explain how techniques from micro-local geometry yield a proof that Donaldson-Thomas type invariants are always equal to certain weighted Euler characteristics of the moduli space of sheaves. We will explain how this yields a proof of a conjecture of Maulik-Nekrasov-Okounkov-Pandharipande on the Hilbert scheme of points of a Calabi-Yau threefold. We will address some attempts at categorifying Donaldson-Thomas theory. This means that we will discuss how one might try to construct a cohomology theory expressing Donaldson-Thomas invariants as alternating sums of dimensions of cohomology groups.