In mathematics, specifically algebraic geometry, DonaldsonâÂÂThomas theory is the theory of DonaldsonâÂÂThomas invariants. Given a compact moduli space of sheaves on a CalabiâÂÂYau threefold, its DonaldsonâÂÂThomas invariant is the virtual number of its points, i.e., the integral of the cohomology class 1 against the virtual fundamental class. The DonaldsonâÂÂThomas invariant is a holomorphic analogue of the Casson invariant. The invariants were introduced by . DonaldsonâÂÂThomas invariants have close connections to GromovâÂÂWitten invariants of algebraic three-folds and the theory of stable pairs due to Rahul Pandharipande and Thomas.
DonaldsonâÂÂThomas theory is physically motivated by certain BPS states that occur in string and gauge theory<sup>pg 5</sup>. This is due to the fact the invariants depend on a stability condition on the derived category of the moduli spaces being studied. Essentially, these stability conditions correspond to points in the Kahler moduli space of a Calabi-Yau manifold, as considered in mirror symmetry, and the resulting subcategory is the category of BPS states for the corresponding SCFT.
The basic idea of GromovâÂÂWitten invariants is to probe the geometry of a space by studying pseudoholomorphic maps from Riemann surfaces to a smooth target. The moduli stack of all such maps admits a virtual fundamental class, and intersection theory on this stack yields numerical invariants that can often contain enumerative information. In similar spirit, the approach of DonaldsonâÂÂThomas theory is to study curves in an algebraic three-fold by their equations. More accurately, by studying ideal sheaves on a space. This moduli space also admits a virtual fundamental class and yields certain numerical invariants that are enumerative.
Whereas in GromovâÂÂWitten theory, maps are allowed to be multiple covers and collapsed components of the domain curve, DonaldsonâÂÂThomas theory allows for nilpotent information contained in the sheaves, however, these are integer valued invariants. There are deep conjectures due to Davesh Maulik, Andrei Okounkov, Nikita Nekrasov and Rahul Pandharipande, proved in increasing generality, that GromovâÂÂWitten and DonaldsonâÂÂThomas theories of algebraic three-folds are actually equivalent. More concretely, their generating functions are equal after an appropriate change of variables. For CalabiâÂÂYau threefolds, the DonaldsonâÂÂThomas invariants can be formulated as weighted Euler characteristic on the moduli space. There have also been recent connections between these invariants, the motivic Hall algebra, and the ring of functions on the quantum torus.
For a CalabiâÂÂYau threefold and a fixed cohomology class there is an associated moduli stack of coherent sheaves with Chern character . In general, this is a non-separated Artin stack of infinite type which is difficult to define numerical invariants upon it. Instead, there are open substacks parametrizing such coherent sheaves which have a stability condition imposed upon them, i.e. -stable sheaves. These moduli stacks have much nicer properties, such as being separated of finite type. The only technical difficulty is they can have bad singularities due to the existence of obstructions of deformations of a fixed sheaf. In particular<blockquote></blockquote>Now because is CalabiâÂÂYau, Serre duality implies<blockquote></blockquote>which gives a perfect obstruction theory of dimension 0. In particular, this implies the associated virtual fundamental class<blockquote></blockquote>is in homological degree . We can then define the DT invariant as<blockquote></blockquote>which depends upon the stability condition and the cohomology class . It was proved by Thomas that for a smooth family the invariant defined above does not change. At the outset researchers chose the Gieseker stability condition, but other DT-invariants in recent years have been studied based on other stability conditions, leading to wall-crossing formulas.