Whereas in Gromov–Witten theory, maps are allowed to be multiple covers and collapsed components of the domain curve, Donaldson–Thomas 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 Gromov–Witten and Donaldson–Thomas theories of algebraic three-folds are actually equivalent.[2] More concretely, their generating functions are equal after an appropriate change of variables. For Calabi–Yau threefolds, the Donaldson–Thomas 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.[clarification needed]
Definition
For a Calabi-Yau threefold [3][4] 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
Now because is Calabi-Yau, Serre duality implies
which gives a perfect obstruction theory of dimension 0. In particular, this implies the associated virtual fundamental class
is in homological degree . We can then define the DT invariant as
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.[5]