Donaldson-Thomas invariants

Kontsevich and Soibelman have discovered tantalizing links between cluster algebras, stability conditions [19] and Donaldson-Thomas invariants. Large parts of their theory are still conjectural but a fascinating overall picture is already emerging. In their construction, they start from a quiver (without loops or $2$-cycles) endowed with a superpotential and associate to it a $3$-Calabi-Yau triangulated category $T$ with a $t$-structure. They then define and study a refinement of Bridgeland's space of stability conditions on $T$ (or more generally on any $3$-Calabi-Yau category with ind-constructible moduli stacks of objects and morphisms). They define abstract Donaldson-Thomas invariants using their putative generating functions on the space of stability conditions, see for example the note [80]. Cluster transformations then govern the way these generating functions behave under wall-crossing in the space of stability conditions.