DonaldsonThomas

Donaldson-Thomas invariants, commonly called DT invariants, are integer-valued invariants in algebraic geometry that count certain subschemes or coherent sheaves on Calabi-Yau threefolds. They were introduced by Simon Donaldson and Richard Thomas in the late 1990s as part of a program to formulate enumerative invariants of curves in threefolds. The counts are defined via moduli spaces of stable sheaves or ideal sheaves with fixed Chern character, and are weighted by the Behrend function to yield integers. The resulting numbers depend on the chosen stability condition and the numerical data describing the objects being counted.

DT invariants are related to other curve-counting theories. The MNOP conjecture posits a precise relationship between DT invariants and Gromov-Witten invariants of Calabi-Yau threefolds after a change of variables, tying together two major approaches to counting curves. The theory has extensions, including motivic and cohomological versions, and adaptations to quivers with potential and other settings.

A key feature of Donaldson-Thomas theory is wall-crossing: as stability conditions vary, the invariants change according to established wall-crossing formulas, such as those of Kontsevich-Soibelman and Joyce-Song. In physics, DT invariants appear in string theory as counts of BPS states associated with Calabi-Yau compactifications, reflecting deep connections between geometry and physics. Donaldson-Thomas theory remains a foundational tool in modern enumerative geometry, with ongoing developments and applications.