Donaldson–Thomas theory

In mathematics, specifically algebraic geometry, Donaldson–Thomas theory is the theory of Donaldson–Thomas invariants. Given a compact moduli space of sheaves on a Calabi–Yau threefold, its Donaldson–Thomas invariant is the virtual number of its points, i.e., the integral of the cohomology class 1 against the virtual fundamental class. The Donaldson–Thomas invariant is a holomorphic analogue of the Casson invariant. The invariants were introduced by Simon Donaldson and Richard Thomas (1998). Donaldson–Thomas invariants have close connections to Gromov–Witten invariants of algebraic three-folds and the theory of stable pairs due to Rahul Pandharipande and Thomas.

Donaldson–Thomas theory is physically motivated by certain BPS states that occur in string and gauge theory^{pg 5}. This is due to the fact the invariants depend on a stability condition on the derived category ${\displaystyle D^{b}({\mathcal {M}})}$ 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 ${\displaystyle {\mathcal {P}}\subset D^{b}({\mathcal {M}})}$ is the category of BPS states for the corresponding SCFT.