Donaldson–Thomas invariant

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.

The basic idea of Gromov–Witten 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 Donaldson–Thomas 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.

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