Donaldson–Thomas theory

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 complex analogue of the Casson invariant. The invariants were introduced by Simon Donaldson and Richard Thomas (1998).

Contents

Examples

  • The moduli space of lines on the quintic threefold is a discrete set of 2875 points. The virtual number of points is the actual number of points, and hence the Donaldson–Thomas invariant of this moduli space is the integer 2875.
  • Similarly, the Donaldson–Thomas invariant of the moduli space of conics on the quintic is 609250.

Facts

  • The Donaldson–Thomas invariant of the moduli space M is equal to the weighted Euler characteristic of M. The weight function associates to every point in M an analogue of the Milnor number of a hyperplane singularity.

Generalizations

  • instead of moduli spaces of sheaves, one considers moduli spaces of derived category objects.
  • instead of integer valued invariants, one considers motivic invariants.

References


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