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).

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

• Donaldson, S. K.; Thomas, R. P. (1998), "Gauge theory in higher dimensions", in Huggett, S. A.; Mason, L. J.; Tod, K. P. et al., The geometric universe (Oxford, 1996), Oxford University Press, pp. 31–47, ISBN 978-0-19-850059-9, MR1634503, http://www2.imperial.ac.uk/~rpwt/skd.pdf 
• Kontsevich, Maxim (2007), Donaldson-Thomas invariants, Mathematische Arbeitstagung, Bonn, http://www.ihes.fr/~maxim/TEXTS/DTinv-AT2007.pdf