In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, specifically algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, Donaldson–Thomas theory is the theory of Donaldson–Thomas invariants. Given a compact

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

 moduli space

Moduli space

In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...

 of sheaves

Sheaf (mathematics)

In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...

 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

Fundamental class

In mathematics, the fundamental class is a homology class [M] associated to an oriented manifold M, which corresponds to "the whole manifold", and pairing with which corresponds to "integrating over the manifold"...

. The Donaldson–Thomas invariant is a complex analogue of the Casson invariant

Casson invariant

In 3-dimensional topology, a part of the mathematical field of geometric topology, the Casson invariant is an integer-valued invariant of oriented integral homology 3-spheres, introduced by Andrew Casson....

. The invariants were introduced by .

Examples

• The moduli space of lines on the quintic threefold
  Quintic threefold
  In mathematics, a quintic threefold is a degree 5 dimension 3 hypersurface in 4-dimensional projective space. Non-singular quintic threefolds are Calabi-Yau manifolds.The Hodge diamond of a non-singular quintic 3-fold is-Rational curves:...
  
   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
  Euler characteristic
  In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...
  
   of M. The weight function associates to every point in M an analogue of the Milnor number
  Milnor number
  In mathematics, and particularly singularity theory, the Milnor number, named after John Milnor, is an invariant of a function germ.If f is a complex-valued holomorphic function germ then the Milnor number of f, denoted μ, is either an integer greater than or equal to zero, or it is infinite. It...
  
   of a hyperplane singularity.

Generalizations

• instead of moduli spaces of sheaves, one considers moduli spaces of derived category
  Derived category
  In mathematics, the derived category D of an abelian category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C...
  
   objects.
• instead of integer valued invariants, one considers motivic
  Motive (algebraic geometry)
  In algebraic geometry, a motive denotes 'some essential part of an algebraic variety'. To date, pure motives have been defined, while conjectural mixed motives have not. Pure motives are triples , where X is a smooth projective variety, p : X ⊢ X is an idempotent correspondence, and m an integer...
  
  invariants.