An Introduction to (Motivic) Donaldson-Thomas Theory

On the wall-crossing formula for quadratic differentials

“…While this is the version of the wall-crossing formula that follows most readily from the one stated in the references [30,9], our main result will ultimately allow us to drop the requirement that the path φ t lies entirely in the domain C τ .…”

“…Since we are mainly interested in the application to quadratic differentials, our discussion will be quite cursory. For other introductory accounts of this material, see the references [24,31,22,30,6].…”

“…We now describe a general framework in which we can formulate the Kontsevich-Soibelman wall-crossing formula. Following [30], we define a motivic theory to be a rule associating to a scheme X an abelian group R(X). This rule is required to be functorial in two ways: If u : X → Y is any morphism of schemes, there is a homomorphism u * : R(Y ) → R(X) called the pullback along u, and if u is of finite type, there is a homomorphism u !…”

“…for each n ∈ N where Sym(X) = X n S n is the nth symmetric product. These data are required to satisfy several axioms listed in Section 4 of [30].…”

“…for any 1-morphism u : X → Y, and a pushforward u ! : R(X) → R(Y) if u is of finite type in the sense of [30]. One also has structures ⊠, 1 ∈ R(Spec C), and σ n for n ∈ N satisfying various properties as in the definition of a motivic theory for schemes.…”