Generalized Donaldson-Thomas invariants - Joyce, D (International Press, 2011)
This is a survey of the book [16] with Yinan Song, Donaldson–Thomas invariants DTα(τ ) ∈ Z ‘count’ τ -(semi)stable coherent sheaves with Chern character α on a Calabi–Yau 3-fold X. They are unchanged under deformations of X. The conventional definition works only for classes α with no strictly τ -semistable sheaves. Behrend showed that DTα(τ) can be written as a weighted Euler characteristic χMαst(τ), νMαst(τ)) of the stable moduli scheme Mαst(τ) by a constructible function νMαst(τ) we call the ‘Behrend function’. We discuss generalized Donaldson–Thomas invariants DTα(τ) ∈ Q. These are defined for all classes α, and are equal to DTα(τ) when it is defined. They are unchanged under deformations of X, and transform according to a known wall-crossing formula under change of stability condition τ . We conjecture that they can be written in terms of integral BPS invariants DTα(τ) ∈ Z when the stability condition τ is ‘generic’. We extend the theory to abelian categories mod-CQ/I of representations of a quiver Q with relations I coming from a superpotential W on Q, and connect our ideas with Szendrői's noncommutative Donaldson–Thomas invariants, and work by Reineke and others on invariants counting quiver representations. The book has significant overlap with a recent, independent paper of Kontsevich and Soibelman.