Joyce survey on generalized Donaldson-Thomas invariants http://arxiv.org/abs/0910.0105
arXiv:0910.4315 Motivic Donaldson-Thomas invariants: summary of results from arXiv Front: math.AG by Maxim Kontsevich, Yan Soibelman This is a short summary of main results of our paper arXiv:0811.2435 where the concept of motivic Donaldson-Thomas invariant was introduced. It also contains a discussion of some open questions from the loc.cit., in particular, the geometry related to the split attractor flow.
http://ncatlab.org/nlab/show/Donaldson-Thomas+invariant
arXiv:1103.4229 Stability conditions and curve counting invariants on Calabi-Yau 3-folds from arXiv Front: math.AG by Yukinobu Toda The purpose of this paper is twofold: first we give a survey on the recent developments of curve counting invariants on Calabi-Yau 3-folds, e.g. Gromov-Witten theory, Donaldson-Thomas theory and Pandharipande-Thomas theory. Next we focus on the proof of the rationality conjecture of the generating series of PT invariants, and discuss its conjectural Gopakumar-Vafa form.
[arXiv:0910.0105] Generalized Donaldson-Thomas invariants from arXiv Front: math.AG by Dominic Joyce This is a survey of the book arXiv:0810.5645 with Yinan Song. Let X be a Calabi-Yau 3-fold over C. The Donaldson-Thomas invariants of X are integers DT^a(t) which count stable sheaves with Chern character a on X, with respect to a Gieseker stability condition t. They are defined only for Chern characters a for which there are no strictly semistable sheaves on X. They have the good property that they are unchanged under deformations of X. Their behaviour under change of stability condition t was not understood until now
We discuss “generalized Donaldson-Thomas invariants” \bar{DT}^a(t). These are rational numbers, defined for all Chern characters a, and are equal to DT^a(t) if there are no strictly semistable sheaves in class a. They are deformation-invariant, and have a known transformation law under change of stability condition. We conjecture they can be written in terms of integral “BPS invariants” \hat{DT}^a(t) when the stability condition t is “generic”
We extend the theory to abelian categories of representations of a quiver with relations coming from a superpotential, and connect our ideas with Szendroi’s “noncommutative Donaldson-Thomas invariants” and work by Reineke and others. There is significant overlap between arXiv:0810.5645 and the independent paper arXiv:0811.2435 by Kontsevich and Soibelman.
Vittoria Bussi (Oxford): Donaldson?Thomas theory: generalizations and related conjectures. Joyce-style abstract: Generalized Donaldson?Thomas invariants defined by Joyce and Song are rational numbers which “count” both ∞-stable and ∞-semistable coherent sheaves with Chern character ? on a Calabi?Yau 3-fold X, where ? denotes Gieseker stability for some ample line bundle on X. These invariants are defined for all classes ?, and are equal to the classical DT defined by Thomas when it is defined. They are unchanged under deformations of X, and transform by a wall-crossing formula under change of stability condition ?. Joyce and Song use gauge theory and transcendental complex analytic methods, so that the theory of generalized Donaldson?Thomas invariants is valid only in the complex case. This also forces them to put constraints on the Calabi?Yau 3-fold they can define generalized Donaldson?Thomas invariants for. This seminar will propose a new algebraic method extending the theory to algebraically closed fields K of characteristic zero. We will describe the local structure of the moduli stack M of coherent sheaves on X, showing that an atlas for M may be written locally as the zero locus of an almost closed 1-form defined on an etale neighborhood in the tangent space of M and use this to deduce identities on the Behrend function. This last statement has yet a gap in its proof, which we are going to fix soon. This will yield also the extension of generalized Donaldson?Thomas theory to compactly supported coherent sheaves on noncompact quasi-projective Calabi?Yau 3-folds. A similar argument, adapted to complexes of sheaves, should yield also to the validity of generalized DT invariants theory in the derived categorical framework, extending an announced result by Behrend and Getzler, and proving an important conjecture by Toda. Finally we depict several ideas on further developments of in the DT theory, suggested by our strategy, in the direction of motivic and categorified DT invariants in the sense of Kontsevich and Soibelman.
nLab page on Donaldson-Thomas invariants