Generalized Donaldson-Thomas invariants
Dominic Joyce
TL;DR
Joyce and Song develop a comprehensive generalization of Donaldson--Thomas theory for Calabi--Yau 3-folds by introducing \\bar{DT}^\\alpha(\\tau) for all classes, proving deformation-invariance and outlining a universal wall-crossing framework. They connect these invariants to stable-pair counts PI^{\\alpha, n}(\\tau') and define integral BPS invariants \\hat{DT}^\\alpha(\\tau) via Möbius inversion, conjecturing integrality under generic stability. The work provides a local model of the moduli problem as critical loci, builds a Lie-algebraic bridge from stack functions to an explicit Lie algebra \\tilde{L}(X), and establishes explicit 0- and 1-dimensional-geometry results, paralleling Gromov–Witten/ GV structures. It further extends the construction to quivers with superpotentials, linking to noncommutative DT invariants and embedding CY3-type wall-crossing phenomena in a broader algebraic framework, with connections to Kontsevich–Soibelman and Szendrőierentiations. Overall, the paper places DT theory in a deformation-invariant, wall-crossing–transparent, and integrality-enriched setting, with practical computational tools via stable pairs and quiver models.
Abstract
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.