Surveys in Differential Geometry

Volume 16 (2011)

Generalized Donaldson-Thomas invariants

Pages: 125 – 160

DOI: http://dx.doi.org/10.4310/SDG.2011.v16.n1.a4

Author

Dominic Joyce (Mathematical Institute, 24-29 St. Giles, Oxford, OX1 3LB, United Kingdom)

Abstract

This is a survey of the book \cite{JoSo} with Yinan Song.Donaldson–Thomas invariants $DT^\al(\tau)\in\Z$ `count'$\tau$-(semi)stable coherent sheaves with Chern character $\al$ on aCalabi–Yau 3-fold $X$. They are unchanged under deformations of$X$. The conventional definition works only for classes $\al$ withno strictly $\tau$-semistable sheaves. Behrend showed that$DT^\al(\tau)$ can be written as a weighted Euler characteristic$\chi\bigl(\M_\st^\al(\tau), \nu_{\M_\st^\al(\tau)}\bigr)$ of thestable moduli scheme $\M_\st^\al(\tau)$ by a constructible function$\nu_{\M_\st^\al(\tau)}$ we call the `Behrend function'.

We discuss {\it generalized Donaldson–Thomas invariants\/}$\bar{DT}{}^\al(\tau)\in\Q$. These are defined for all classes$\al$, and are equal to $DT^\al(\tau)$ when it is defined. They areunchanged under deformations of $X$, and transform according to aknown wall-crossing formula under change of stability condition$\tau$. We conjecture that they can be written in terms of integral{\it BPS invariants\/} $\hat{DT}{}^\al(\tau)\in\Z$ when thestability condition $\tau$ is `generic'.

We extend the theory to abelian categories $\modCQI$ ofrepresentations of a quiver $Q$ with relations $I$ coming from asuperpotential $W$ on $Q$, and connect our ideas with Szendr\H oi'snoncommutative Donaldson–Thomas invariants, and work by Reineke andothers on invariants counting quiver representations. The book\cite{JoSo} has significant overlap with a recent, independent paperof Kontsevich and Soibelman~\cite{KoSo1}.

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