Generalized Donaldson-Thomas invariants

TL;DR

This paper surveys the theory of Donaldson-Thomas invariants for Calabi-Yau 3-folds, introduces generalized invariants that are rational and deformation-invariant, and explores their behavior under stability changes and connections to BPS invariants.

Contribution

It extends Donaldson-Thomas theory to generalized invariants applicable to all Chern characters and relates them to BPS invariants, also connecting to noncommutative invariants and quiver representations.

Findings

Generalized DT invariants are rational and deformation-invariant.

Transformation laws for invariants under stability changes are established.

Connections to noncommutative DT invariants and BPS invariants are demonstrated.

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.