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A theory of generalized Donaldson-Thomas invariants

Dominic Joyce, Yinan Song

TL;DR

This work extends Donaldson-Thomas theory to generalized, Behrend-weighted invariants applicable to all Chern classes on Calabi–Yau 3-folds, including cases with strictly semistable objects. It builds a comprehensive framework using constructible/stack functions, Ringel–Hall algebras, and a Behrend-weighted Lie algebra morphism to define bar{DT}^α(τ) and PI^{α,n}(τ'), establishing deformation-invariance and explicit wall-crossing laws. A key advance is the beaming of Behrend function identities through a local critical locus description of the moduli stack, enabling a consistent counting scheme that yields integer “BPS-like” invariants hat{DT}^α(τ) in generic stability. The theory is extended to quivers with superpotentials, connecting to noncommutative DT invariants and KS-type motivic formalisms, and it includes extensive examples (including noncompact CY3s) and analysis of integrality conjectures. This provides a robust, deformation-stable paradigm linking DT theory, stability conditions, and quiver representations, with potential applications to curve counting and BPS state enumeration in string theory.

Abstract

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. This book defines and studies a generalization of Donaldson-Thomas invariants. Our new invariants \bar{DT}^a(t) 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. To prove all this we study the local structure of the moduli stack M of coherent sheaves on X. We show that an atlas for M may be written locally as Crit(f) for f a holomorphic function on a complex manifold, and use this to deduce identities on the Behrend function of M. We compute our invariants in examples, and make a conjecture about their integrality properties. 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. This book is surveyed in the paper arXiv:0910.0105.

A theory of generalized Donaldson-Thomas invariants

TL;DR

This work extends Donaldson-Thomas theory to generalized, Behrend-weighted invariants applicable to all Chern classes on Calabi–Yau 3-folds, including cases with strictly semistable objects. It builds a comprehensive framework using constructible/stack functions, Ringel–Hall algebras, and a Behrend-weighted Lie algebra morphism to define bar{DT}^α(τ) and PI^{α,n}(τ'), establishing deformation-invariance and explicit wall-crossing laws. A key advance is the beaming of Behrend function identities through a local critical locus description of the moduli stack, enabling a consistent counting scheme that yields integer “BPS-like” invariants hat{DT}^α(τ) in generic stability. The theory is extended to quivers with superpotentials, connecting to noncommutative DT invariants and KS-type motivic formalisms, and it includes extensive examples (including noncompact CY3s) and analysis of integrality conjectures. This provides a robust, deformation-stable paradigm linking DT theory, stability conditions, and quiver representations, with potential applications to curve counting and BPS state enumeration in string theory.

Abstract

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. This book defines and studies a generalization of Donaldson-Thomas invariants. Our new invariants \bar{DT}^a(t) 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. To prove all this we study the local structure of the moduli stack M of coherent sheaves on X. We show that an atlas for M may be written locally as Crit(f) for f a holomorphic function on a complex manifold, and use this to deduce identities on the Behrend function of M. We compute our invariants in examples, and make a conjecture about their integrality properties. 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. This book is surveyed in the paper arXiv:0910.0105.

Paper Structure

This paper contains 79 sections, 105 theorems, 258 equations.

Table of Contents

  1. Introduction
  2. Brief sketch of background from Joyc1Joyc2Joyc3Joyc4Joyc5Joyc6Joyc7
  3. Behrend functions of schemes and stacks, from §\ref{['dt4']}
  4. Summary of the main results in §\ref{['dt5']}
  5. Examples and applications in §\ref{['dt6']}
  6. Extension to quivers with superpotentials in §\ref{['dt7']}
  7. Relation to the work of Kontsevich and Soibelman KoSo1
  8. Constructible functions and stack functions
  9. Artin stacks and (locally) constructible functions
  10. Stack functions
  11. Operators $\Pi^\mu$ and projections $\Pi^{\rm vi}_n$
  12. Stack function spaces ${\bar{\text{\underline{\rm SF\!}\,}},\bar{\rm SF}({\mathfrak F},\chi,{\mathbb Q})}$
  13. Background material from Joyc3Joyc4Joyc5Joyc6
  14. Ringel--Hall algebras of an abelian category
  15. (Weak) stability conditions on ${\mathbin{\mathcal{A}}}$
  16. ...and 64 more sections

Key Result

Theorem 2.4

Let ${\mathbin{\mathbb K}}$ have characteristic zero, ${\mathbin{\mathfrak E}},{\mathbin{\mathfrak F}},{\mathbin{\mathfrak G}},{\mathbin{\mathfrak H}}$ be Artin ${\mathbin{\mathbb K}}$-stacks with affine geometric stabilizers, and $\beta:{\mathbin{\mathfrak F}}\rightarrow{\mathbin{\mathfrak G}},$$\g \xymatrix@C=35pt{ \mathop{\rm CF}\nolimits({\mathbin{\mathfrak E}}) \ar[r]_{\mathop{\rm CF}\nolimit

Theorems & Definitions (232)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Definition 2.10
  • ...and 222 more