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Stability structures, motivic Donaldson-Thomas invariants and cluster transformations

Maxim Kontsevich, Yan Soibelman

TL;DR

The paper develops a comprehensive framework linking stability conditions on 3d Calabi–Yau categories to generalized Donaldson–Thomas invariants, using motivic Hall algebras and motivic quantum tori. It constructs both motivic and numerical DT-invariants, proves multiplicative wall-crossing formulas, and establishes integrality conjectures that connect to Behrend's microlocal approach. The work unifies geometric DT theory, noncommutative and ind-constructible categories, and quiver/cluster transformations, while connecting to Seiberg–Witten physics and integrable systems. Italso provides a program for extending DT-type counts to broader noncommutative geometries, with orientation data and a quasi-classical limit yielding integral BPS-like counts.

Abstract

We define new invariants of 3d Calabi-Yau categories endowed with a stability structure. Intuitively, they count the number of semistable objects with fixed class in the K-theory of the category ("number of BPS states with given charge" in physics language). Formally, our motivic DT-invariants are elements of quantum tori over a version of the Grothendieck ring of varieties over the ground field. Via the quasi-classical limit "as the motive of affine line approaches to 1" we obtain numerical DT-invariants which are closely related to those introduced by Behrend. We study some properties of both motivic and numerical DT-invariants including the wall-crossing formulas and integrality. We discuss the relationship with the mathematical works (in the non-triangulated case) of Joyce, Bridgeland and Toledano-Laredo, as well as with works of physicists on Seiberg-Witten model (string junctions), classification of N=2 supersymmetric theories (Cecotti-Vafa) and structure of the moduli space of vector multiplets. Relating the theory of 3d Calabi-Yau categories with distinguished set of generators (called cluster collection) with the theory of quivers with potential we found the connection with cluster transformations and cluster varieties (both classical and quantum).

Stability structures, motivic Donaldson-Thomas invariants and cluster transformations

TL;DR

The paper develops a comprehensive framework linking stability conditions on 3d Calabi–Yau categories to generalized Donaldson–Thomas invariants, using motivic Hall algebras and motivic quantum tori. It constructs both motivic and numerical DT-invariants, proves multiplicative wall-crossing formulas, and establishes integrality conjectures that connect to Behrend's microlocal approach. The work unifies geometric DT theory, noncommutative and ind-constructible categories, and quiver/cluster transformations, while connecting to Seiberg–Witten physics and integrable systems. Italso provides a program for extending DT-type counts to broader noncommutative geometries, with orientation data and a quasi-classical limit yielding integral BPS-like counts.

Abstract

We define new invariants of 3d Calabi-Yau categories endowed with a stability structure. Intuitively, they count the number of semistable objects with fixed class in the K-theory of the category ("number of BPS states with given charge" in physics language). Formally, our motivic DT-invariants are elements of quantum tori over a version of the Grothendieck ring of varieties over the ground field. Via the quasi-classical limit "as the motive of affine line approaches to 1" we obtain numerical DT-invariants which are closely related to those introduced by Behrend. We study some properties of both motivic and numerical DT-invariants including the wall-crossing formulas and integrality. We discuss the relationship with the mathematical works (in the non-triangulated case) of Joyce, Bridgeland and Toledano-Laredo, as well as with works of physicists on Seiberg-Witten model (string junctions), classification of N=2 supersymmetric theories (Cecotti-Vafa) and structure of the moduli space of vector multiplets. Relating the theory of 3d Calabi-Yau categories with distinguished set of generators (called cluster collection) with the theory of quivers with potential we found the connection with cluster transformations and cluster varieties (both classical and quantum).

Paper Structure

This paper contains 50 sections, 34 theorems, 435 equations.

Table of Contents

  1. Introduction
  2. Counting problems for $3$-dimensional Calabi-Yau varieties
  3. Non-commutative varieties with polarization
  4. Donaldson-Thomas invariants for non-commutative $3d$ Calabi-Yau varieties
  5. Multiplicative wall-crossing formula
  6. Some analogies and speculations
  7. About the content of the paper
  8. Stability conditions for graded Lie algebras
  9. Stability data
  10. Reformulation of the stability data
  11. Topology and the wall-crossing formula
  12. Crossing the wall of second kind
  13. Invariants $\Omega(\gamma)$ and the dilogarithm
  14. Symplectic double torus
  15. Complex integrable systems and stability data
  16. ...and 35 more sections

Key Result

Theorem 1

The forgetting map $Stab({\mathcal{C}})\to {\bf C}^n\simeq { \operatorname{Hom}}(\Gamma,{\bf C})$ given by $( Z,{\mathcal{C}}^{ss}, (\operatorname{Log} Z(E))_{E\in {\mathcal{C}}^{ss}})\mapsto Z$, is a local homeomorphism.

Theorems & Definitions (100)

  • Example 1
  • Remark 1
  • Remark 2
  • Theorem 1
  • Conjecture 1
  • Remark 3
  • Definition 1
  • Remark 4
  • Remark 5
  • Theorem 2
  • ...and 90 more