[Paper Review] Stability structures, motivic Donaldson-Thomas invariants and cluster transformations

Maxim Kontsevich, Yan Soibelman|ArXiv.org|Nov 16, 2008

Algebraic structures and combinatorial models71 references730 citations

TL;DR

This paper establishes a framework for generalized Donaldson-Thomas (DT) invariants in non-commutative 3-dimensional Calabi-Yau categories using stability structures and motivic Hall algebras. It derives a wall-crossing formula for these invariants under changes in stability conditions and proves that quantum DT-invariants transform covariantly under quiver mutations via cluster-type transformations, with the quasi-classical limit recovering known cluster transformations.

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).

Motivation & Objective

• To generalize Donaldson-Thomas invariants to non-commutative 3d Calabi-Yau categories using stability conditions.
• To define motivic DT-invariants via the motivic Hall algebra and establish their integrality and wall-crossing behavior.
• To show that quantum DT-invariants transform covariantly under quiver mutations, linking them to cluster transformations.
• To establish a quasi-classical limit of the invariants that recovers classical cluster transformations on the symplectic double torus.
• To prove that the conjugacy class of the automorphism $Φ_{\mathcal{C}} = \operatorname{Ad}_{A_{\mathcal{C}}}^{-1} \circ \tau$ is invariant under mutation, providing a categorical invariant of quivers.

Proposed method

• Uses stability data on graded Lie algebras to define motivic DT-invariants via the motivic Hall algebra construction.
• Introduces a motivic Milnor fiber and motivic functions in an equivariant setting to handle singularities and construct invariants.
• Applies the Behrend microlocal formula to define virtual counts in symmetric obstruction theory settings.
• Constructs the quantum torus associated to a quiver and defines the automorphism $\Phi_Q = \operatorname{Ad}_{\mathbf{E}_Q}^{-1} \circ \tau$ to encode DT-invariants.
• Derives the action of $C_{Q,0}$, a composition of isomorphisms and conjugations, to describe the transformation of quantum torus generators under mutation.
• Takes the quasi-classical limit to recover cluster transformations in terms of coordinates on the symplectic double torus, matching known formulas.

Experimental results

Research questions

• RQ1 How do motivic DT-invariants behave under wall-crossing in non-commutative 3d Calabi-Yau categories?
• RQ2 What is the transformation rule for quantum DT-invariants under quiver mutation?
• RQ3 How does the motivic Hall algebra construction lead to integrality of DT-invariants?
• RQ4 What is the relationship between the automorphism $\Phi_Q$ and cluster transformations in the quasi-classical limit?
• RQ5 Is the conjugacy class of $\Phi_{\mathcal{C}}$ invariant under changes of stability condition in a $t$-structure with finitely generated heart?

Key findings

• The automorphism $\Phi_Q = \operatorname{Ad}_{\mathbf{E}_Q}^{-1} \circ \tau$ encodes the quantum DT-invariant and transforms covariantly under quiver mutation.
• The map $C_{Q,0}$ satisfies $C_{Q,0} \circ \Phi_Q = \Phi_{Q'} \circ C_{Q,0}$, proving invariance of the DT-invariant structure under mutation.
• In the quasi-classical limit, the action of $C_{Q,0}$ on generators reproduces the standard cluster transformation formulas in terms of $y_i$ and $x_i$ coordinates.
• The conjugacy class of $\Phi_Q$ is an invariant of the quiver under mutation, providing a categorical invariant in the non-commutative setting.
• The quasi-classical limit of the automorphism $\Phi_Q$ preserves the variety $N$ defined by $y_i = -\prod_j x_j^{a_{ij}}$, linking the algebraic structure to the geometric cluster variety.
• For a $t$-structure with finitely generated heart, the conjugacy class of $\Phi_{\mathcal{C}}$ is independent of the choice of stability condition, establishing a robust invariant.