Stability structures, motivic Donaldson-Thomas invariants and cluster transformations

TL;DR

This paper introduces new motivic and numerical Donaldson-Thomas invariants for 3d Calabi-Yau categories with stability structures, exploring their properties, wall-crossing behavior, and connections to cluster transformations and physical theories.

Contribution

It defines novel motivic DT-invariants, relates them to existing invariants, and links the theory to cluster transformations and physical models, advancing the understanding of stability and invariants in Calabi-Yau categories.

Findings

Motivic DT-invariants are elements of quantum tori over Grothendieck rings.

Numerical DT-invariants relate closely to Behrend's invariants.

Wall-crossing formulas and integrality properties are established.

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).