Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
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发表时间：
2008-11
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通讯作者：
M. Kontsevich;Y. Soibelman
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作者：
M. Kontsevich;Y. Soibelman
定义了具有稳定结构的三维Calabi-Yau范畴的新的不变量。直观地说，他们在范畴的K-理论中计算具有固定类的半稳定对象的数量（物理学语言中的“具有给定电荷的BPS状态的数量”）。形式上，我们的动机DT-不变量是量子环面的元素在Grothendieck环的品种在地面场的一个版本。通过准经典极限“仿射线逼近1”，我们得到了与Behrend引入的DT-不变量密切相关的数值DT-不变量。我们研究了运动和数值DT-不变量的一些性质，包括跨壁公式和可积性。我们讨论了与数学作品（在非三角形的情况下）的乔伊斯，Bridgeland和Toledano-Laredo，以及与物理学家的Seiberg-Witten模型（弦结），分类的N=2超对称理论（Cecotti-Vafa）和结构的向量多重模空间的工作。将3D Calabi-Yau范畴的理论与具有特殊生成元的集合（称为簇集合）与具有潜在性的颤动的理论相关联，我们发现了与簇变换和簇种类（经典和量子）的连接。
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).