Cohomological Hall algebra and motivic Donaldson-Thomas invariants

这个项目是物理学与数学思想成功互动的一个例子。在数学上，该项目致力于基于新的数学对象（称为上同调霍尔代数）的动机Donaldson-Thomas不变量的方法。它是在PI和马克西姆·康采维奇的联合工作中引入的。提出的工作简化了该领域的一些旧结果（例如，它给出了所谓的墙壁穿越公式的透明解释，该公式显示了Donaldson-Thomas不变量如何依赖于稳定条件）。它还开辟了新的工作方向。其中一些已经引起了数学界的关注（如对称颤振的上同调霍尔代数的结构猜想）。该方法是在具有势的颤振的框架内发展起来的。后者产生由势的临界点产生的三维Calabi-Yau范畴。Donaldson-Thomas不变量是根据势的消失循环定义的。上同调霍尔代数编码了密尔诺纤维在临界位置附近的电位的上同调结构。这与PI和Kontsevich在利用动机整合的动机Donaldson-Thomas不变量方法上的早期工作有关。新的方法更直接、更容易。颤栗和上述范畴之间的相互作用在两个方向上都有应用，例如在研究Donaldson-Thomas不变量对突变的行为过程中。这导致了一个有趣的应用程序到集群转换。此外，与chen - simons理论的类比为三维流形的拓扑不变量提供了一种新的应用。从另一个角度来看，上同调霍尔代数是弦理论家在90年代中期设想的BPS状态代数的数学化身。在一些超对称理论中，动机Donaldson-Thomas不变量对应于改进的BPS状态。从某种意义上说，该项目首次给出了“与模型无关”的BPS状态（以及精炼的BPS状态）概念的数学严格定义。BPS态的过壁公式，在黑洞熵的猜想中起着重要的作用，可以用一种新的非平凡的方式来写，并在数学上证明。也许这就是不同物理学家团体关注这个项目结果的原因之一。对这种新方法日益增长的兴趣已经产生了由物理学和数学界的资深和年轻研究人员撰写的大量论文。最近在美国、欧洲和日本组织的几次会议和讲习班受到与该项目有关的发展的影响。

The project is an example of a successful interaction of ideas originated in physics with those in mathematics. Mathematically, the project is devoted to an approach to motivic Donaldson-Thomas invariants based on the new mathematical object, called Cohomological Hall algebra. It was introduced in the joint work of PI and Maxim Kontsevich. Proposed work simplifies some old results in the area (e.g. it gives a transparent explanation of the so-called wall-crossing formulas which show how Donaldson-Thomas invariants depend on a stability condition). It also opens new directions of work. Some of them have already attracted attention of mathematical community (e.g. the conjecture about the structure of Cohomological Hall algebra for symmetric quiver). The approach is developed in the framework of quivers with potential. The latter give rise to 3-dimensional Calabi-Yau categories generated by critical points of the potential. Donaldson-Thomas invariants are defined in terms of the sheaf of vanishing cycles of the potential. Cohomological Hall algebra encodes the structure of the cohomology of Milnor fiber of the potential near the critical locus. This makes a link with the earlier work of the PI and Kontsevich on the approach to motivic Donaldson-Thomas invariants which utilizes motivic integration. The new approach is more direct and easier. The interplay between quivers and categories mentioned above is used in both directions, e.g. in the course of study of the behavior of Donaldson-Thomas invariants with respect to mutations. This leads to an interesting application to cluster transformations. Furthermore, the analogy with Chern-Simons theory suggests a new application to topological invariants of 3-dimensional manifolds. From another perspective, Cohomological Hall algebra is a mathematical incarnation of the algebra of BPS states envisioned by string theorists in the middle of 90's. Motivic Donaldson-Thomas invariants correspond to refined BPS states in some supersymmetric theories. In a sense the project gives first mathematically rigorous definition of the notion of BPS state (and refined BPS state) which is ``model independent". Wall-crossing formulas for BPS states, which play a role e.g. in the conjectures about the entropy of black holes, can be written in a new non-trivial way and proven mathematically. Maybe this is one of the reasons for the attention of different groups of physicists to the results of the project. Growing interest to the new approach has already generated a flow of papers written by senior and young researchers in both physics and mathematics communities. Several conferences and workshops recently organized in the US, Europe and Japan were influenced by the developments related to the project.