Cohomological Hall algebra and motivic Donaldson-Thomas invariants

上同调霍尔代数和动机唐纳森-托马斯不变量

• 批准号: 1101554
• 负责人: Yan Soibelman
• 金额: $ 12.99万
• 依托单位: Kansas State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101554&HistoricalAwards=false
• 关键词: Cohomological; Hall; algebra; motivic; Donaldson

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.

该项目是一个成功的例子，互动的想法起源于物理学与数学。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）的动机唐纳森-托马斯不变量的方法。它是在PI和Maxim Kontsevich的联合工作中引入的。建议的工作简化了一些旧的结果在该地区（例如，它给出了一个透明的解释，所谓的跨壁公式，显示如何唐纳森-托马斯不变量依赖于稳定性条件）。它还开辟了新的工作方向。其中一些猜想已经引起了数学界的关注（如关于对称环的上同调Hall代数结构的猜想）。该方法是在具有潜力的颤抖的框架中开发的。后者产生的三维卡-丘范畴所产生的临界点的潜力。唐纳森-托马斯不变量是根据势的消失圈层定义的。上同调Hall代数描述了临界点附近势的Milnor纤维的上同调结构。这与PI和Kontsevich关于利用动机整合的动机Donaldson-Thomas不变量的方法的早期工作有关。新方法更直接、更容易。上面提到的颤抖和范畴之间的相互作用可以在两个方向上使用，例如在研究唐纳森-托马斯不变量关于突变的行为的过程中。这就引出了一个有趣的集群转换应用。此外，与陈-西蒙斯理论的类比提出了一个新的应用，以拓扑不变量的三维流形。从另一个角度看，上同调Hall代数是90年代中期弦理论家设想的BPS态代数的数学化身。Motivic Donaldson-Thomas不变量对应于某些超对称理论中的精化BPS态。从某种意义上说，该项目首先给出了BPS状态（和改进的BPS状态）概念的数学严格定义，这是“模型独立”的。BPS态的过壁公式，例如在黑洞熵的理论中起作用，可以用一种新的非平凡的方式来写，并在数学上得到证明。也许这就是不同物理学家群体对该项目结果的关注的原因之一。对这种新方法的兴趣越来越大，已经产生了物理和数学界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研讨会受到了与该项目有关的发展的影响。