Algebra of the infrared, Fukaya-Seidel categories and wall-crossing formulas

红外代数、Fukaya-Seidel 范畴和穿墙公式

• 批准号: 1507316
• 负责人: Yan Soibelman
• 金额: $ 15万
• 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: 2015
• 资助国家: 美国
• 起止时间: 2015-08-01 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1507316&HistoricalAwards=false
• 关键词: Algebra; infrared; Fukaya; Seidel; categories

The last 40 years have been seeing an extremely fruitful interaction between mathematics and quantum physics. One of the main parts in this progress has been played by the String Theory, with its main idea of interpreting elementary particles as small circles (called strings) rather than point-like objects. On the mathematical side it has led to appearance not only of new problems in classical areas, but also a fast development of completely new areas. Even the terminology reflects that: Mathematicians speak about "quantum groups", "mirror symmetry", "instantons", "branes" - all those terms are borrowed from physics. In the opposite direction, many mathematical concepts (some of them were considered esoteric at the time) now become powerful tools for a working physicist. String theorists speak about "derived categories of branes", "motivic invariants" - the terminology borrowed from mathematics. There is no doubt that the above-mentioned interaction will affect not only some fields in mathematics and physics but will influence the way of thinking about the real world. Current project fits nicely in the above paradigm. It is highly motivated by the recent important developments in the 2-dimensional supersymmetric massive theories, but its scope is purely mathematical, and its applications are important for quantum physics. The project connects new and active area of research in theoretical physics with deep mathematical questions in homological algebra, toric geometry, Floer theory and Topological Field Theory. In more technical terms, the project is devoted to the new approach to the concept of Fukaya-Seidel category, which is the mathematical counterpart of the so-called Landau-Ginzburg model in physics. Main motivation for the ideas developed in the project was the work of physicists D. Gaiotto, G. Moore, E. Witten on 2-dimensional supersymmetric gauge theories. By replacing the language of plane webs proposed by physicists by the dual language of polygons, the PI connects their formalism with the geometry of secondary polytopes of the polygon of critical values of the Landau-Ginzburg potential. The categorical structure which appears in the work of physicists is then explained in the language of factorization sheaves on the secondary polytope. It opens the door to generalizations to the case of higher-dimensional Topological Field Theories. A conjectural new description of the Fukaya-Seidel category will be investigated by the PI. It generalizes the earlier ideas of Haydys as well as the more recent proposal of Gaitto, Moore and Witten. It expresses the Fukaya-Seidel category as a deformation of the category which has combinatorial nature. The deformation is given by some Maurer-Cartan element, which is defined in terms of solutions to the Witten (or zeta-instanton) equation. A conjectural relation of the degenerations of the moduli space of solutions to Witten equation with faces of the secondary polytope will also be studied. The relation of webs with Gromov-Hausdorff limit of the solutions is stressed. Several applications of the proposed formalism will be studied, such as:a) wall-crossing formulas of Cecotti-Vafa and Kontsevich-Soibelman;b) complexified and holomorphic Chern-Simons theory;c) Morse theory of functions which are Morse but not Morse-Smale. According to earlier work of Kapranov and Saito that story is intrinsically connected to Steinberg relations in the algebraic K-theory and Stasheff polytopes;d) relation to the theory of spectral networks of Gaiotto, Moore and Neitzke.

在过去的40年里，数学和量子物理之间的互动取得了极其丰硕的成果。这一进展的主要部分之一是弦理论，它的主要思想是将基本粒子解释为小圆圈(称为弦)，而不是点状物体。在数学方面，它不仅导致了经典领域中新问题的出现，而且导致了全新领域的快速发展。甚至连术语也反映了这一点：数学家们谈论的是“量子群”、“镜像对称性”、“瞬子”、“膜”--所有这些术语都是从物理学借来的。在相反的方向上，许多数学概念(其中一些在当时被认为是深奥的)现在成为工作的物理学家的强大工具。弦理论家谈到了“膜的派生范畴”、“动机不变量”--这是从数学中借用来的术语。毫无疑问，上述相互作用不仅会影响数学和物理的一些领域，而且会影响人们对现实世界的思考方式。当前的项目很好地符合上述范例。它是由二维超对称质量理论最近的重要发展高度推动的，但它的范围是纯数学的，它的应用对量子物理很重要。该项目将理论物理中新的和活跃的研究领域与同调代数、环面几何、Floer理论和拓扑场论中的深层次数学问题联系起来。在更专业的术语中，该项目致力于对Fukaya-Seidel范畴概念的新方法，这是物理学中所谓的Landau-Ginzburg模型的数学对应。该项目中形成这些想法的主要动机是物理学家D.Gaiotto，G.Moore，E.Witten在二维超对称规范理论上的工作。通过用多边形的对偶语言代替物理学家提出的平面网的语言，PI将它们的形式化与Landau-Ginzburg势的临界值的多边形次多面体的几何联系起来。在物理学家的工作中出现的范畴结构然后用次要多面体上的因式分解层的语言来解释。它为高维拓扑场论的推广打开了大门。PI将对Fukaya-Seidel范畴进行猜想的新描述。它概括了海代早期的观点以及盖托、摩尔和维腾最近提出的观点。它将Fukaya-Seidel范畴表示为具有组合性的范畴的变形。形变由一些Maurer-Cartan元给出，该元是根据Witten(或Zeta-瞬子)方程的解定义的。还将研究Witten方程的解的模空间的退化与二次多面体的面的猜想关系。强调了Webs与解的Gromov-Hausdorff极限的关系。我们将研究所提出的形式主义的几个应用，例如：a)Cecotti-Vafa和Kontsevich-Soibelman的跨壁公式；b)复化和全纯的Chern-Simons理论；c)Morse但不是Morse-Smer的函数的Morse理论。根据Kapranov和Saito的早期工作，这个故事与代数K理论中的Steinberg关系和Stasheff多面体有内在的联系；d)与Gaiotto，Moore和Neitzke的谱网络理论有关系。