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

在过去的40年里，数学和量子物理的相互作用卓有成效。弦理论在这一进程中扮演了一个主要角色，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：数学家谈论“量子群”、“镜像对称”、“瞬子”、“膜”——所有这些术语都是从物理学中借用来的。相反，许多数学概念（其中一些在当时被认为是深奥的）现在成为工作物理学家的强大工具。弦理论家谈论“膜的派生范畴”，“动机不变量”——从数学中借用的术语。毫无疑问，上述相互作用不仅会影响数学和物理的某些领域，而且会影响对现实世界的思考方式。当前的项目很好地符合上述范例。它受到最近二维超对称质量理论的重要发展的高度激励，但它的范围是纯数学的，它的应用对量子物理学很重要。该项目将理论物理的新兴和活跃的研究领域与同调代数、环几何、花理论和拓扑场论等深层次的数学问题联系起来。用更专业的术语来说，该项目致力于对Fukaya-Seidel类别概念的新方法，这是物理学中所谓的朗道-金兹堡模型的数学对应物。该项目的主要动机是物理学家D. Gaiotto， G. Moore， E. Witten对二维超对称规范理论的研究。通过用多边形的双重语言取代物理学家提出的平面网语言，PI将它们的形式化与朗道-金兹堡势的临界值多边形的二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，然后用二次多面体上的分解层的语言加以解释。它为高维拓扑场论的推广打开了大门。PI将对福冈-赛德尔范畴的推测性新描述进行研究。它概括了Haydys早期的思想以及Gaitto， Moore和Witten最近的建议。它将福冈-赛德尔范畴表达为具有组合性质的范畴的变形。变形是由一些毛雷尔-卡坦元素给出的，它是根据Witten（或ζ -instanton）方程的解来定义的。本文还研究了具有二次多面体面的Witten方程解的模空间退化的推测关系。着重讨论了腹板与解的Gromov-Hausdorff极限的关系。本文将研究所提出的形式主义的几种应用，例如：a) Cecotti-Vafa和kontsevic - soibelman的过墙公式；b)复变全纯chen - simons理论；c)莫尔斯函数的莫尔斯理论，它是莫尔斯函数但不是莫尔斯小函数。根据Kapranov和Saito的早期工作，这个故事与代数k理论和Stasheff多面体中的Steinberg关系具有内在联系；d)与Gaiotto、Moore和Neitzke的谱网络理论的关系。

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.