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

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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.