题目: An Introduction to Algebra of the Infrared

报告人：李龙飞（Longfei Li）(Kansas State University)

时间：8月31日（星期一）、9月2日（星期三）、9月4日（星期五），10:00-11:30

地点：东区五教5201教室

摘要:

The algebra of the infrared (AoI) was introduced by Gaiotto–Moore–Witten in the study of massive two-dimensional field theories and their infrared behavior, and was later developed mathematically by Kapranov–Kontsevich–Soibelman. It lies at the intersection of combinatorics, homotopy algebra, deformation theory, and symplectic geometry. One of its striking features is that geometric data coming from configurations of points and their polygonal decompositions naturally give rise to (L_\infty)- and (A_\infty)-structures, which in turn are closely related to Fukaya–Seidel categories.

The first talk will introduce the combinatorial and algebraic background. We will begin with configurations of points in the plane, triangulations, polygonal subdivisions, and the secondary polytope. These objects provide a convenient way to organize degenerations and factorizations of polygons, and their face structure encodes many of the higher algebraic identities that appear later. We will then review the basic ideas of (A_\infty)- and (L_\infty)-algebras and briefly discuss their interpretation from the viewpoint of deformation theory. The goal is to explain why combinatorial decompositions naturally lead to higher operations and homotopy-coherent algebraic structures.

The second talk will focus on the construction of the algebra of the infrared developed by Kapranov–Kontsevich–Soibelman. Starting from a finite configuration of points in the complex plane, we will describe the (L_\infty)-algebra associated with convex polygons and their subdivisions, as well as the directed (A_\infty)-algebra determined by a choice of half plane. We will discuss how the factorization of polygonal configurations produces the relevant (L_\infty)- and (A_\infty)-relations, and how deformation-theoretic ideas enter the construction.

The third talk will discuss the symplectic-geometric meaning of the construction. We will recall the basic idea of Fukaya–Seidel categories associated with Lefschetz fibrations and explain how the algebra of the infrared is expected to provide an algebraic model for structures arising from vanishing paths, thimbles, and their interactions. We will also briefly describe the physical motivation for the theory, in particular the appearance of solitons and infrared data in two-dimensional massive theories.

Finally, I will briefly mention my recent work extending the construction from ordinary complex-valued potentials to curve-valued potentials. In this setting, the base of a Lefschetz-type fibration is a complex curve rather than the complex plane. If time permits, I will briefly mention perverse schobers and their role in organizing categorical monodromy and wall-crossing phenomena.