Perverse schobers and cluster categories

Abstract

We introduce a framework for the description of categorified perverse sheaves, called perverse schobers, on surfaces with boundary in terms of constructible sheaves of stable ∞-categories on ribbon graphs. We show that the global sections of some of these sheaves describe the derived categories of a class of relative Calabi–Yau dg-algebras, called relative Ginzburg algebras, associated with n-angulated surfaces. We use local-to-global principles to study the representation theory of these Ginzburg algebras, relating it with the geometry of the underlying surface. Using the derived categories of these relative Ginzburg algebras, we also construct a novel class of additive categorifications of cluster algebras associated with marked surfaces without punctures with coefficients in the boundary arcs. We show that these cluster categories coincide with the topological Fukaya categories of the surfaces with values in the derived category of 1-periodic chain complexes. We further study the relation between perverse schobers, relative Calabi–Yau structures and exact ∞-structures on ∞-categories.