Relative Calabi–Yau structures and perverse schobers on surfaces

Abstract

We give a treatment of relative Calabi–Yau structures on functors between $R$-linear stable $\infty$-categories, with $R$ any ${\mathbb{E}}_{\infty }$-ring spectrum, generalizing previous treatments in the setting of dg categories. Using their gluing properties, we further construct relative Calabi–Yau structures on the global sections of perverse schobers, that is, categorified perverse sheaves, on surfaces with boundary. We treat examples coming from Fukaya categories and representation theory. In a related direction, we define the monodromy of a perverse schober parametrized by a ribbon graph on a framed surface and show that it forms a local system of stable $\infty$-categories.