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, i.e. categorified perverse sheaves, on surfaces with boundary. We treat examples related to 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.

arXiv comment62 pages, v3) final version