Fukaya categories of surfaces with DG coefficients

Abstract

Given a compact surface $S$ admitting a Liouville form, a finite set $Q \subset S$, a line field on $S$, and a local system of dg-categories on $S \setminus Q$ together with a section of the monodromy functor around each interior $q \in Q$, we construct a Fukaya category. We give examples where such Fukaya categories are quasi-equivalent to derived categories of projective completions of line bundles and of iterated blow-ups of the affine plane; and to wrapped Fukaya categories of cotangent bundles of spheres. We define two versions of the Fukaya category, a topological, which is constructed combinatorially from a full arc system, and a geometric, which includes all Lagrangian branes supported on immersed curves. We prove that the topological category embeds into the geometric category in an essentially unique way. To define Lagrangian branes we develop a generalisation of the Chekanov--Eliashberg algebra incorporating dg-coefficients and wrapping along the boundary.