Ginzburg algebras of triangulated surfaces and perverse schobers

Abstract

Ginzburg algebras associated to triangulated surfaces provide a means to categorify the cluster algebras of these surfaces. As shown by Ivan Smith, the finite derived category of such a Ginzburg algebra can be embedded into the Fukaya category of the total space of a Lefschetz fibration over the surface. Inspired by this perspective we provide a description of the full derived category in terms of a perverse schober. The main novelty is a gluing formalism describing the Ginzburg algebra as a colimit of certain local Ginzburg algebras associated to discs. As a first application we give a new proof of the derived invariance of these Ginzburg algebras under flips of an edge of the triangulation. Finally, we note that the perverse schober as well as the resulting gluing construction can also be defined over the sphere spectrum.

Geometric models for the derived categories of Ginzburg algebras of n-angulated surfaces via local-to-global principles

“…This paper continues the study of Ginzburg algebras of triangulated surfaces, or more generally n-angulated surfaces, using their description in terms of perverse schobers started in [Chr21].…”

“…In Section 3, we introduce some preliminaries from higher category theory, most importantly the Grothendieck construction. In Section 4, we recall the notion of a parametrized perverse schober from [Chr21] and discuss the perverse schobers describing relative Ginzburg algebras. In Sections 5 and 6, we construct the global sections associated to curves and describe the morphisms objects in terms of intersections.…”

“…We are further concerned with a generalization of this class of Ginzburg algebras which are called relative Ginzburg algebras, c.f. [Chr21]. The difference to the non-relative versions is that these dg-algebras incorporate additional information about the boundary of the surface, and thus exhibit a gluing behavior.…”

Spectral networks and stability conditions for Fukaya categories with coefficients

“…schober [29]. The full theory is still under development [4,10,25,30,8,11]. For much of this section we restrict to the following special class of schobers.…”

“…The categories E p,q are the fibers of a schober E on S with singular fibers at the branch points of S → S. The stability conditions on the fibers combine to give a holomorphic family of stability conditions on E. (We have not defined what this means in the case of singular fibers, so strictly speaking the statement only makes sense on the complement of the branch locus.) The global categories F(S; E) K should recover the 3CY quiver categories considered in [21,49], at least for certain choices of S. In the case n = 1, the work [8] recovers the 3CY quiver category from the schober using a definition of the Fukaya category with coefficients based on ribbon graphs and homotopy limits of DG-categories.…”