Relative critical loci and quiver moduli

TL;DR

This paper establishes a deep connection between the cotangent complex of quiver representation stacks and derived moduli of Ginzburg dg-algebra modules, extending to dg-categories and deformations, with applications to Lagrangian subvarieties.

Contribution

It generalizes the identification of cotangent stacks to derived moduli of Ginzburg algebra modules for quivers, extending to dg-categories and deformations, and uncovers new Lagrangian subvarieties.

Findings

Identification of cotangent to derived stack with derived moduli of Ginzburg algebra modules

Extension of results to finite type dg-categories and deformations

Discovery of new Lagrangian subvarieties in the Hilbert scheme of points

Abstract

In this paper we identify the cotangent to the derived stack of representations of a quiver $Q$ with the derived moduli stack of modules over the Ginzburg dg-algebra associated with $Q$. More generally, we extend this result to finite type dg-categories, to a relative setting as well, and to deformations of these. It allows us to recover and generalize some results of Yeung, and leads us to the discovery of seemingly new lagrangian subvarieties in the Hilbert scheme of points in the plane.