Let $Q$ be a finite quiver. As far as I know there's a great amount of work concerning the so-called quiver varieties one can associate to it. Loosely speaking, these are obtained by taking GIT quotients of semistable representations of the (deformed) preprojective algebra one get from $Q$.


These varieties turn out to have a wide range of applications to representation theory and algebraic geometry.


Is there any work going on pursuing the "stacky" approach? Like taking already the stack of semistable representations or maybe even the full stack of representations of the preprojective algebra?



Moduli stack of quiver representations
There is a differential graded version of the preprojective algebra, called *Ginzburg dg-algebra*, that is a special case of the Calabi-Yau completion introduced by [Keller](https://arxiv.org/abs/0908.3499).


The derived moduli stack of object (as defined by [Toen-Vaquié](https://arxiv.org/abs/math/0503269)) of a Calabi-Yau completion becomes a ($0$-shifted) symplectic derived stack. This is more generally true for the moduli of object of a finite type $2$-Calabi-Yau dg-category, as shown by [Brav-Dyckerhoff](https://arxiv.org/abs/1812.11913), and also [Toen](https://arxiv.org/abs/1401.1044) in a little less general situation.


These ideas have been used in a recent [paper](https://arxiv.org/pdf/2006.01069.pdf) of mine with Bozec and Scherotzke to introduce and study (apparently new) lagrangian subvarieties in the Hilbert scheme of points in the plane.


There are also [two](https://arxiv.org/abs/1612.06352) [papers](https://arxiv.org/abs/1802.05398) of Yeung on related topics, that are very interesting.


