Calabi-Yau categories from Gorenstein algebras

Abstract

Let $(Q,W)$ be a quiver with potential having finite-dimensional Jacobian algebra. I will construct from $(Q,W)$ a new algebra, which will be Gorenstein with 2-Calabi-Yau singularity category whenever it is Noetherian, this singularity category being conjecturally equivalent to Amiot's cluster category of $(Q,W)$. Both the Noetherianity and the equivalence are proved to hold when $Q$ is acyclic. The construction appears to be related to the Ginzburg dg-algebra of $(Q,W)$, and I will discuss both precise and conjectural connections to this dg-algebra. Time permitting, I will also discuss results in other Calabi-Yau dimensions.