K-theoretic Hall algebras for quivers with potential

Abstract

Given a quiver with potential $(Q,W)$, Kontsevich-Soibelman constructed a Hall algebra on the cohomology of the stack of representations of $(Q,W)$. As shown by Davison-Meinhardt, this algebra comes with a filtration whose associated graded algebra is supercommutative. A special case of this construction is related to work of Nakajima, Varagnolo, Maulik-Okounkov etc. about geometric constructions of Yangians and their representations; indeed, given a quiver $Q$, there exists an associated pair $(\widetilde{Q},\widetilde{W})$ for which the CoHA is conjecturally the positive half of the Yangian $Y_{\text{MO}}(\mathfrak{g}_Q)$. The goal of this article is to extend these ideas to K-theory. More precisely, we construct a K-theoretic Hall algebra using category of singularities, define a filtration whose associated graded algebra is a deformation of a symmetric algebra, and compare the $\text{KHA}$ and the $\text{CoHA}$ using the Chern character. As before, we expect our construction for the special class of quivers $(\widetilde{Q},\widetilde{W})$ to recover the positive part of quantum affine algebra $U_q(\hat{\mathfrak{g}_Q})$ defined by Okounkov-Smirnov, but for general pairs $(Q,W)$ we expect new phenomena.