Counting using Hall algebras III: Quivers with potentials

Jiarui Fei

Vol. 300 (2019), No. 2, 347–373

DOI: 10.2140/pjm.2019.300.347

Abstract

For a quiver with potential, we can associate a vanishing cycle to each representation space. If there is a nice torus action on the potential, the vanishing cycles can be expressed in terms of truncated Jacobian algebras. We study how these vanishing cycles change under the mutation of Derksen, Weyman and Zelevinsky. The wall-crossing formula leads to a categorification of quantum cluster algebras under some assumption. This is a special case of A. Efimov’s result, but our approach is more concrete and down-to-earth. We also obtain a counting formula relating the representation Grassmannians under sink-source reflections.

Keywords

quiver representation, quiver with potential, Ringel–Hall algebra, Donaldson–Thomas invariants, vanishing cycle, virtual motive, moduli space, representation grassmannian, quantum cluster algebra, cluster character, quantum dilogarithm, wall-crossing, BB-tilting, mutation, Jacobian algebra, polynomial count

Mathematical Subject Classification 2010

Primary: 16G20

Secondary: 13F60, 14N35, 16G10

Milestones

Received: 14 July 2017

Revised: 6 July 2018

Accepted: 14 September 2018

Published: 30 July 2019