Vol. 300, No. 2, 2019

Download this article
Download this article For screen
For printing
Recent Issues
Vol. 309: 1  2
Vol. 308: 1  2
Vol. 307: 1  2
Vol. 306: 1  2
Vol. 305: 1  2
Vol. 304: 1  2
Vol. 303: 1  2
Vol. 302: 1  2
Online Archive
The Journal
Editorial Board
Submission Guidelines
Submission Form
Policies for Authors
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author Index
To Appear
Other MSP Journals
Counting using Hall algebras III: Quivers with potentials

Jiarui Fei

Vol. 300 (2019), No. 2, 347–373

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.

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
Received: 14 July 2017
Revised: 6 July 2018
Accepted: 14 September 2018
Published: 30 July 2019
Jiarui Fei
Department of Mathematics
Shanghai Jiao Tong University
Department of Mathematics
University of California
United States