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Counting using Hall
algebras III: Quivers with potentials
Jiarui Fei |
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Vol. 300 (2019), No. 2, 347–373
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Abstract |
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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. |
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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
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Mathematical Subject Classification 2010
Primary: 16G20
Secondary: 13F60, 14N35, 16G10
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Milestones
Received: 14 July 2017
Revised: 6 July 2018
Accepted: 14 September 2018
Published: 30 July 2019
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Authors |
| Jiarui Fei | |
| Department of Mathematics Shanghai Jiao Tong University Shanghai China |
Department of Mathematics University of California Riverside California United States |
