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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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