Counting using Hall Algebras III. Quivers with Potentials

2013/07/10 by Fei, Jiarui
#13F60 #14N10 #Algebraic Geometry (math.AG) #FOS: Mathematics #Primary 16G20 #Quantum Algebra (math.QA) #Rings and Algebras (math.RA) #Secondary 16G10

paper · doi:10.48550/arxiv.1307.2667

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-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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