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Categorified canonical bases and framed BPS states. (English) Zbl 1505.13032

Summary: We consider a cluster variety associated to a triangulated surface without punctures. The algebra of regular functions on this cluster variety possesses a canonical vector space basis parametrized by certain measured laminations on the surface. To each lamination, we associate a graded vector space, and we prove that the graded dimension of this vector space gives the expansion in cluster coordinates of the corresponding basis element. We discuss the relation to framed BPS states in \({\mathcal{N}}=2\) field theories of class \({\mathcal{S}}\).

MSC:

13F60 Cluster algebras
16G20 Representations of quivers and partially ordered sets
81T60 Supersymmetric field theories in quantum mechanics
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