Categorified canonical bases and framed BPS states

Abstract

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 $\mathcal{S}$.

51 pages. Version 2: Improved presentation and fixed treatment of loops with weight >1. Version 3: Minor changes

References in corpus (8)

• A duality map for quantum cluster varieties from surfaces
• Gentle algebras arising from surface triangulations
• Discrete Integrable Systems, Supersymmetric Quantum Mechanics, and Framed BPS States - I
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• Quivers, Line Defects and Framed BPS Invariants
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Cited by in corpus (7)

• Strong positivity for quantum theta bases of quantum cluster algebras
• Schemes of modules over gentle algebras and laminations of surfaces
• Quantization of canonical bases and the quantum symplectic double
• An expansion formula for type A and Kronecker quantum cluster algebras
• ${\mathrm{SL}}_3$-laminations as bases for ${\mathrm{PGL}}_3$ cluster varieties for surfaces
• Cluster algebras and their bases
• Quantum traces for ${\mathrm{SL}}_n(\mathbb{C})$: The case $n=3$