Selecta Mathematica

, 25:69 | Cite as

Categorified canonical bases and framed BPS states

  • Dylan G. L. AllegrettiEmail author


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 \({\mathcal {S}}\).

Mathematics Subject Classification

13F60 16G20 81T60 



In writing this paper, I have benefitted from conversations with many people, including Tom Bridgeland, Michele Cirafici, Michele Del Zotto, Joseph Karmazyn, Daniel Labardini-Fragoso, Sven Meinhardt, Andrew Neitzke, Harold Williams, and Yu Zhou.


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Mathematical Sciences Research InstituteBerkeleyUSA

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