Laurent Positivity of Quantized Canonical Bases for Quantum Cluster Varieties from Surfaces

  • So Young Cho
  • Hyuna Kim
  • Hyun Kyu KimEmail author
  • Doeun Oh


In 2006, Fock and Goncharov constructed a nice basis of the ring of regular functions on the moduli space of framed PGL2-local systems on a punctured surface S. The moduli space is birational to a cluster \({\mathcal{X}}\)-variety, whose positive real points recover the enhanced Teichmüller space of S. Their basis is enumerated by integral laminations on S, which are collections of closed curves in S with integer weights. Around ten years later, a quantized version of this basis, still enumerated by integral laminations, was constructed by Allegretti and Kim. For each choice of an ideal triangulation of S, each quantum basis element is a Laurent polynomial in the exponential of quantum shear coordinates for edges of the triangulation, with coefficients being Laurent polynomials in q with integer coefficients. We show that these coefficients are Laurent polynomials in q with positive integer coefficients. Our result was expected in a positivity conjecture for framed protected spin characters in physics and provides a rigorous proof of it, and may also lead to other positivity results, as well as categorification. A key step in our proof is to solve a purely topological and combinatorial ordering problem about an ideal triangulation and a closed curve on S. For this problem we introduce a certain graph on S, which is interesting in its own right.


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This research was supported by the 2017 UREP program of Ewha Womans University, Department of Mathematics. We thank the referee for helpful comments. Hyun Kyu Kim: This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant number 2017R1D1A1B03030230). H.K. thanks Dylan Allegretti and Thang Le for help, discussion, questions, comments, and encouragements.


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Authors and Affiliations

  1. 1.Department of MathematicsEwha Womans UniversitySeoulRepublic of Korea

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