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

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 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.