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 ${\rm PGL}_2$-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 Teichm\"uller 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.