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.