3-d Calabi--Yau categories for Teichm\"uller theory

For g, n ≥ 0 a 3-dimensional Calabi-Yau A∞-category Cg,n is constructed such that a component of the space of Bridgeland stability conditions, Stab(Cg,n), is a moduli space of quadratic differentials on a genus g surface with simple zeros and n simple poles. For a generic point in the moduli space the corresponding quantum/refined Donaldson–Thomas invariants are computed in terms of counts of finite-length geodesics on the flat surface determined by the quadratic differential. As a consequence, these counts satisfy wall-crossing formulas.

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