3-d Calabi–Yau categories for Teichmüller theory

Abstract

For $g,n\ge 0$, we construct a 3-dimensional Calabi–Yau $A_{\mathrm{\infty }}$-category ${\mathcal{C}}_{g,n}$ such that a component of the space of Bridgeland stability conditions, $\mathrm{Stab}({\mathcal{C}}_{g,n})$, 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, we compute the corresponding quantum/refined Donaldson–Thomas (DT) invariants in terms of counts of finite-length geodesics on the flat surface determined by the quadratic differential. As a consequence, we find that these counts satisfy wall-crossing formulas.