3-d Calabi--Yau categories for Teichmüller theory

Abstract

For $g,n\ge 0$ a 3-dimensional Calabi-Yau $A_{\infty }$-category ${\mathcal{C}}_{g,n}$ is constructed 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 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.

arXiv commentv3: more details added in some proofs, typos corrected, to appear in Duke Math J