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

TL;DR

This paper constructs 3-dimensional Calabi-Yau categories linked to Teichmüller theory, connecting stability conditions with quadratic differentials and computing Donaldson-Thomas invariants via geodesic counts, revealing wall-crossing phenomena.

Contribution

It introduces a new class of Calabi-Yau categories associated with Teichmüller theory and relates their stability spaces to moduli of quadratic differentials, enabling explicit invariant computations.

Findings

Stability conditions form a moduli space of quadratic differentials.

Donaldson-Thomas invariants are computed via geodesic counts.

Wall-crossing formulas are satisfied by these counts.

Abstract

For $g,n\geq 0$ 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.