ncts

The problem of counting saddle connections and closed loops on Riemann surfaces with quadratic differential (equivalently: half-translation surfaces) can be, somewhat surprisingly, reformulated in terms of counting semistable objects in a 3-d Calabi-Yau category with stability condition. Here "counting" happens within the powerful framework of motivic Donaldson-Thomas theory as developed by Kontsevich-Soibelman, Joyce, and others. For meromorphic quadratic differentials with simple zeros, this reformulation is due to the work of Bridgeland-Smith. The case of quadratic differentials without higher order poles - in particular holomorphic ones - requires entirely different methods, based on deformation of A-infinity categories and transfer of stability conditions. As an application, counts of saddle connections and closed loops are related by the wall-crossing formula as one moves around in the moduli space. Based on 2104.06018 and 2303.18249.