Spectral networks and stability conditions

Spectral networks are certain labeled graphs drawn on a Riemann surface which generalize the saddle connections and closed loops for the flat metric of a quadratic differential. They were introduced by physicists Gaiotto, Moore, and Neitzke as a way of computing BPS spectra. A conjectural mathematical theory was suggested by Kontsevich, in which spectral networks are supports of semistable objects for a stability condition in the sense of Bridgeland. This is joint work in progress with Katzarkov, Kontsevich, Pandit, and Simpson. An example in which everything can be described explicitly is the A2xA5 category, where a recursive procedure constructs all spectral networks which have a "spider web" like shape.