Spectral networks compute certain enumerative invariants associated with Hitchin systems, by focusing on the interplay of certain geometric and combinatorial data within them. In physics, spectral networks count BPS states of class S theories through 2d-4d wall crossing. I will describe a 3d-5d uplift of this based on exponential networks, that computes generalized Donaldson-Thomas invariants of toric Calabi Yau threefolds.

The BPS spectra of Class S theories are among the best understood, thanks in part to a construction known as Spectral Networks. We will review this framework and recent developments of it, and present results obtained through their applications.