Spectral Networks

We introduce new geometric objects called spectral networks. Spectral networks are networks of trajectories on Riemann surfaces obeying certain local rules. Spectral networks arise naturally in four-dimensional $${\mathcal{N} = 2}$$ theories coupled to surface defects, particularly the theories of class S. In these theories, spectral networks provide a useful tool for the computation of BPS degeneracies; the network directly determines the degeneracies of solitons living on the surface defect, which in turn determines the degeneracies for particles living in the 4d bulk. Spectral networks also lead to a new map between flat $${{\rm GL}(K, \mathbb{C})}$$ connections on a two-dimensional surface C and flat abelian connections on an appropriate branched cover $${\Sigma}$$ of C. This construction produces natural coordinate systems on moduli spaces of flat $${{\rm GL}(K, \mathbb{C})}$$ connections on C, which we conjecture are cluster coordinate systems.