Wall-Crossing Invariants from Spectral Networks

A new construction of BPS monodromies for 4d ${\mathcal N}=2$ theories of class S is introduced. A novel feature of this construction is its manifest invariance under Kontsevich-Soibelman wall crossing, in the sense that no information on the 4d BPS spectrum is employed. The BPS monodromy is encoded by topological data of a finite graph, embedded into the UV curve $C$ of the theory. The graph arises from a degenerate limit of spectral networks, constructed at maximal intersections of walls of marginal stability in the Coulomb branch of the gauge theory. The topology of the graph, together with a notion of framing, encode equations that determine the monodromy. We develop an algorithmic technique for solving the equations, and compute the monodromy in several examples. The graph manifestly encodes the symmetries of the monodromy, providing some support for conjectural relations to specializations of the superconformal index. For $A_1$-type theories, the graphs encoding the monodromy are "dessins d'enfants" on $C$, the corresponding Strebel differentials coincide with the quadratic differentials that characterize the Seiberg-Witten curve.