Spectral networks

TL;DR

Spectral networks are geometric structures on Riemann surfaces that help compute BPS degeneracies in 4D N=2 theories and establish a new link between flat connections, providing natural coordinate systems on moduli spaces.

Contribution

Introduction of spectral networks as a new geometric tool for analyzing BPS states and flat connections in 4D N=2 theories of class S.

Findings

Spectral networks determine BPS degeneracies of solitons and particles.

They establish a new correspondence between flat GL(K,C) connections and abelian connections on branched covers.

Propose that the natural coordinate systems are cluster coordinates on moduli spaces.

Abstract

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 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 determine the degeneracies for particles living in the 4d bulk. Spectral networks also lead to a new map between flat GL(K,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 GL(K,C) connections on C, which we conjecture are cluster coordinate systems.