Spectral networks

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.

87 pages, 48 figures; v2: typos, correction to general rule for signs of BPS counts

References in corpus (9)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• The quantum dilogarithm and representations quantum cluster varieties
• R-Twisting and 4d/2d Correspondences
• Wild Wall Crossing and BPS Giants
• BPS Wall Crossing and Topological Strings
• Spectral Networks and Snakes
• 2d Wall-Crossing, R-twisting, and a Supersymmetric Index
• Asymptotics for general connections at infinity

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