Abstract
We define “BPS graphs” on punctured Riemann surfaces associated with AN −1 theories of class mathcal{S} . BPS graphs provide a bridge between two powerful frameworks for studying the spectrum of BPS states: spectral networks and BPS quivers. They arise from degenerate spectral networks at maximal intersections of walls of marginal stability on the Coulomb branch. While the BPS spectrum is ill-defined at such intersections, a BPS graph captures a useful basis of elementary BPS states. The topology of a BPS graph encodes a BPS quiver, even for higher-rank theories and for theories with certain partial punctures. BPS graphs lead to a geometric realization of the combinatorics of Fock-Goncharov N - triangulations and generalize them in several ways.
Highlights
Studying the spectrum of BPS states has led to many exact non-perturbative results about four-dimensional gauge theories with N = 2 supersymmetry
While the BPS spectrum is ill-defined at such intersections, a BPS graph captures a useful basis of elementary BPS states
BPS states are defined for a theory on the Coulomb branch B of the moduli space of vacua, where the gauge symmetry is spontaneously broken to a Cartan torus U(1)r
Summary
For theories with full punctures (k = 1), certain candidate BPS graphs are dual to the N -triangulations of C used by Fock and Goncharov to study higher Teichmuller theory [12], which were shown to be related to spectral networks in [5]. They are part of a much more general class of BPS graphs, related to each other by sequences of elementary local transformations (flip and cootie moves).
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