Constraints on the BPS Spectrum of N = 2, D = 4 Theories with A-D-E Flavor Symmetry
Oliver DeWolfe, Tamas Hauer, Amer Iqbal, Barton Zwiebach
TL;DR
The paper introduces a self-intersection based selection rule for BPS junctions in four-dimensional ${\cal N}=2$ theories with ADE flavor symmetry, deriving $ (\mathbf{J}, \mathbf{J}) - \mathrm{GCD}(p,q) \ge -2$ from holomorphic curve considerations. This constraint, together with the junction-to-weight-lattice mapping, reproduces the familiar Seiberg-Witten spectra for $SU(2)$ with $N_f=0$–$4$ and then predicts arbitrarily large flavor representations for $D_{n\ge 5}$ and $E_n$ theories, shown to be consistent with decoupling limits. The authors analyze brane decoupling and duality constraints, revealing that higher ADE theories require new, large representations and exhibit intricate symmetry structures under $\mathrm{SL}(2,\mathbb{Z})$ reductions to subgroups $M$ that act with outer automorphisms on the flavor algebras. Overall, the work provides a concrete, lattice-based framework for constraining and understanding the BPS spectrum across a broad class of ${\cal N}=2$ theories, including non-Lagrangian fixed points, and outlines several open questions related to multiplicities and quantization of zero modes.
Abstract
BPS states of N=2, D=4 Super Yang-Mills theories with ADE flavor symmetry arise as junctions joining a D3-brane to a set of 7-branes defining the enhanced flavor algebra. We show that the familiar BPS spectrum of SU(2) theories with N_f <= 4 is simply given by the set of junctions whose self-intersection is bounded below as required by supersymmetry. This constraint, together with the relations between junction and weight lattices, is used to establish the appearance of arbitrarily large flavor representations for the case of D_{n>=5} and E_n symmetries. Such representations are required by consistency with decoupling down to smaller flavor symmetries.