Framed BPS States, Moduli Dynamics, and Wall-Crossing

TL;DR

The paper develops a nonrelativistic, ${\cal N}=4$ supersymmetric quantum-mechanical framework for framed BPS states in strongly coupled ${\cal N}=2$ Seiberg–Witten theories by treating a heavy core dyon surrounded by light halo probes. It constructs a massive ${\cal N}=4$ mechanics on a 3N-dimensional moduli space, deriving a duality-invariant probe-core interaction and a controllable potential near marginal stability walls, and then quantizes the system to count BPS bound states. The central results are first-principles primitive and semi-primitive wall-crossing formulae, $\Delta\Omega(\gamma_c+\gamma_h)$ and the generating function for $\Omega(\gamma_c+n\gamma_h)$, compatible with Denef–Moore and KS formalisms. The framework links to line operators and Darboux coordinates, suggesting broader applicability to framed BPS objects, two-dimensional/ four-dimensional hybrids, and possible extensions to supergravity.

Abstract

We formulate supersymmetric low energy dynamics for BPS dyons in strongly-coupled N=2 Seiberg-Witten theories, and derive wall-crossing formulae thereof. For BPS states made up of a heavy core state and n probe (halo) dyons around it, we derive a reliable supersymmetric moduli dynamics with 3n bosonic coordinates and 4n fermionic superpartners. Attractive interactions are captured via a set of supersymmetric potential terms, whose detail depends only on the charges and the special Kaehler data of the underlying N=2 theories. The small parameters that control the approximation are not electric couplings but the mass ratio between the core and the probe, as well as the distance to the marginal stability wall where the central charges of the probe and of the core align. Quantizing the dynamics, we construct BPS bound states and derive the primitive and the semi-primitive wall-crossing formulae from the first principle. We speculate on applications to line operators and Darboux coordinates, and also about extension to supergravity setting.