Quantum Quivers and Hall/Hole Halos

TL;DR

This work develops a unified framework for counting BPS bound-state degeneracies in Calabi–Yau compactifications by modeling the system as ${\mathcal N}=4$ quiver quantum mechanics, arising from dimensional reduction of ${\mathcal N}=1$ theories. A crucial non-renormalization theorem fixes the first-order part of the multi-particle Lagrangian, enabling a precise bridge between the Coulomb (multicentered particle) and Higgs (fused D-brane) pictures as the string coupling ${\rm g}_s$ is varied. Ground-state counts are then computed from the cohomology of quiver moduli spaces, with explicit results for Hall halos and dyons in ${\mathcal N}=2$ theories, and general counting formulas (e.g. Reineke’s) for loopless quivers. The framework illuminates deep connections between physical BPS spectra and geometric representation theory, providing exact degeneracy predictions and a path to broader generalizations, including potential extensions to quivers with loops and superpotentials.

Abstract

Two pictures of BPS bound states in Calabi-Yau compactifications of type II string theory exist, one as a set of particles at equilibrium separations from each other, the other as a fusion of D-branes at a single point of space. We show how quiver quantum mechanics smoothly interpolates between the two, and use this, together with recent mathematical results on the cohomology of quiver varieties, to solve some nontrivial ground state counting problems in multi-particle quantum mechanics, including one arising in the setup of the spherical quantum Hall effect, and to count ground state degeneracies of certain dyons in supersymmetric Yang-Mills theories. A crucial ingredient is a non-renormalization theorem in N=4 quantum mechanics for the first order part of the Lagrangian in an expansion in powers of velocity.