Effective Theory on Non-Abelian Vortices in Six Dimensions

TL;DR

This work analyzes non-Abelian BPS vortices in a six-dimensional ${\cal N}=1$ $U(N)$ gauge theory with $N$ fundamental hypermultiplets, identifying NG and quasi-NG moduli on vortex world-volumes and constructing their four-dimensional effective theories. It frames the moduli space via symmetry and Kähler-quotient structures, showing ${\cal M}_{1,N} \simeq {\bf C}\times {\bf CP}^{N-1}$ for a single vortex and ${\cal M}_{k,N}$ as a fibered quotient ${\cal M}_{k,N} = {\bf C}^k \times {\bf C}^{\tfrac{1}{2}k(k-1)} \times G_{N,k}$ (locally), with coincident vortices adding a ${\bf C}^{k^2-1}$ fiber. A central contribution is a systematic method to construct the most general effective Lagrangian on the vortex world-volume that respects symmetry and SUSY, including deformations that account for quasi-NG modes, potentially reconciling discrepancies between Hanany-Tong and Manton metrics. The paper also discusses the geometry of the moduli space, its deformation, and implications for brane-world scenarios, such as possible localized gauge bosons on vortices and extensions to other gauge groups. Overall, it provides a symmetry-driven framework for understanding low-energy vortex dynamics and their role in higher-dimensional model building.

Abstract

Non-Abelian vortices in six spacetime dimensions are obtained for a supersymmetric U(N) gauge theory with N hypermultiplets in the fundamental representation. Massless (moduli) fields are identified and classified into Nambu-Goldstone and quasi-Nambu-Goldstone fields. Effective gauge theories for the moduli fields are constructed on the four-dimensional world volume of vortices. A systematic method to obtain the most general form of the effective Lagrangian consistent with symmetry is proposed. The moduli space for the multi-vortices is found to be a vector bundle over the complex Grassmann manifold.