Noncommutative Vortex Solitons
Dongsu Bak, Kimyeong Lee, Jeong-Hyuck Park
TL;DR
This work studies noncommutative vortex solitons in the Abelian-Higgs theory, focusing on exact multi-vortex solutions, BPS structure, fluctuation spectra, and their low-energy dynamics. It shows that self-dual BPS vortices exist only for $θ v^2 ≤ 1$ and that non-BPS configurations become unstable below the threshold due to tachyonic modes, while for $θ v^2 > 1$ the non-BPS branch remains stable. The paper identifies moduli as vortex positions via a covariant position operator and demonstrates that low-energy dynamics are governed by a flat moduli-space metric, with the full system describable by a matrix model of $m$ D0-branes in two dimensions. It also constructs exact time-dependent moving vortices and discusses the matrix nature of the moduli coordinates, highlighting the role of noncommutative geometry in shaping vortex interactions and dynamics.
Abstract
We consider the noncommutative Abelian-Higgs theory and investigate general static vortex configurations including recently found exact multi-vortex solutions. In particular, we prove that the self-dual BPS solutions cease to exist once the noncommutativity scale exceeds a critical value. We then study the fluctuation spectra about the static configuration and show that the exact non BPS solutions are unstable below the critical value. We have identified the tachyonic degrees as well as massless moduli degrees. We then discuss the physical meaning of the moduli degrees and construct exact time-dependent vortex configurations where each vortex moves independently. We finally give the moduli description of the vortices and show that the matrix nature of moduli coordinates naturally emerges.