Nielsen-Olesen Vortices in Noncommutative Abelian Higgs Model

TL;DR

The paper develops a nonperturbative study of vortices in the noncommutative Abelian Higgs model, proving that Nielsen-Olesen vortices possess quantized magnetic flux $I=n$ independent of the NC parameter $\theta$. By formulating Bogomolnyi equations in the operator framework, it constructs a robust large-$\theta$ vortex with $\phi_{\infty}$ and $B_{\infty}$ yielding $B_{\infty}=n|0\rangle\langle 0|$, and demonstrates that the topological charge remains $I=n$ even after including $1/\theta$ corrections. The Seiberg-Witten map is analyzed, showing the gauge-field map remains unchanged to $O(\theta)$ while the Higgs map is linear in $\phi$, connecting NC and commutative descriptions at large distances. The work further shows that small- and large-$\theta$ expansions preserve integer charge, with higher-order terms contributing only total derivatives, and reports an exact vortex in a diagonal left-right NC $U(1)$ theory, highlighting broader topological structures and their brane-theoretic implications via RR charge and index theory.

Abstract

We construct Nielsen-Olesen vortex solution in the noncommutative abelian Higgs model. We derive the quantized topological flux of the vortex solution. We find that the flux is integral by explicit computation in the large $θ$ limit as well as in the small $θ$ limit. In the context of a tachyon vortex on the brane-antibrane system we demonstrate that it is this topological charge that gives rise to the RR charge of the resulting BPS D-brane. We also consider the left-right-symmetric gauge theory which does not have a commutative limit and construct an exact vortex solution in it.