# Periodic Maxwell-Chern-Simons vortices with concentrating property

## TL;DR

This paper analyzes the Maxwell-Chern-Simons model to establish the existence of periodic vortices with concentrating properties, providing new insights into the uniform Chern-Simons limit and addressing open problems in vortex theory.

## Contribution

It derives the uniform Chern-Simons limit of the Maxwell-Chern-Simons model without restrictions and proves the existence of periodic vortices with concentrating density properties.

## Key findings

- Established the uniform (CS) limit of the (MCS) model.
- Proved existence of periodic Maxwell-Chern-Simons vortices with concentration properties.
- Provided a key relation between Higgs and neutral scalar fields.

## Abstract

In order to study electrically and magnetically charged vortices in fractional quantum Hall effect and anyonic superconductivity, the Maxwell-Chern-Simons (MCS) model was introduced by [Lee, Lee, Min (1990)] as a unified system of the classical Abelian-Higgs model (AH) and the Chern-Simons (CS) model. In this article, the first goal is to obtain the uniform (CS) limit result of (MCS) model with respect to the Chern-Simons parameter without any restriction on either a particular class of solutions or the number of vortex points. The most important step for this purpose is to derive the relation between the Higgs field and the neutral scalar field. Our (CS) limit result also provides the critical clue to answer the open problems raised by [Ricciardi,Tarantello (2000)] and [Tarantello (2004)], and we succeed to establish the existence of periodic Maxwell-Chern-Simons vortices satisfying the concentrating property of the density of superconductive electron pairs. Furthermore, we expect that the (CS) limit analysis in this paper would help to study the stability, multiplicity, and bubbling phenomena for solutions of the (MCS) model.