The Dynamics of Chern-Simons Vortices

TL;DR

This paper investigates the dynamics of vortices in three-dimensional Chern-Simons theories, revealing how magnetic fields relate to geometric properties of the moduli space and deriving these results through fermion zero mode analysis.

Contribution

It provides a geometric characterization of vortex dynamics in Chern-Simons theories, connecting magnetic fields to Ricci forms and Chern characters, with derivations via fermion zero modes.

Findings

Magnetic field for Abelian vortices is the Ricci form on moduli space.

Magnetic field for non-Abelian vortices is the first Chern character of an index bundle.

Derived results by integrating out fermions and analyzing zero modes.

Abstract

We study vortex dynamics in three-dimensional theories with Chern-Simons interactions. The dynamics is governed by motion on the moduli space M in the presence of a magnetic field. For Abelian vortices, the magnetic field is shown to be the Ricci form over M; for non-Abelian vortices, it is the first Chern character of a suitable index bundle. We derive these results by integrating out massive fermions and following the fate of their zero modes.