Exotic vortices and twisted holomorphic maps

TL;DR

This paper studies exotic vortex equations on compact Riemann surfaces, generalizing known vortex models, and explores their solutions, moduli space, and quantum dynamics, revealing new geometric and physical insights.

Contribution

It constructs vortex solutions from gauged holomorphic maps into complex space forms and conjectures all such solutions arise this way, advancing the understanding of exotic vortex moduli spaces.

Findings

Vortex solutions can be constructed from gauged holomorphic maps.

Identified selection rules for nonAbelian exotic vortices on the sphere.

Computed the Witten index for low-temperature vortex quantum mechanics.

Abstract

We consider the exotic vortex equations on compact Riemann surfaces. These generalise the well-known Jackiw-Pi and Ambj{\o}rn-Olesen vortex equations and arise as equations for Bogomolny-Prasad-Sommerfield-like configurations in nonrelativistic Chern-Simons-matter theories. We show that (exotic) vortex solutions in $U(N)$ gauge theories with a single fundamental flavour on compact Riemann surfaces can be constructed from gauged holomorphic maps into complex space forms of dimension $N$. We conjecture, with some evidence, that all such solutions arise this way. This leads us to insights regarding the moduli theory of exotic vortices, including the identification of interesting selection rules for nonAbelian exotic vortices on the sphere. We also consider the classical and quantum dynamics of exotic vortices. Using recent localisation results, we give the Witten index of the quantum mechanics describing low-temperature Abelian exotic vortex dynamics.