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Cartan connections for an infinite family of integrable vortices

Sven Bjarke Gudnason, Calum Ross

Abstract

An infinite family of integrable vortex equations is studied and related to the Cartan geometry of the underlying Riemann surfaces. This Cartan picture gives an interpretation of the vortex equations as the flatness of a non-Abelian connection. Solutions of the vortex equations also give rise to magnetic zero-modes for a certain Dirac operator on the lifted geometry. The family of integrable vortex equations is parametrised by a positive number $n$, that is equal to unity in the standard case and an integer in the case of polynomial vortex equations; finally, it may be extended to any positive real number.

Cartan connections for an infinite family of integrable vortices

Abstract

An infinite family of integrable vortex equations is studied and related to the Cartan geometry of the underlying Riemann surfaces. This Cartan picture gives an interpretation of the vortex equations as the flatness of a non-Abelian connection. Solutions of the vortex equations also give rise to magnetic zero-modes for a certain Dirac operator on the lifted geometry. The family of integrable vortex equations is parametrised by a positive number , that is equal to unity in the standard case and an integer in the case of polynomial vortex equations; finally, it may be extended to any positive real number.

Paper Structure

This paper contains 7 sections, 3 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Fixed geometry with infinite family of vortex equations
  3. Local geometry on the Riemann surface
  4. Cartan geometry with circle bundles over Riemann surfaces
  5. Zero-modes
  6. Normalised vortex equations with infinite family of geometries
  7. Conclusion and outlook

Key Result

Lemma 1

If $\Psi:\mathbb{H}^{1}_{C_0}\to \mathbb{C}^{2}$ is a magnetic Dirac mode of the Dirac operator $D_{\mathbb{H}^{1}_{C_0}, A}$, for the $\mathop{\mathrm{U}}\nolimits(1)$ gauge field $A$, then is a magnetic Dirac mode on $\mathbb{R}^{3}_{C_0}$ with the Dirac operator $D_{\mathbb{R}^{3}_{C_0},H^{*}A}$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (6)

  • Definition 1
  • Lemma 1
  • Definition 2
  • Proposition 1
  • proof
  • Corollary 1