A Fueter operator for 3/2-spinors

TL;DR

This paper investigates the moduli space of solutions to monopole equations for 3/2-spinors on 3-manifolds, linking its non-compactness to special solutions called 3/2-Fueter sections of a fiber bundle.

Contribution

It introduces the concept of 3/2-Fueter sections and establishes their role in characterizing the non-compactness of the monopole moduli space.

Findings

Non-compactness of the moduli space is equivalent to existence of 3/2-Fueter sections.

3/2-Fueter sections satisfy an overdetermined non-linear elliptic PDE.

The fiber bundle of sections lacks a hyperk"ahler structure.

Abstract

We show the non-compactness of moduli space of solutions of the monopole equations for $3/2$-spinors on a closed 3-manifold is equivalent to the existence of `3/2-Fueter sections' that are solutions of an overdetermined non-linear elliptic differential equation. These are sections of a fiber bundle whose fiber is a special 4-dimensional submanifold of the hyperk\"ahler manifold of center-framed charged one $SU(2)$-instantons on $\mathbf{R}^4$. This fiber bundle does not inherit a hyperk\"ahler structure.