Pin(2)-equivariance of the Rarita-Schwinger-Seiberg-Witten Equations

TL;DR

This paper introduces a variant of the Seiberg-Witten equations involving Rarita-Schwinger operators on 4-manifolds, revealing a Pin(2) symmetry that influences the structure of the solution space.

Contribution

It defines a new set of equations with Pin(2) symmetry and analyzes their solution space, showing non-compactness under certain topological conditions.

Findings

Moduli space of solutions is always non-compact under specific assumptions.

The equations exhibit a Pin(2) symmetry beyond the standard U(1).

Finite dimensional approximations are used to analyze eigenvalue problems.

Abstract

We define a variant of the Seiberg-Witten equations using the Rarita-Schwinger operators for closed simply connected spin smooth 4-manifold X. The moduli space of solutions to the system of non-linear differential equations consist of harmonic 3/2-spinors and connections satisfying certain curvature condition. Beside having an obvious U(1)-symmetry, these equations also have a symmetry by Pin(2). We exploit this additional symmetry to perform finite dimensional approximations for the eigenvalue problem of the 3/2-monopole map and show that under a certain topological assumption, the moduli space of solutions is always non-compact, and thus non-empty.