Symmetric calorons and the rotation map

TL;DR

This paper explores symmetric $SU(2)$ calorons, focusing on their invariance under specific isometries and the rotation map, using a monad matrix approach to construct new solutions and derive Nahm data for charge 2 cases.

Contribution

It introduces a monad matrix construction for calorons based on ADHM-like methods and demonstrates how symmetry analysis can lead to new caloron solutions, including explicit Nahm data for charge 2.

Findings

Identification of fixed points under cyclic symmetry groups

Development of a monad matrix construction for calorons

Construction of Nahm data for charge 2 calorons

Abstract

We study $SU(2)$ calorons, also known as periodic instantons, and consider invariance under isometries of $S^1\times\mathbb{R}^3$ coupled with a non-spatial isometry called the rotation map. In particular, we investigate the fixed points under various cyclic symmetry groups. Our approach utilises a construction akin to the ADHM construction of instantons -- what we call the monad matrix data for calorons -- derived from the work of Charbonneau and Hurtubise. To conclude, we present an example of how investigating these symmetry groups can help to construct new calorons by deriving Nahm data in the case of charge $2$.